How to succeed in human modified environments

Open… access code

 

Affiliations: 1) Max Planck Institute for Evolutionary Anthropology, Germany; 2) University of California Santa Barbara, USA; 3) Victoria University of Wellington, New Zealand. Corresponding authors: corina_logan@eva.mpg.de, rachael.shaw@vuw.ac.nz, kelseybmccune@gmail.com

 

This registered report has been pre-study peer reviewed and received an In Principle Acceptance on 8 Sep 2022 by:

Chris Chambers (2022) The role of behavioural flexibility in promoting resilience to human environmental impacts. Peer Community in Registered Reports. https://rr.peercommunityin.org/articles/rec?id=200. Reviewers: Gloriana Chaverri, Vedrana Šlipogor, and Alizée Vernouillet

Cite as: Logan CJ, Shaw R, Lukas D, McCune KB. 2022. How to succeed in human modified environments (http://corinalogan.com/ManyIndividuals/mi1.html) In principle acceptance by PCI Registered Reports of the version on 25 Aug 2022 https://github.com/ManyIndividuals/ManyIndividuals/blob/573b4b5802550f47f246fb8b71c8efc4f853445c/Files/rrs/mi1.Rmd.

See the reproducible manuscript (Rmd) version for the code

 

ABSTRACT

Human modifications of environments are increasing, causing global changes that other species must adjust to or suffer from. Behavioral flexibility (hereafter ‘flexibility’) could be key to coping with rapid change. Behavioral research can contribute to conservation by determining which behaviors can predict the ability to adjust to human modified environments and whether these can be manipulated. When research that manipulates behavior in a conservation context occurs, it primarily trains a specific behavior to improve individual success in the wild. However, training a domain general cognitive ability, such as flexibility, has the potential to change a whole suite of behaviors, which could have a larger impact on influencing success in adjusting to human modified environments. This project asks whether flexibility can be increased by experimentally increasing environmental heterogeneity and whether such an increase can help species succeed in human modified environments. We explore whether it is possible to take insights from highly divergent species and apply them to address critical conservation challenges. This pushes the limits in terms of understanding how conserved these abilities may be and to what extent they can be shaped by the environment. We aim to 1) conduct flexibility interventions in flexible species that are successful in human modified environments (great-tailed grackles and California scrub-jays or blue jays) to understand how flexibility relates to success; and 2) implement these interventions in two vulnerable species (toutouwai and Florida scrub-jays) to determine whether flexibility as a generalizable cognitive ability can be trained and whether such training improves success in human-modified environments. This research will significantly advance our understanding of the causes and consequences of flexibility, linking behavior to environmental change, cognition, and success in human modified environments through a comparative and global framework. This registered report launches our reproducible research program, ManyIndividuals (https://github.com/ManyIndividuals/ManyIndividuals), which is a global network of researchers with field sites investigating hypotheses that involve generalizing across many individuals.

REGISTERED REPORT DETAILS

• Level of bias = 6: This registered report was written (Jul 2021-May 2022), and revised after round one of peer review at Peer Community in Registered Reports (Jul 2022) prior to collecting any data.
• Programmatic registered report: Three Stage 2 articles will result from this one Stage 1 registered report: one for toutouwai, one for grackles, and one for jays.
• Deviations from the Stage 1 registered report: [to be filled in as needed after data collection begins]

INTRODUCTION

Human modified environments are increasing (Goldewijk, 2001; X. Liu et al., 2020; Wu et al., 2011), causing global changes that other species must adjust to or suffer from (Alberti, 2015; Chejanovski et al., 2017; Ciani, 1986; Federspiel et al., 2017). Behavioral flexibility (hereafter ‘flexibility’) could be key for adjusting to such change: individuals interact with their environment through behavior, making it crucial to an ecologically valid understanding of how species adjust to environmental changes (Lee & Thornton, 2021). One of the top priorities for behavioral research to maximize conservation progress is to determine which cognitive abilities and behaviors can predict the ability to adjust to human modified environments and whether these can be manipulated (Moseby et al., 2016). The rare research that manipulates behavior in a conservation context usually focuses on training specific behaviors (for example, predator recognition through predator exposure) to improve individual success in the wild (Jolly et al., 2018; Moseby et al., 2012; Ross et al., 2019; West et al., 2018; see review in Tetzlaff et al., 2019). However, training a general cognitive ability, such as flexibility – the ability to rapidly adapt behavior to changes through learning throughout the lifetime (see the theory behind this definition in Mikhalevich et al., 2017) – has the potential to change a whole suite of behaviors and more broadly influence success in adjusting to human modified environments. Recent evidence supports this hypothesis: as far as we are aware, we were the first to show that flexibility can be manipulated using serial reversal learning of color preferences, and that the manipulated individuals were more flexible in a new context (locus switching on a puzzlebox) as well as being more innovative (solved more loci on a puzzlebox) (C. Logan et al., 2022).

Environments where informational cues about resources vary in a heterogeneous (but non-random) way across space and time are hypothesized to open a pathway for species to functionally detect and react to such cues via flexibility (Mikhalevich et al., 2017). Human modified environments likely provide a different set of informational cues that vary heterogeneously across space and time, and the species that are successful in such environments are likely those who are able to detect and track such cues. Because heterogeneous environments are hypothesized to select for flexibility (Wright et al., 2010), we expect that experimentally manipulating environments to be more heterogeneous will result in an increase in flexibility in individuals, which will then increase their success in such environments (Figure 1). Success can relate to any number of variables regarding the usage of and investment in resources and response to threats, from improved foraging efficiency to increased dispersal and survival within human modified environments, to placing nests in more protective locations. Whether a measure of success is predicted to relate to flexibility depends on what is already known about the particular population and their particular environment.

 

Figure 1. The theory behind this research illustrated by a directed acyclic graph (DAG), which is a theoretical model of the causal relationships among the key variables in our investigation. Based on the theoretical background provided by Mikhalevich et al. (2017), we assume that more heterogeneity causes more flexibility, which then causes more success in human modified environments.

 

This investigation asks whether flexibility can be increased by experimentally increasing environmental heterogeneity (via serial reversal learning) and whether such an increase can help species succeed in human modified environments. We explore whether it is possible to take insights from highly divergent species and apply them to address critical conservation challenges. Serial reversal learning tasks have been performed with a wide diversity of species (birds: Bond et al., 2007; bumblebees: Strang & Sherry, 2014; stingrays: Daniel & Schluessel, 2020). There is variation across individuals and species in their performance, however almost all previous studies show that individuals improve their flexibility if the reversal intervention is given multiple times in sequence (rats: Mackintosh et al., 1968; guppies: Lucon-Xiccato & Bisazza, 2014; poison frogs: Y. Liu et al., 2016). We aim to conduct a flexibility intervention in flexible species that are successful in human modified environments (great-tailed grackles and California scrub-jays or blue jays) to understand how flexibility relates to success, and implement these interventions in two vulnerable species (toutouwai and Florida scrub-jays) to determine whether flexibility as a generalizable cognitive ability can be trained and whether such training improves success in human modified environments (Figure 2).

While we do not examine the potential spread of the post-manipulation success behaviors from manipulated individuals to individuals that are not involved in our studies, we acknowledge that this is a possibility worthy of future investigation. Manipulating the flexibility of a few individuals could have population-level effects because significant research on social information use in birds (e.g., Valente et al., 2021) demonstrates the potential for the manipulated behavior to disseminate to conspecifics (for example, if manipulated individuals are faster at locating new resources, which could attract the attention of conspecifics, or if unmanipulated individuals copy the manipulated individuals’ nesting or foraging locations). In the event that social learning is not used by a given population to spread the behaviors of manipulated individuals, investing in the training of specific individuals to increase their success in the wild could still have conservation impacts. In some cases, it is possible to train many individuals in a population or a species because there are not many individuals left (Greggor et al., 2021). It is also possible to train all individuals involved in a conservation management event such as a translocation (Greggor et al., 2021). Therefore, there can still be significant population consequences even if each individual needs to be trained to achieve the goal.

This comparative approach will ultimately reveal how conserved these abilities may be and to what extent they can be shaped by the environment. To increase the generalizability of the conclusions from the ManyIndividuals project, we here also provide multiple methodological options that other researchers can use to test these questions in additional species. The results will substantially advance our understanding of the causes and consequences of flexibility, linking behavior to environmental change, cognition, and success in human modified environments through a comparative and global framework.

RESEARCH QUESTIONS

Can behavioral flexibility in individuals be increased by increasing environmental heterogeneity? If so, does increased flexibility help individuals succeed in human modified environments?

Prediction 1: Flexibility can be increased in individuals and such an increase improves the likelihood of success in human modified environments. This would indicate that the abilities involved in tracking changing resources in the environment are the same as or related to the abilities involved in succeeding in human modified environments. It would also indicate that flexibility is trainable and that such training could be a useful conservation tool for threatened and endangered species.

Prediction 1 alternative 1: Flexibility can be increased in individuals, but such an increase does not improve the likelihood of success in human modified environments. This would indicate that species associated with human modified environments form this association for reasons other than their flexibility, and that threatened species are likely not very successful in human modified environments for reasons unrelated to their ability to change their behavior with changing circumstances. An alternative could be that the changes induced by the increase in flexibility do not persist for sufficiently long times to make a difference on the subsequent likelihood of success (changes in grackles were still present for four weeks after the manipulation and longer time periods were not attempted so the threshold is unknown C. Logan et al., 2022).

Prediction 1 alternative 2: Flexibility can be increased in some populations, but not others. This would indicate that flexibility manipulations may not work for all populations, and that the effectiveness of such experiments should first be tested in the population of interest before including such an intervention in a conservation plan. If flexibility is not manipulatable in threatened populations, this would indicate that they are likely not very successful in human modified environments because of their inability to change their behavior with changing circumstances, and that flexibility is not trainable. If flexibility is not manipulatable in populations that are successful in human modified environments, this could indicate that they might have used flexibility in the past when originally forming the association, but the need to maintain flexibility in their repertoire is no longer necessary (Wright et al., 2010). In populations where flexibility is not manipulatable, this would indicate that the abilities involved in tracking changing resources in the environment are independent of the abilities involved in succeeding in human modified environments.

Population-specific background and tailored research questions

 

Figure 2. Comparing the species involved in this investigation relative to their geographic range and association with human modified habitats. The yellow dots represent field site locations. Photo credit: grackle and CASJ, Corina Logan; blue jay, Rhododendrites; FLSJ, VvAndromedavV; toutouwai, Rachael Shaw.

 

Great-tailed grackles (Quiscalus mexicanus)

Background

Great-tailed grackles are flexible (Logan, 2016; Logan, MacPherson, et al., 2019), highly associated with human modified environments (Johnson & Peer, 2001), and have been rapidly expanding their geographic range across North America over the past 140 years (Wehtje, 2003). They are social and polygamous, and eat a diversity of human foods as well as foraging on insects and on substrates for other natural food items (Johnson & Peer, 2001). Males tend to be the socially dominant sex (Johnson et al., 2000) and also the sex that disperses away from their natal area (Sevchik et al., 2019). Rodrigo et al. (2021) found that more grackles are present and more foraging events occur during garbage pick ups when garbage tends to spill out of the bags, thus increasing food availability. Attending to garbage trucks, potentially across space and time, is an example of how flexibility can help individuals meet foraging needs in the context of changing environmental cues. Great-tailed grackle behavioral flexibility is manipulatable using serial reversal learning, and this manipulation improves their flexibility in a new context as well as their innovativeness (Logan, MacPherson, et al., 2019), which shows that training a general cognitive ability can affect more behaviors than the behavior that was trained and potentially make them more successful in human modified environments. We are in the middle of long-term data collection on grackles to answer questions about how flexibility relates to exploration (McCune KB et al., 2019), the predictability of their space use (McCune KB et al., 2020) and their foraging breadth (Logan, Lukas, et al., 2019).

Research questions

• G.Q1: Do flexibility manipulated individuals differ in the proportion of time spent at cafes and garbage dumpsters when food is present? We will investigate this question by tracking their presence at cafes and dumpsters when food is present versus when it is absent before and after manipulating their flexibility using serial reversal learning in the wild (Figure 3). Table 1 summarizes our predictions, analysis plans, interpretations for the various directions the results could go, and the hypotheses that could be contradicted given the various outcomes.

• G.Q2: Does manipulating behavioral flexibility alter the number of microhabitats used? We will investigate this question by tracking their presence in a variety of microhabitats before and after manipulating their flexibility using serial reversal learning in the wild. We only count that a microhabitat was used if the individual had at least 5% of their data points there. This prevents a microhabitat from being counted even if an individual was simply moving through it, and therefore not necessarily using it.

• G.Q3: Does manipulating flexibility alter the number of different food items taken by grackles? We will investigate this question by tracking the various food items they take before and after manipulating their flexibility using serial reversal learning in the wild.

Note: we may not have the time or personnel to collect data for G.Q2 and G.Q3, however we will attempt to answer these questions if possible.

 

Figure 3. The reversal learning experiment in a group context (Design 2) tailored to the great-tailed grackle research question.

 

Table 1. Study design for the great-tailed grackle research. References that were not already cited in the introduction: Duckworth (2009), Dingemanse & Wolf (2013), Peterson et al. (2011), Grinnell (1917).

 

Jays (Aphelocoma californica and/or A. coerulescens and/or Cyanocitta cristata)

We are not yet sure which jay species will be feasible to test as part of this investigation. We aim to test a disturbance resilience species like the California scrub-jay or blue jay. We also aim to test the disturbance sensitive Florida scrub-jay, but we are still waiting for permits on the Florida scrub-jay and blue jay. Regardless of the species that end up in the investigation, we will use the same hypotheses and predictions below.

Background

Jay species exhibit a diversity of social systems and success in colonizing suburban and urban areas. California scrub-jays (Aphelocoma californica, hereafter “CASJ”) and blue jays (Cyanocitta cristata, hereafter “BLJA”) are singular, monogamous breeders that are increasing in abundance, expanding their range sizes, and highly successful in natural, suburban, and urban areas (Blair, 1996; Curry et al., 2017). We therefore consider these “disturbance-resilient” (DR) jay species. In contrast, the Florida scrub-jay (A. coerulescens; hereafter “FLSJ”) is a “disturbance-sensitive” (DS) jay species that is threatened, endemic, and range-restricted to xeric oak scrub habitat in Florida (Woolfenden & Fitzpatrick, 1996).

These species forage primarily on mast (acorns, hazelnuts, etc.) that they cache throughout their territory, which makes it available to eat year-round. They are also opportunistic omnivores and specifically need high-fat and high-protein arthropods to feed to nestlings and fledglings (Curry et al., 2017). Nesting and foraging substrates can be drastically different in human modified environments compared to natural areas [e.g., predominance of non-native vegetation; Tuomainen & Candolin (2011)], and it is unknown whether suburban and urban jays are able to persist in these environments through behavioral adjustments. The DS jay species, the FLSJ, can persist in suburban habitats after conversion from xeric oak scrub, however suburban populations of FLSJ steadily decline (Bowman pers. comm.). This is potentially due to the presence of suboptimal habitat resulting from fire suppression (Woolfenden & Fitzpatrick, 1996), higher rates of brood reduction through nestling starvation (Shawkey et al., 2004), and the lack of nutritionally complete prey items (Shawkey et al., 2004) in suburban habitats. It is possible that behavioral flexibility in habitat use and foraging breadth underlies the ability of some FLSJ to persist in human-dominated areas.

We aim to compare behavioral flexibility within species, between suburban and natural populations to determine whether variation in flexibility relates to variation in presence in these habitats. Subsequently, we will compare flexibility between DS and DR jay species to determine whether this trait is related to the greater success of DR jay species, like the CASJ and BLJA, in human-dominated areas. Lastly, we will test whether manipulating flexibility increases the foraging and microhabitat breadth of jays in human modified environments. Manipulating the flexibility of a subset of individuals has the potential to affect the population because previous research demonstrates that both species have the capacity to use foraging information discovered by others (social learning) to flexibly change their behavior (K. B. McCune, 2018; Midford et al., 2000).

While we have begun to acquire permits to work with these species, the processing time for permits to research threatened and endangered species can be very long and unpredictable. Therefore, while we aim to collect data on the FLSJ during the 5-year timeline of this project, it is possible this species will have to be omitted if the permits are not approved in time. If this occurs, we will not be able to compare CASJ/BLJA and FLSJ (Question 2 in Table 2), but we will still be able to compare behavioral flexibility of CASJ/BLJA populations in human modified and natural areas, and assess whether the behavioral flexibility manipulation affects success in human modified environments.

Research questions

For all research questions, Table 2 summarizes our predictions, analysis plans, interpretations for the various directions the results could go, and the hypotheses that could be contradicted given the various outcomes.

• J.Q1: Do jay populations in human modified areas differ in baseline behavioral flexibility compared to populations in natural areas? We will investigate this question by comparing performance on serial reversal learning in the wild between jays in natural areas and jays in human modified areas.
• J.Q2: Are disturbance-resilient (DR) jays more behaviorally flexible than disturbance-sensitive (DS) jays? We will investigate this question by comparing performance on serial reversal learning in the wild between DR and DS jay species.
• J.Q3: Does manipulating behavioral flexibility alter the number of microhabitats used? We will investigate this question by tracking their presence in a variety of microhabitats before and after manipulating their flexibility using serial reversal learning in the wild. We only count that a microhabitat was used if the individual had at least 5% of their data points there. This prevents a microhabitat from being counted even if an individual was simply moving through it, and therefore not necessarily using it.
• J.Q4: Does manipulating flexibility alter the number of different food items taken by jays? We will investigate this question by tracking the various food items they take before and after manipulating their flexibility using serial reversal learning in the wild.

 

Figure 4. The reversal learning experiment in a group context (Design 2) tailored to the jay research questions. The white rectangles represent feeder locations, the feeder with the X is in the unrewarded location while the feeder with the green check is the rewarded location.

 

Table 2. Study design for the jay research. References that were not already cited in the introduction: Galbraith et al. (2015), Lapiedra et al. (2017), Rice et al. (2003), Emery & Clayton (2004), Sol et al. (2002), Peterson et al. (2011), Grinnell (1917).

 

Toutouwai (North Island robin, Petroica longipes)

Background

Toutouwai are a small insectivorous passerine species that is endemic to the North Island of New Zealand. Prior to the arrival of European settlers, they were abundant and widespread. However, European colonization brought the introduction of mammalian predators, which had devastating consequences for New Zealand’s avifauna. As a result, the national conservation status of the toutouwai is now at risk and in decline (Robertson et al., 2016). In response to avifauna declines in New Zealand, many offshore islands have now been established as predator free sanctuaries. Threatened endemic birds have also been reintroduced into ‘mainland island’ sanctuaries; areas on New Zealand’s North and South Islands that are designed to eliminate the threat posed by non-native mammalian predators (Saunders & Norton, 2001). Yet, despite predator control and fences protecting sanctuary populations, these vulnerable species still come into contact with invasive mammalian predators beyond mainland sanctuary boundaries.

At Zealandia, a 225 hectare predator-proof fenced sanctuary located in central Wellington, less than 20 years have passed since the introduction of toutouwai (Miskelly et al., 2005). This species now thrives inside the sanctuary with some individuals surviving up to 14 years, but is struggling to gain a foothold in the surrounding reserves. Toutouwai may undergo juvenile dispersal from 5 weeks post-fledging onwards, traveling anywhere from a few meters to several kilometers from their natal territory (Richard & Armstrong, 2010). Many juvenile toutouwai are sighted establishing territories outside of the sanctuary each year, yet few persist for more than a few weeks (Shaw & Harvey, 2018). One potential reason for the failure to persist, is that toutouwai tend to forage on the ground and are thus at high risk from invasive mammalian predators, which they fail to recognize. It is possible that the more flexible individuals that disperse outside the sanctuary might forage at a diversity of heights and/or more readily learn to recognize novel predators, and therefore have a higher likelihood of surviving post-dispersal, but currently this hypothesis is untested.

Previous research on the cognitive abilities of toutouwai reveals that this species can retain a learned skill for several months in the wild, without reinforcement (Shaw & Harvey, 2020). Moreover, their bold and curious nature means that they will interact with novel objects and readily engage in a suite of cognitive tests (Shaw et al., 2015). As such, they are an ideal species in which to examine whether flexibility manipulations might influence the dispersal decisions made by juveniles, or enhance the survival of juveniles that attempt to establish beyond the sanctuary fence.

Research questions

For all research questions, Table 3 summarizes our predictions, analysis plans, interpretations for the various directions the results could go, and the hypotheses that could be contradicted given the various outcomes.

• T.Q1: Does a flexibility manipulation alter dispersal timing and distance, as well as the likelihood that juvenile toutouwai will disperse beyond the protection of Zealandia’s fence and attempt to establish in the adjacent urban reserves? We will investigate this question by measuring their age at dispersal, dispersal distance, and habitat dispersed to after either manipulating their flexibility using serial reversal learning in the wild (manipulated group) or not manipulating their flexibility by giving them only one reversal rather than serial reversals (control group).
  
• T.Q2: Compared to control individuals, are flexibility manipulated individuals more likely to survive their first 16 weeks post-fledging, particularly if they disperse into the urban reserves outside the sanctuary fence? We will investigate this question by tracking their dispersal destination after either manipulating their flexibility using serial reversal learning in the wild (manipulated group) or not manipulating their flexibility by giving them only one reversal rather than serial reversals (control group).

 

Figure 5. The reversal learning experiment in a group context (Design 2) tailored to the toutouwai research questions.

 

Table 3. Study design for the toutouwai research. References that were not already cited in the introduction: Pasinelli et al. (2004); Howard (1960).

 

METHODS

Our approach involves individuals participating in a serial reversal learning task and measuring success in natural behavior in the wild. In the following, we outline the procedure we are planning to use with wild birds: the serial reversal learning will involve a feeder setup (see design 2 below), and the success measures will be matched to the relevant behavior in the respective population. The framework we provide is general to facilitate its adaptation and replication to other populations (i.e., another researcher could adapt our hypotheses, methods, and analysis plans to their system). We start each section of the methods and analysis plans with general considerations (e.g., minimum sample size), before providing the specific details for each of the systems we plan to study.

We present one experimental design for the flexibility intervention (Figure 6) that can be conducted in two ways: in visual isolation (design 1) and in a group context (design 2). Experimenters can decide which (or both) they want to conduct in their population. Only one experiment must be conducted per population to be able to test these hypotheses. Conducting more than one of these experiments per population is acceptable, but not necessary. Before we present the designs, we first validated the reversal passing criteria and made them generalizable to a variety of species (see the next two sections). Depending on the response variable, there is the option to conduct a within- or between-subjects design:

1. Within-subjects: run the manipulation on all individuals and compare pre- and post-manipulation success measures
2. Between-subjects: with manipulated and control groups and compare post-manipulation success measures. In this case, 50% of the individuals will be assigned to the control condition and 50% to the flexibility manipulation condition. Assignment to condition will be random (using the random number generator random.org).

 

Figure 6. Template design for the reversal learning experiment.

Determining when to switch each individual to the next reversal: reversal passing criterion

Different criteria exist to decide whether an individual has learned an association between the presence of a reward and some other feature (e.g., color or shape). The two main two criteria used are to switch an individual after it either has chosen 10 out of 12 choices correct (e.g., Shaw et al., 2015) or 17 out of 20 choices correct (e.g., Logan, 2016). The criteria are further modified depending on whether choices are assessed continuously or grouped in predetermined blocks.

Here, we assess whether achieving 10 correct choices out of the last 12 continuously counted choices can be used as a reliable reversal passing criterion. To determine reliability and suitability, we investigated five questions (see below) by generalizing previously simulated reversal learning data from Logan CJ et al. (2020), based on data from great-tailed grackles. We simulated the choices individuals with different learning rates (phi) and rates of deviating from learned associations (lambda) would make in the initial discrimination and in the first reversal. Grackles are fast to reverse preferences compared with many other species (Logan, 2016), therefore we generalized the simulations to other species by setting the parameters that guide performance (phi and lambda) to lead to slower performances.

The findings from these simulated data indicate that deciding that an individual has passed the reversal when they choose 10 out of the last 12 consecutive trials correctly is functional and reliable because of the following:

1) individuals will be finished after fewer trials than with other criteria

With the 10 out of 12 criterion, individuals pass the reversal 8 trials faster (median) than with the 17 out of 20 criterion. This means that, for most individuals, the two rules are equally effective because they will pass both in the same amount of trials (i.e., the individual who met the 17/20 criterion in 50 trials would have met the 10/12 criterion in 42 trials), but because the 10 out of 12 criterion is restricted to 12 trials instead of 20, individuals need 8 fewer trials to meet the passing criterion. No individual needs more trials with the 10 out of 12 criterion. When trials are grouped into blocks of 10 such that they could only pass on trial 20, 30, etc., individuals need a median of 5 more trials compared to when choices are assessed continuously.

2) classification of individuals using the 10/12 criterion is less noisy because there is less of a chance for individuals to approach the criterion and not pass or never pass

The average improvement in the number of trials individuals need to reach the respective criterion is larger than the median of 8 trials. This occurs because there are no individuals who are faster with the 17 out of 20 criterion, and because there is a subset of individuals who need considerably fewer trials with the 10/12 criterion (Figure 7). Individuals who require a larger number of trials (>100) to pass almost never occur with the 10/12 criterion, whereas they are more common with the 17/20. With more trials, there is a higher chance that an individual will deviate from their preference by chance. This is also reflected in that 65 of the 626 simulated individuals never reached the 17/20 criterion within the maximum 300 trials, whereas there were only 4 individuals with the 10/12 criterion. Accordingly, an additional benefit of choosing the 10/12 criterion is that it is more likely that data for all individuals, even those who are slow to learn an association, can be collected.

Figure 7. There is less variation with the 10/12 reversal learning passing criterion and it requires fewer trials to reach than the 17/20 passing criterion. The lines represent the densities of individuals (estimated with smoothing which means values can go down to zero) across the 626 simulated individuals that needed a certain number of trials to either reach the 10/12 (red) or the 17/20 (black) criterion. With the 10/12 criterion, most individuals need 8 fewer trials (indicated by the lines showing the mode of the two density distributions). In addition, there are only very few individuals who need 100 or more trials with the 10/12 criterion, while there are several individuals that needed such large numbers with the 17/20 criterion.

3) variation among individuals with the 10/12 criterion is still present and similar to the variation detected with other criteria

As described in point 1, when changing the criterion from 17/20 to 10/12, most individuals need 8 fewer trials. This also means that the differences among individuals, which might contain relevant information about variation among them, is preserved. When transforming performance with the two criteria to ranks, individuals are sorted essentially in the same order independent of which criterion is used. This is shown in Figure 7: most points are shifted up by exactly 8 trials.

4) individuals can be assumed to have reliably learned the association using the 10/12 criterion

Based on the two reversal passing criteria (10/12 and 17/20), we can extract the attractions that simulated individuals have formed toward both the rewarded and the unrewarded option at the point at which they meet each of these criteria. Comparing the two attractions (to the rewarded and unrewarded options), we can determine whether individuals are likely to have learned an association or not. Independent of the criterion, individuals generally formed a preference for the rewarded option: 89% of individuals favor the rewarded option between 2.5 and 14 times more than the unrewarded option. With both criteria, individuals always have a stronger attraction to the rewarded than the unrewarded option. The smallest difference between the attraction scores to the rewarded and unrewarded options we observe at the point of passing is the same with both criteria. With the 10/12 criterion, individuals would in the next trial, on average, choose the rewarded option with a probability of 76% (3 times more likely to choose rewarded over unrewarded option), whereas this is 84% with the 17/20 criterion (5 times more likely).

5) the learned association means that individuals who move to the next reversal are unlikely to solve the reversed association by chance

As expected, based on the relative attraction scores at the end of the previous reversal, most individuals are unlikely to choose the now rewarded option. We expect that, on average, individuals will choose the newly rewarded option in 4 or fewer trials out of the first 12 trials (red line in Figure 8). This is a lower number of trials compared to individuals who have no association with either option (gray line in Figure 8), and a slightly higher number compared to individuals who use the 17/20 criterion (black line in Figure 8). The probability that an individual would, after a reversal, immediately choose the rewarded option 10 times during the first 12 trials (and pass) by chance is 0.001. However, even such rare individuals will have actually reversed their preference during their first 12 trials because they update their attractions on every trial.

Figure 8. Individuals form strong enough preferences using the 10/12 passing criterion as indicated by the fact that they are unlikely to pass in the first 12 trials of their next reversal (red line). These individuals would take longer to switch their preference than individuals who have no preference (gray line), and they would be slightly faster at switching their preference than individuals who formed their previous association using the 17/20 criterion (black line).

# Load previously simulated data from xpop

# There are two sets of simulated data, with initial
# attractions towards both options set at 0.1, and eight
# different phi and four different lambda combinations
simulatedreversaldata_attractionscores_1 <- read.csv(url("https://raw.githubusercontent.com/corinalogan/grackles/master/Files/Preregistrations/gxpopbehaviorhabitat_SimulatedReversalData_Grackles_PhiLambda_Attraction02_Aug2021.csv"),
    header = T, sep = ",", stringsAsFactors = F)

simulatedreversaldata_attractionscores_2 <- read.csv(url("https://raw.githubusercontent.com/corinalogan/grackles/master/Files/Preregistrations/gxpopbehaviorhabitat_SimulatedReversalData_Grackles_PhiLambda_Attraction04_Aug2021.csv"),
    header = T, sep = ",", stringsAsFactors = F)

# In both simulations, sites (which differ by the phi and
# lambda assigned to individuals) were counted from 1-16;
# for the second simulation we change this to 17-32
simulatedreversaldata_attractionscores_2$Site <- simulatedreversaldata_attractionscores_2$Site +
    16

# In both simulations, individuals were counted from 1-320;
# for the second population we change the ids to start at
# 321
simulatedreversaldata_attractionscores_2$Bird_ID <- simulatedreversaldata_attractionscores_2$Bird_ID +
    320

# We combine the two data sets for the further analyses
library(dplyr)
simulatedreversaldata_attractionscores <- bind_rows(simulatedreversaldata_attractionscores_1,
    simulatedreversaldata_attractionscores_2)


# Preparing simulated data for analyses

# In the simulations, trials were counted continuously for
# each bird. We now want to change this so that it restarts
# counting trials from 1 upward once a bird switches to
# reversal.

for (birds in 1:length(unique(simulatedreversaldata_attractionscores$Bird_ID))) {
    currentbird <- unique(simulatedreversaldata_attractionscores$Bird_ID)[birds]
    maximuminitial <- max(simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$Bird_ID ==
        currentbird & simulatedreversaldata_attractionscores$Reversal ==
        "initial", ]$Trial)
    simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$Bird_ID ==
        currentbird & simulatedreversaldata_attractionscores$Reversal ==
        "reversal", ]$Trial <- simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$Bird_ID ==
        currentbird & simulatedreversaldata_attractionscores$Reversal ==
        "reversal", ]$Trial - maximuminitial
}

# We need to adjust the coding during the reversal learning
# so that 'correct' now matches whether it is correct or
# not.
simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$Choice ==
    0, ]$Choice <- 2

# To use the model to estimate the phi and lambda
# parameters, we first need to change the column names to
# match these to the specifications in the model: change
# Bird_ID to id; change Reversal to Choice, change
# CorrectChoice to Correct, change Site to Expid

colnames(simulatedreversaldata_attractionscores) <- c("counter",
    "id", "Session", "Trial", "Reversal", "Choice", "Correct",
    "Phi_mean", "Lambda_mean", "Site", "Phi_sd", "Lambda_sd",
    "ThisBirdsPhi", "ThisBirdsLambda", "Attraction1", "Attraction2")

# There are several simulated individuals who never reached
# any criterion during the initial learning phase (in xpop
# we worked with the criterion of 17 correct out of the 20
# choices given as one block). We need to remove these
# individuals from the dataset

birdswithreversal <- as.data.frame(simulatedreversaldata_attractionscores %>%
    group_by(id) %>%
    summarise(experiments = length(unique(Reversal))))
birdswithreversal <- birdswithreversal[birdswithreversal$experiments ==
    2, ]
simulatedreversaldata_attractionscores <- simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$id %in%
    birdswithreversal$id, ]

# Next, we need to change the ids of the birds to be
# continuous again for easier further processing in loops
simulatedreversaldata_attractionscores$id <- as.integer(as.factor(simulatedreversaldata_attractionscores$id))



# 1) With the 10 correct out of 12 continuous trials
# criterion, individuals finish after fewer trials than
# with other criteria

# We use the data from the first reversal to check after
# how many trials an individual reaches the 10 out of 12
# trials solved correctly rule. We are using a sliding
# window, rather than assuming that the experiment is split
# into discrete blocks of trials. For this, we check for
# each individual at what trial number it first reached the
# criterion that it had correctly solved 10 out of the last
# 12 trials.

# We create a matrix to store the output. We have two
# columns, one with the id of the individual the other with
# the number of trials they needed with this rule. Having
# this setup makes the comparison with the alternative rule
# easier. In addition, it will keep track of individuals
# that never reached the criterion
trialnumber_initiallearning_reachedtwelveten <- matrix(ncol = 2,
    nrow = length(unique(simulatedreversaldata_attractionscores$id)))

# We loop through all unique individuals in the simulated
# dataset (626 individuals) and for each check when they
# reached the criterion - at that point we can stop
# ('break') and ignore the remaining performance. In the
# initial association learning, wrong choices are coded as
# 2 and correct choices as 1, so individuals reached the
# criterion if they chose 10*1 + 2*2 (equals 14) - so
# individuals reached criterion when the sum in choices
# across 12 trials is lower than 15.
for (i in 1:length(unique(simulatedreversaldata_attractionscores$id))) {

    thisindividualsinitaltrials <- simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$id %in%
        i & simulatedreversaldata_attractionscores$Reversal %in%
        "initial", ]
    trialnumber_initiallearning_reachedtwelveten[i, 1] <- i
    for (k in 1:(nrow(thisindividualsinitaltrials) - 12)) {
        if (sum(thisindividualsinitaltrials[c(k:(k + 11)), ]$Choice) <
            15) {
            trialnumber_initiallearning_reachedtwelveten[i, 2] <- (k +
                11)
            break
        }
    }
}

# Next, we repeat this for the alternative rule of 17 out
# of 20 trials, again using a sliding window approach
# across trials (17*1 + 3*2 = 23).
trialnumber_initiallearning_reachedseventeentwenty <- matrix(ncol = 2,
    nrow = length(unique(simulatedreversaldata_attractionscores$id)))

for (i in 1:length(unique(simulatedreversaldata_attractionscores$id))) {

    thisindividualsinitaltrials <- simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$id %in%
        i & simulatedreversaldata_attractionscores$Reversal %in%
        "initial", ]
    trialnumber_initiallearning_reachedseventeentwenty[i, 1] <- i
    for (k in 1:(nrow(thisindividualsinitaltrials) - 20)) {
        if (sum(thisindividualsinitaltrials[c(k:(k + 19)), ]$Choice) <
            24) {
            trialnumber_initiallearning_reachedseventeentwenty[i,
                2] <- (k + 19)
            break
        }
    }
}

# We now have the number of trials each individual would
# have needed with the two rules to reach the respective
# criterion. We can check whether individuals needed fewer
# trials with the 10 out of 12 rule than the 17 out of 20
# rule
twelvetenfaster <- trialnumber_initiallearning_reachedseventeentwenty[,
    2] - trialnumber_initiallearning_reachedtwelveten[, 2]

# The median improvement is 8 trials
median(twelvetenfaster, na.rm = T)
# This means that for most individuals the two rules are
# equally effective, but because the 10 out of 12 rule is
# restricted to 12 trials instead of 20 individuals need 8
# fewer trials.

# The same is seen in a regression
summary(lm(trialnumber_initiallearning_reachedseventeentwenty[,
    2] ~ trialnumber_initiallearning_reachedtwelveten[, 2]))
# The intercept is 7 (similar to the mean listed above),
# and the slope is 1.3 (suggesting that deviations are
# slightly more when individuals need a large number of
# trials).

# We can also check whether counting trials continously,
# instead of grouping them into blocks, means that
# individuals finish sooner. In the original simulation,
# trials where stopped when birds reached the 17 out of 20
# criterion based on blocks of 10 trials. We can check
# whether counting trials continously means they would have
# finished sooner:

# We now get the number of trials they needed to pass the
# block criterion
trialnumber_initiallearning_reachedseventeentwentyblock <- matrix(ncol = 2,
    nrow = length(unique(simulatedreversaldata_attractionscores$id)))

for (i in 1:length(unique(simulatedreversaldata_attractionscores$id))) {
    thisindividualsinitaltrials <- simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$id %in%
        i & simulatedreversaldata_attractionscores$Reversal %in%
        "initial", ]
    trialnumber_initiallearning_reachedseventeentwentyblock[i,
        1] <- i
    trialnumber_initiallearning_reachedseventeentwentyblock[i,
        2] <- max(thisindividualsinitaltrials$Trial)
}

plot(trialnumber_initiallearning_reachedseventeentwentyblock[,
    2] ~ trialnumber_initiallearning_reachedseventeentwenty[,
    2])
abline(a = 0, b = 1)
median(trialnumber_initiallearning_reachedseventeentwentyblock[,
    2] - trialnumber_initiallearning_reachedseventeentwenty[,
    2], na.rm = T)
# 5 trials faster
mean(trialnumber_initiallearning_reachedseventeentwentyblock[,
    2] - trialnumber_initiallearning_reachedseventeentwenty[,
    2], na.rm = T)
# 7 trials faster


# 2) With the 10 correct out of 12 continuous trials
# criterion, variation among individuals is still present
# and similar to the variation detected with other criteria

# Changing the criterion from 17/20 to 10/12 generally
# means that most individuals need 8 fewer trials. That
# also means that the differences among individuals, which
# might contain relevant information, is preserved. We can
# see this when we plot their ranks in how many trials they
# need (individuals who need the same number of trials have
# the same rank; there are fewer ranks with the 10/12
# criterion because large values do not appear)
plot(as.integer(as.factor(trialnumber_initiallearning_reachedseventeentwenty[,
    2])) ~ as.integer(as.factor(trialnumber_initiallearning_reachedtwelveten[,
    2])), xlim = c(0, 100), ylim = c(0, 100))
abline(a = 0, b = 1)



# 3) With the 10 correct out of 12 continuous trials
# criterion, classification is less noisy because there is
# less of a chance of individuals missing the criterion
# repeatedly or even never reaching it, meaning that
# individuals slow to learn will be more reliably
# classified than with a different criterion

# While the median improvement is 8 trials, the average
# improvement is 15 trials
mean(twelvetenfaster, na.rm = T)
# The average improvement is larger than the median
# because, first, there are no individuals who are faster
# with the 17 out of 20 rule, and secondly because there is
# a subset of individuals who need considerably fewer
# trials with the 10 out of 12 rule. We can see this in a
# plot:

seventeentwenty <- trialnumber_initiallearning_reachedseventeentwenty[,
    2]
seventeentwenty <- seventeentwenty[is.na(seventeentwenty) ==
    F]

twelveten <- trialnumber_initiallearning_reachedtwelveten[, 2]
twelveten <- twelveten[is.na(twelveten) == F]

plot(density(seventeentwenty), ylim = c(0, 0.05), ann = F, bty = "n",
    xlim = c(0, 300), lwd = 4, xaxt = "n", yaxt = "n")
lines(density(twelveten), col = "red", lwd = 4)
axis(side = 1, at = c(0, 50, 100, 150, 200, 250, 300))
segments(x0 = 16, y0 = 0, x1 = 16, y1 = 0.05, col = "red", lwd = 1.5,
    lty = 2)
segments(x0 = 24, y0 = 0, x1 = 24, y1 = 0.045, col = "black",
    lwd = 1.5, lty = 2)
mtext("Trials needed to reach criterion", side = 1, cex = 1.25,
    line = 3)
mtext("Density of individuals", side = 2, cex = 1.25, line = 1)
legend(x = 160, y = 0.045, legend = c(lty1 = "17 / 20 criterion",
    lty1 = "10 / 12 criterion"), lwd = c(4, 4), col = c("black",
    "red"), cex = 1.25, bty = "n")
abline(v = 0, lty = 1)
mtext("16 trials", side = 1, cex = 1, line = -16, at = 32, col = "red",
    bg = "white")
mtext("24 trials", side = 1, cex = 1, line = -14.5, at = 40,
    col = "black", bg = "white")

# m1Fig_ReversalCriterionComparison

# With the 17 out of 20 rule, some individuals need a very
# large number of trials. This is presumably because with
# more trials there is a higher chance that an individual
# will deviate from their preference by chance, leading to
# them not reaching the criterion. Such individuals with
# very larger number of trials (>100) almost never occur
# with the 10 out 12 rule. There are also 65 individuals
# (out of the 626) with the 17 out of 20 criterion who
# never reached the criterion within the maximum 300 trials
# that were simulated - while there are only 4 with the 10
# out of 12 criterion.
sum(is.na(trialnumber_initiallearning_reachedseventeentwenty))
sum(is.na(trialnumber_initiallearning_reachedtwelveten))

# Accordingly, an additional benefit of choosing the 10/12
# rule is that it is more likely that data for all
# individuals, even those slow to learn an association, can
# be collected.



# 4) With the 10 correct out of 12 continuous trials
# criterion, individuals can be assumed to have reliably
# learned the association

# The difference in attraction, the mechanisms of the
# assumed learning processes mean that changes in
# attraction scores are non-linear. New information that an
# option is rewarded leads initially to large increases in
# the attraction score towards that option, but each
# additional information about the reward leads to smaller
# and smaller additional increases; new information that a
# previously rewarded option is now unrewarded lead
# initially to large drops in the attraction score, but
# additional information that this option is no longer
# rewarded leads to smaller declines in the attraction
# score. This means that with the 10 out of 12 criterion,
# individuals might have already reached the part in the
# curve of change of attraction scores where there are
# large differences that have essentially plateaued. We can
# check this by assessing both the actual attraction scores
# at that trial when individuals would have reached the
# criterion, and how high their attraction score is in
# relation to the maximum attraction score they obtained
# during the experiment.
simulatedreversaldata_attractionscores$relativeattraction <- simulatedreversaldata_attractionscores$Attraction1/simulatedreversaldata_attractionscores$Attraction2

# We create a matrix to store the output. We have two
# columns, one with the id of the individual the other with
# the number of trials they needed with this rule. Having
# this setup makes the comparison with the alternative rule
# easier. In addition, it will keep track of individuals
# that never reached the criterion
trialnumber_initiallearning_reachedtwelveten <- matrix(ncol = 6,
    nrow = length(unique(simulatedreversaldata_attractionscores$id)))
colnames(trialnumber_initiallearning_reachedtwelveten) <- c("Id",
    "TrialAtReachingCriterion", "AttractionScore", "RelativeAttractionScore",
    "AttractionOutOfMax", "RelativeAttractionOutOfMax")
trialnumber_initiallearning_reachedtwelveten <- as.data.frame(trialnumber_initiallearning_reachedtwelveten)

# We loop through all unique individuals in the simulated
# dataset (626 individuals) and for each check when they
# reached the criterion - at that point we can stop
# ('break') and ignore the remaining performance. In the
# initial association learning, wrong choices are coded as
# 2 and correct choices as 1, so individuals reached the
# criterion if they chose 10*1 + 2*2 (equals 14) - so
# individuals reached criterion when the sum in choices
# across 12 trials is lower than 15.
for (i in 1:length(unique(simulatedreversaldata_attractionscores$id))) {
    thisindividualsinitaltrials <- simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$id %in%
        i & simulatedreversaldata_attractionscores$Reversal %in%
        "initial", ]
    trialnumber_initiallearning_reachedtwelveten[i, 1] <- i
    for (k in 1:(nrow(thisindividualsinitaltrials) - 12)) {
        if (sum(thisindividualsinitaltrials[c(k:(k + 11)), ]$Choice) <
            15) {
            trialnumber_initiallearning_reachedtwelveten[i, 2] <- (k +
                11)
            trialnumber_initiallearning_reachedtwelveten[i, 3] <- thisindividualsinitaltrials[k +
                11, ]$Attraction1
            trialnumber_initiallearning_reachedtwelveten[i, 4] <- thisindividualsinitaltrials[k +
                11, ]$relativeattraction
            trialnumber_initiallearning_reachedtwelveten[i, 5] <- thisindividualsinitaltrials[k +
                11, ]$Attraction1/max(thisindividualsinitaltrials$Attraction1,
                na.rm = T)
            trialnumber_initiallearning_reachedtwelveten[i, 6] <- thisindividualsinitaltrials[k +
                11, ]$relativeattraction/max(thisindividualsinitaltrials$relativeattraction,
                na.rm = T)
            break
        }
    }
}

# Next, we repeat this for the alternative rule of 17 out
# of 20 trials, again using a sliding window approach
# across trials (17*1 + 3*2 = 23).
trialnumber_initiallearning_reachedseventeentwenty <- matrix(ncol = 6,
    nrow = length(unique(simulatedreversaldata_attractionscores$id)))
colnames(trialnumber_initiallearning_reachedseventeentwenty) <- c("Id",
    "TrialAtReachingCriterion", "AttractionScore", "RelativeAttractionScore",
    "AttractionOutOfMax", "RelativeAttractionOutOfMax")
trialnumber_initiallearning_reachedseventeentwenty <- as.data.frame(trialnumber_initiallearning_reachedseventeentwenty)

for (i in 1:length(unique(simulatedreversaldata_attractionscores$id))) {
    thisindividualsinitaltrials <- simulatedreversaldata_attractionscores[simulatedreversaldata_attractionscores$id %in%
        i & simulatedreversaldata_attractionscores$Reversal %in%
        "initial", ]
    trialnumber_initiallearning_reachedseventeentwenty[i, 1] <- i
    for (k in 1:(nrow(thisindividualsinitaltrials) - 20)) {
        if (sum(thisindividualsinitaltrials[c(k:(k + 19)), ]$Choice) <
            24) {
            trialnumber_initiallearning_reachedseventeentwenty[i,
                2] <- (k + 19)
            trialnumber_initiallearning_reachedseventeentwenty[i,
                3] <- thisindividualsinitaltrials[k + 19, ]$Attraction1
            trialnumber_initiallearning_reachedseventeentwenty[i,
                4] <- thisindividualsinitaltrials[k + 19, ]$relativeattraction
            trialnumber_initiallearning_reachedseventeentwenty[i,
                5] <- thisindividualsinitaltrials[k + 19, ]$Attraction1/max(thisindividualsinitaltrials$Attraction1,
                na.rm = T)
            trialnumber_initiallearning_reachedseventeentwenty[i,
                6] <- thisindividualsinitaltrials[k + 19, ]$relativeattraction/max(thisindividualsinitaltrials$relativeattraction,
                na.rm = T)
            break
        }
    }
}

# We can plot the relative attraction scores (prefer
# rewarded option relative to unrewarded option) at the
# points where each individual reached either the 10/12 or
# the 17/20 criterion.
plot(trialnumber_initiallearning_reachedtwelveten$RelativeAttractionScore ~
    trialnumber_initiallearning_reachedseventeentwenty$RelativeAttractionScore,
    ylim = c(0, 50), xlim = c(0, 50))
abline(a = 0, b = 1)
library(rethinking)
PI(trialnumber_initiallearning_reachedtwelveten[is.na(trialnumber_initiallearning_reachedtwelveten$RelativeAttractionScore) ==
    F, ]$RelativeAttractionScore)
PI(trialnumber_initiallearning_reachedseventeentwenty[is.na(trialnumber_initiallearning_reachedseventeentwenty$RelativeAttractionScore) ==
    F, ]$RelativeAttractionScore)

# We can see that (i) the lowest relative attraction scores
# are very similar, suggesting that independent of the
# criterion, individuals generally have formed a preference
# for the rewarded option (89% of the values lie between
# 2.5 and 14); and that (ii) for most individuals, the
# relative attraction scores are very similar for the two
# criteria.

# We can convert the relative attraction scores into the
# probability that an individual will choose the rewarded
# option in the trial following the cut-off by converting
# it back with the respective phi and lambda values
attractionlightgrey_twelveten <- median(trialnumber_initiallearning_reachedtwelveten$AttractionScore,
    na.rm = T)
attractiondarkgrey_twelveten <- (median(trialnumber_initiallearning_reachedtwelveten$AttractionScore,
    na.rm = T))/(median(trialnumber_initiallearning_reachedtwelveten$RelativeAttractionScore,
    na.rm = T))
bird_lambda <- median(simulatedreversaldata_attractionscores$ThisBirdsLambda)
probability_choose_lightgrey_tenoftwelve <- exp(bird_lambda *
    attractionlightgrey_twelveten)/(exp(bird_lambda * attractionlightgrey_twelveten) +
    exp(bird_lambda * attractiondarkgrey_twelveten))

attractionlightgrey_seventeentwenty <- median(trialnumber_initiallearning_reachedseventeentwenty$AttractionScore,
    na.rm = T)
attractiondarkgrey_seventeentwenty <- (median(trialnumber_initiallearning_reachedseventeentwenty$AttractionScore,
    na.rm = T))/(median(trialnumber_initiallearning_reachedseventeentwenty$RelativeAttractionScore,
    na.rm = T))
bird_lambda <- median(simulatedreversaldata_attractionscores$ThisBirdsLambda)
probability_choose_lightgrey_seventeenoftwenty <- exp(bird_lambda *
    attractionlightgrey_seventeentwenty)/(exp(bird_lambda * attractionlightgrey_seventeentwenty) +
    exp(bird_lambda * attractiondarkgrey_seventeentwenty))

probability_choose_lightgrey_tenoftwelve
# 0.7612336
probability_choose_lightgrey_seventeenoftwenty
# 0.839137

# While individuals are more likely to choose the rewarded
# option if they needed the 17/20 criterion to pass, also
# with the 10/12 criterion individuals show a clear
# preference for the rewarded option


# 5) this learned association means that individuals who
# move to the next reversal with this criteria are unlikely
# to solve the reversed association by chance.

# We can simulate the performance in the first 12 trials
# after switching the rewarded option for sets of 10000
# individuals who either have no learned association, have
# been stopped after the 10/12 rule, or stopped after the
# 17/20 rule.
choselightgrey_seventeenoftwenty <- matrix(nrow = 10000, ncol = 12)
choselightgrey_seventeenoftwenty <- as.data.frame(choselightgrey_seventeenoftwenty)

for (i in 1:10000) {
    for (j in 1:12) choselightgrey_seventeenoftwenty[i, j] <- if (runif(1,
        0, 1) < probability_choose_lightgrey_seventeenoftwenty) {
        1
    } else {
        0
    }
}

# We can plot how often these individuals choose the now
# rewarded option in the first 12 trials: individuals with
# no learned associations are in the grey distribution,
# individuals with the 10/12 rule in the blue distribution,
# and individuals with the 17/20 rule are in the red
# distribution.
plot(NA, xlim = c(1, 12), ylim = c(0, 0.3))
lines(dbinom(1:12, 12, 1 - probability_choose_lightgrey_seventeenoftwenty) ~
    c(1:12), col = "red")
lines(dbinom(1:12, 12, 1 - probability_choose_lightgrey_tenoftwelve) ~
    c(1:12), col = "blue")
lines(dbinom(1:12, 12, 0.5) ~ c(1:12), col = "grey")

# As expected based on the relative attraction scores
# (probability to choose the now rewarded option of 24% for
# 10/12 and 16% for 17/20), most individuals choose the now
# rewarded option in less than 6 out of the first 12
# trials, whereas that is the mean for individuals with no
# association

# We can make this more explicit and incorporate that
# individuals will also change their association as they
# learn that the rewarded option has switched assume that a
# bird made 10/12 choices correct
attractiondarkgrey <- 0.388985
attractionlightgrey <- 0.08549744
bird_phi <- median(simulatedreversaldata_attractionscores$ThisBirdsPhi)

# We give the individual 12 trials. We assume that at first
# they chose the previously rewarded option (dark grey) and
# switch to the now rewarded option halfway through
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1

probability_choose_lightgrey_twelvetrials <- exp(bird_lambda *
    attractionlightgrey)/(exp(bird_lambda * attractionlightgrey) +
    exp(bird_lambda * attractiondarkgrey))

attractionlightgrey_seventeentwenty
attractiondarkgrey <- attractionlightgrey_seventeentwenty
attractionlightgrey <- attractiondarkgrey_seventeentwenty
bird_phi <- median(simulatedreversaldata_attractionscores$ThisBirdsPhi)

# We give the individual 12 trials. We assume that at first
# they chose the previously rewarded option (dark grey) and
# switch to the now rewarded option halfway through
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractiondarkgrey <- (1 - bird_phi) * attractiondarkgrey + bird_phi *
    0
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1
attractionlightgrey <- (1 - bird_phi) * attractionlightgrey +
    bird_phi * 1

probability_choose_lightgrey_twentytrials <- exp(bird_lambda *
    attractionlightgrey)/(exp(bird_lambda * attractionlightgrey) +
    exp(bird_lambda * attractiondarkgrey))

# While the learning during the 12 trials changes what they
# choose compared to what they would choose just based on
# their attractions at the end of the previous phase (black
# line is shifted up compared to the blue line), this is
# still not sufficient for them to pass the 10/12 rule
plot(NA, xlim = c(0, 12), ylim = c(0, 0.3), bty = "n", ann = F)
lines(dbinom(0:12, 12, 0.5) ~ c(0:12), col = "grey", lwd = 2)
lines(dbinom(0:12, 12, probability_choose_lightgrey_twelvetrials) ~
    c(0:12), col = "red", lwd = 2)
lines(dbinom(0:12, 12, probability_choose_lightgrey_twentytrials) ~
    c(0:12), col = "black", lwd = 2)
mtext("Choice of the newly rewarded option in the first 12 trials after a reversal",
    side = 1, line = 3, cex = 1.25)
mtext("Percentage of individuals", side = 2, line = 2.5, cex = 1.5)
mtext("Strength of association at a reversal with different criteria",
    side = 3, line = 1.25, cex = 1.5)
legend(x = "topright", legend = c(lty1 = "17 / 20", lty1 = "10 / 12",
    lty3 = "random"), lwd = c(2, 2, 2), col = c("black", "red",
    "grey"), cex = 1, bty = "n")

dbinom(1:12, 12, probability_choose_lightgrey_twelvetrials)[10]
# The probability that a bird would, after a reversal,
# reach the 10 out of 12 rule by chance is 0.001

Determining after which reversal an individual has completed the experiment: serial reversal passing criterion

Data from previous serial reversal experiments suggests that individuals who go through multiple reversals will end up with a performance that is similar to the individuals who needed the fewest trials on the first reversal (C. Logan et al., 2022; Lucon-Xiccato & Bisazza, 2014). This suggests that the manipulation changes individuals within their natural range of variation rather than pushing them to new limits. This means that we can use the performance of the fastest individuals in the first reversal to set the criterion for passing the serial reversal experiment. Accordingly, we can only set the serial reversal passing criterion after the data from the first reversal begins to become available. Some species might already have data from previous studies on reversal learning, however it is important to set the passing criterion for this experiment using this particular setup. Therefore, the criterion must be established from scratch for each species using this setup.

The serial reversal passing criterion: reach the reversal passing criterion (10 out of 12 trials correct) in X trials or fewer in two consecutive reversals.

X = the number of trials required that marks the fastest 20% of individuals in the first reversal. For example, if you test 20 individuals, the number of trials for the 4th fastest individual will be the criterion. For 10 individuals, use the number of trials for the 2nd fastest individual. The fastest 20% was validated using the grackle data (C. Logan et al., 2022): it aligns with the one sigma rule from a normal distribution, indicating the percentage of individuals who are faster than the mean number of trials minus one standard deviation. If more than 20% of individuals reach this number of trials in their first reversal (because there might be a tie), choose the next fastest number of trials to pass. Particularly near the beginning of the experiment, it will be important to set the passing criterion to a lower number to ensure that individuals will be overtrained rather than undertrained.

As the data for additional individuals becomes available, this number can change accordingly. If the number changes across the experiment, we will check whether any currently participating individuals would have already passed according to this criterion and end their experiment.

Individuals need to meet this criterion in two consecutive reversals to pass the serial reversal experiment to ensure that their behavior is consistent and that their speedy performance did not occur by chance. Previous serial reversal experiments show that reversal performance plateaus after a certain number of reversals (e.g., 6-8 reversals in great-tailed grackles C. Logan et al., 2022). If individuals show no consistent improvement after 12 reversals and have not yet met the serial reversal passing criterion, they will be excluded from the experiment. We will plan to start with many more individuals than the minimum sample size to allow for potential drop outs.

We do not expect that the serial reversal manipulation will introduce new negative effects because the passing criterion is set such that the manipulated individuals are only as fast as the fastest 20% of tested individuals. This means that we are not introducing an unnatural amount of flexibility because we are not making any individuals more flexible than what already exists in their population.

There will be individual variation in terms of baseline flexibility before the manipulation such that the flexibility training might influence individuals differently. For example, individuals who are already flexible before the manipulation will not benefit much from the manipulation, while the less flexible individuals will benefit more. Individuals who are already flexible and pass the serial reversals in fewer reversals will still meet the experiment’s passing criterion and be considered to have completed the manipulation, even if they did not improve. Baseline flexibility differences could also be reflected in their pre-manipulation success measures (i.e., individuals with high baseline flexibility might already be successful before the manipulation) if these success measures relate to flexibility. Our statistical models account for these baseline individual differences in success as they might relate to performance on the flexibility manipulation because they include an interaction between the intercept (the value at which individuals start) and slope (by how much they change).

Planned Sample

For each population, depending on the response variable, we ran separate power analyses to determine the planned sample size (see Analysis Plan). For each population, we will aim to reach the minimum sample size required to detect the expected effects of the intervention on the response variables. However, given the difficulties of working with wild individuals, there might be instances where we might not reach a particular target. In such a case, we will interpret the result in light of the power that the particular sample size provides, as indicated by the power analyses. The minimum sample size depends on whether the intervention is performed as a within-subject design (higher power), measuring the response for the same individuals before and after the intervention, or whether it is performed as a between-subjects design, where half the individuals are randomly assigned to the intervention group. In addition, it will depend on whether the response variable has a binary or a continuous outcome (higher power), and in the latter case whether the measure is open-ended (lower power) and therefore individuals will show a large range of values (e.g., dispersal distance).

We will stop collecting data for the flexibility manipulation experiment when a buffer above the minimum sample size is reached or when the season in which the minimum sample size is reached comes to an end, or when the minimum sample sizes for the success measures have been reached. When conducting the manipulation experiment, it is important to aim to test more than the minimum number of individuals because some might not have data in the post-intervention stage.

DESIGN 1 - Reversal learning experiment in visual isolation

Although we do not use this design in our planned studies, we present this design as an option for researchers interested in using the ManyIndividuals framework. Half of the individuals (manipulated group) undergo serial reversal learning until they meet the passing criterion, while the other half (control group) receive only one reversal (Figure 6). A reversal of where the reward is placed represents environmental heterogeneity, and those individuals who have enough experience in this heterogeneous environment through multiple reversals are able to learn to improve their flexibility (the number of trials it takes to change a preference). This flexibility intervention has been shown to work in great-tailed grackles where 8 out of 9 individuals in the serial reversal group achieved the passing criterion (passing 2 consecutive reversals in 50 trials or less) within their given time frame (Logan, MacPherson, et al., 2019). Individuals are presented with two options that differ in color, shape, or in some other way, with one option being the rewarded option. The first rewarded option (i.e., color, shape, etc.) in reversal learning is counterbalanced across individuals at each site. The rewarded option location is pseudorandomized for side. Pseudorandomization consists of alternating location of the rewarded option for the first two trials of a session and then keeping the same color on the same side for at most two consecutive trials thereafter. A list of all 88 unique trial sequences for a 10-trial session, following the pseudorandomization rules, will be generated in advance for experimenters to use during testing (e.g., a randomized trial sequence might look like: LRLLRRLRLR, where L and R refer to the location, left or right, of the rewarded tube). Randomized trial sequences will be assigned randomly to any given 10-trial session using a random number generator (random.org) to generate a number from 1-88. The individual is only allowed one choice per trial and the option on the left is always placed first, with the experimenter always turning to the right when setting up and taking down each trial (if a live experimenter is involved). Once a preference for the rewarded option is reached (10/12 trials correct), the reward is then always placed in the previously non-rewarded option until a preference is reached (using the same criterion). After the control group’s first reversal, they receive a similar amount of experience as the manipulated group, but without the functionality: two apparatuses are still used in every trial, but they are the same color/shape/etc. and both contain food. As with the manipulated group, only one choice, the first choice, is allowed per trial. Choices are scored as: 1=chose the correct option (even if they do not eat the food), 0=chose the incorrect option, and -1=did not make a choice.

Passing criteria:

• Habituation: leave one color/shape/etc. apparatus that is not used in the experiment in the enclosure overnight and feed the individual off of it until they readily approach it for food.
• Training to look for non-visible food: obtain the food from the habituation apparatus (only one presented per trial) in 5 consecutive trials when the food is hidden inside the apparatus (and not visible to the individual unless they approach the apparatus and choose to look inside). Choices are scored as: 1=ate first from inside the apparatus, 0=ate food from around the apparatus but not inside it, and -1=did not eat any food.
• Training to eliminate any color/shape/etc. preference: choose one color/shape, of the two possible colored/shaped apparatuses, first 8 or fewer times out of 10 trials, which indicates no preference. Use both experimental apparatuses (e.g., two colors or shapes) per trial with the food openings taped over so the individual cannot look inside the tubes. Place both apparatuses in the test area at the same time and place food on the outside of both apparatuses at the same time (if you need to spend more time on one apparatus because the food falls off or something, make mirror movements on the other apparatus at the same time so both apparatuses get the same amount of attention and in the same way). Choices are scored as: 1=ate first from the color/shape that will be the rewarded option, 0=ate first from the color/shape that will be the non-rewarded option, and -1=did not eat any food.
• Reversal (including initial discrimination): using the same apparatuses as above but with the openings untaped, must obtain the food from the rewarded color/shape on at least 10 of the latest 12 trials, calculated in 1-trial windows (i.e., the individual can pass on trial 21, 35, 44, etc.).
• Serial reversal manipulation group: across reversals, must achieve the reversal criterion for two consecutive reversals in 50 trials or less. This passing criterion was generated using great-tailed grackles (Logan, MacPherson, et al., 2019) and might need to be adjusted depending on the population.
• Control group: receives as many trials with the control apparatuses (two containers of identical color/shape, both rewarded) as the average number of trials that manipulated individuals require to pass serial reversals. If this is unknown at the beginning of the experiment, test a manipulated individual first and match the control individuals to this number until an average can be obtained. The average can continue to be updated as more manipulated individuals complete testing.

Protocols and data sheet templates

Protocol for reversal learning of a color preference used by Logan, MacPherson, et al. (2019) for great-tailed grackles.

See the data sheet templates in Logan, MacPherson, et al. (2019).

Interobserver reliability

We conduct interobserver reliability by having hypothesis-blind video coders code all of the videos for 20% of the individuals in the experiment: instructions for video coders. Live coder data is then compared with video coder data by analyzing whether the individual made the correct choice (1) or not (0) or whether they did not make a choice (-1) using Cohen’s unweighted kappa (Landis & Koch (1977), with the psych package in R Revelle (2017)). This measure indicates how replicable the experiment and the coding instructions are. Before a video coder is approved to begin coding, they must first pass training where they code a different set of videos and reach an unweighted kappa of 0.89 or above.

library(irr)  #ICC package

# did video coder pass interobserver reliability training?
data <- read.csv("", header = TRUE, sep = ",", stringsAsFactors = FALSE)
head(data)  #Check to make sure it looks right
# Note: c(3,5) is telling R to look at columns 2 and 3 and
# compare them. Double check this:
data[, 3]  #coder 1 (live coder)
data[, 5]  #coder 2 (video coder)
cohen.kappa(data[, c(3, 5)], w = NULL, n.obs = NULL, alpha = 0.05,
    levels = NULL)

# video coder score for 20% of videos =
data <- read.csv("", header = TRUE, sep = ",", stringsAsFactors = FALSE)
head(data)  #Check to make sure it looks right
data[, 3]  #coder 1 (live coder)
data[, 5]  #coder 2 (video coder)
cohen.kappa(data[, c(3, 5)], w = NULL, n.obs = NULL, alpha = 0.05,
    levels = NULL)

DESIGN 2 - Reversal learning experiment in a group context

Feeders are set up in the field, counterbalanced for color/location/etc, and will be available for opening when the experiment is being conducted (Figure 7). The feeders will be fitted with technology to automatically record which individual visits which feeder and when (unless a particular population is easy to track visually without such automated technology). Individuals are habituated to the open feeders filled with food until at least half of the minimum sample size has visited at least 1 feeder. All individuals start on the same rewarded option for the initial discrimination to improve motivation for participating in the experiment (e.g., if the first rewarded option was counterbalanced across individuals, subjects might be slower to learn their rewarded option if they use social information about which option is rewarded). If the subject visits the rewarded feeder, the feeder will automatically, or via a remote controlled by the experimenter, deliver a small amount of food, and then close and reset more food in preparation for the next opening. If the subject visits the non-rewarded feeder, the presence data will be recorded, but the feeder will not open. All feeders will contain one type of high value food. Ideally, feeders will be programmed to automatically switch which feeder type is rewarded as soon as an individual passes criterion in the middle of a test session. If automation is not possible, then the data sheets will be coded live by the experimenter and if that is not feasible, then the data sheet will be generated after each test session or at the end of the day by coding the videos to determine which individuals have passed criterion, and their rewarded feeder type will be changed in the next test session.

Individuals in the control condition (if there is one), will receive 1 reversal and, after they pass criterion on reversal 1, feeders of both options (previously rewarded and previously non-rewarded options) will open for these individuals. This will help keep the whole group interested in visiting the feeders while the individuals in the manipulated group complete their serial reversals. Data are collected on success measures (see below) either before and after the flexibility intervention or only after the intervention (depending on feasibility and how the study design needs to be tailored for each population).

Passing criteria:

• Feeder habituation: all feeders at all locations will have food and be open for several hours daily or until at least half of the minimum sample size in each condition (control and manipulated) have visited at least one of the feeders.
• Reversal passing criterion: an individual is considered to have a preference when they choose 10 of the most recent 12 trials (choices) correct (the rewarded option). This criterion applies to the initial discrimination, and to each reversal.
• Manipulation passing criterion: pass two consecutive reversals in X trials or less (see Serial reversal passing criterion above).

Protocols

Food in the feeders: All feeders are opaque and always have food in them to eliminate the confound due to olfactory differences between the feeders that could be introduced if only the rewarded feeders have food in them. If a feeder needs to be refilled, refill all feeders consecutively in the same time period and refill each for the same amount of time even if that feeder does not need much or any food (in these cases, pretend to fill the feeder as you normally would). This eliminates confounds from cues provided by a differential amount of attention experimenters give the feeders depending on which needs refilling.

Reliability of feeder data

During the experiment, we will determine the reliability of the automated feeder data. We will verify that the feeders are correctly detecting tags by using a (motion-activated, Go-Pro, or similar) camera focused on 20% of the feeder perches. Hypothesis-blind video coders will then code who landed on the perch: individual ID, time of day, and whether it ate food. We will then compare the automated feeder data sheets with the video coder’s data sheets by calculating the Cohen’s unweighted kappa for individual ID and the intra-class correlation coefficient (ICC) on the time of day (kappa: Landis & Koch (1977), with the psych package in R Revelle (2017); ICC: Hutcheon et al. (2010), with the irr package in R: Gamer et al. (2012)). Cohen’s kappa is used when the distance between measurements is not quantifiable numerically, and the ICC is used for continuous variables with equal distances between units.

Assessment of the likelihood of success in human modified environments with regard to the flexibility manipulation

After the manipulation, compare control individuals with individuals in the flexibility manipulation and/or pre-manipulation and post-manipulation measures on the same individuals using one or more of the following success measures. Choose population-relevant success variables that are predicted to be the most likely to be used in human modified environments. This list is not exhaustive - it serves as a place to generate ideas about what the best measures could be in a given population. Observational methods to collect these data may vary among populations and we describe below the methods that we will use (i.e., focal follows and all occurrences sampling).

• Fitness variables: nest success, number of offspring who survived to independence or adulthood, longevity, etc.
• Foraging variables: diet breadth, number of foraging techniques used, etc.
• Movement variables: predictability of movement behavior [e.g., step length and turning angles; see McCune KB et al. (2020)], ability to disperse from a lower risk environment (e.g. a sanctuary, or largely intact natural habitat) to a higher risk, more heavily human modified environment (assess success/survival after dispersal if possible), etc.
• Habitat use variables: foraging substrate (ground, bushes, trees, human modified substrates, human-provided supplemental food), nesting substrate (high or low, tree, bush or reeds), etc.

Begin collecting post-manipulation data on an individual as soon as it passes the manipulation because it is unknown for how long any potential effects of the manipulation will last. For the more social species and for populations who participate in the experiment in groups, there is the potential that individuals who were not in the manipulation condition or who have not yet passed the manipulation condition will learn about post-manipulation success behaviors from the manipulated individuals who passed the experiment. We are ultimately interested in determining whether we can change success behaviors as a result of the flexibility manipulation. If part of this change is the result of social learning from some of the manipulated individuals, it will still result in a change even if we do not quantify what percentage of the mechanism comes from individual learning after the manipulation or social learning after the manipulation. If there is a change in success measures between before and after the manipulation, the manipulation will have been the cause in either case.

If other researchers are interested in beginning to quantify whether social learning is involved in the spread of success behaviors, then they could collect data on post-manipulation success measures from the individuals who did not pass the manipulation and compare them with individuals who passed the manipulation. If there is no difference between both groups’ post-manipulation success measures, this indicates that social learning was involved in the spread of success behaviors. They could then use this data in a future registered report examining the role of social learning in the spread of success behaviors. If researchers are not planning on a social learning component in future research and/or do not have the time or resources to collect more data, they can refrain from collecting post-manipulation data on success measures on the individuals who did not pass the manipulation criterion.

Observational methods

We will use slightly different observational methods for grackles and jays to collect data on foraging and microhabitat use. However, the categorization of food and habitat types will be the same:

Microhabitat types in the suburban habitat (<100m from human structure) include: vertical human structure (e.g. building, bench), native vegetation, non-native vegetation, grass, impervious surface, and dirt. In the natural habitat (>100m from human structure), microhabitat types include all previous categories, but not human structure or impervious surface All categories can be further defined by whether the subject was high (>3m) or low (<3m) (for example, grass and impervious surfaces can occur above 3m if grass is on the roof of a building, and if an individual is walking on the impervious surface of an upper floor parking garage).

Food types are broken down into plant (seed, fruit, human-provided, or unknown plant) and animal (insect larva, adult insect, amphibian, reptile, mammal, bird, egg, human-provided, or unknown animal). “Human-provided” indicates any food item that was acquired from a store at some point and is left out by humans. For example, sunflower seeds would be considered human-provided if they are in the form of bird seed or a human snack. Sunflower seeds would only be counted in the “seed” category if the bird is seen eating it from a plant. Data will be collected on the four plant subcategories and nine animal subcategories and used in the analyses.

During follows, we will record each microhabitat the individual is present in and all food items consumed. Before data analysis, to ensure that we are only including the microhabitats individuals use (rather than just pass through), we will filter the data to include only microhabitats that account for at least 5% of their data points. Although we may not observe every possible microhabitat or food item the individual may use, by equally sampling before and after the manipulation we can detect changes in habitat use.

Additionally, for all observational methods we use binoculars so that we can always attempt to stay far enough from the focal individual that our presence is not affecting their behavior. Because that distance can be different for each individual and species, we hesitate to give a specific number. However, if at any point the focal individual shows it is affected by our presence by looking directly at the researcher, alarm calling or startling, then we end the focal immediately, drop all data from that follow and attempt the follow again the next day.

We will attempt to balance follows for each observational method in the morning and afternoon for all individuals in the study. In this way, we will collect a random sample of data from active and inactive time periods for all individuals. Additionally, when automated tracking technology is used, it will record daily movement patterns, which will identify whether there is temporal or spatial variation in bird behavior.

Focal follows

We will collect data on foraging and microhabitat use of grackles during 10 minute (if using tracking technology in conjunction) or long (if tracking technology is not available) focal follows and our minimum sample size will be 20 individuals. If using the 10 minute focal folows, we will do 4 focal follows per individual before the flexibility manipulation and 4 focal follows per individual after the manipulation (8 total 10-min follows for 80 minutes of follow per bird). If conducting long focal follows, follow the focal for as long as possible (this might require multiple long focal follows across multiple days) until it has exhibited the number of behaviors needed (this depends on the specific criteria involved in the research question). In the interest of minimizing the temporal and spatial autocorrelation of behavior, the focal follow duration and minimum time between focal follows will vary according to the species, the local pattern of persistence of food sources in the environment, and how quickly the species consumes food piles. For example, a study on dolphins (Tursiops cf. aduncus) found that focal follow data that was separated by 10.5 min was functionally independent (Karniski et al., 2015). Therefore, we suggest adjusting the focal follow duration and spacing to accommodate the species under study.

All occurrences sampling

We will collect data on the foraging and microhabitat use of jays during spatial movement tracking (the latter as part of another investigation). We will follow jays for 60 minutes and record at 1-minute intervals the spatial location (GPS coordinates), any food items consumed, the microhabitat the jay is present in, and breeding behaviors if it is the breeding season. We will do 4 tracks per individual before, and 4 tracks per individual after the flexibility manipulation.

Open data

The data will be published in the Knowledge Network for Biocomplexity’s data repository.

Great-tailed grackles

• Grackle protocol and data sheet templates
• Protocol for applying radio tags and conducting GPS tracks from McCune KB et al. (2020)

Planned sample

We will catch at least 20 grackles using walk in traps, mist nets, and bow nets; collect their biometric data, blood, and feathers; apply colored leg bands for individual identification, one band will have a PIT tag that will interface with the automated feeders (if automated feeders are used); attach a radio tag using a leg loop harness (if tracking gear is used); and release them at their point of capture. We will collect data on pre- and post-manipulation success measures and conduct the experiment within the non-breeding season to control for potential temporal differences in the environment and behavior.

Before and after manipulation success measures

The proportion of time spent at dumpsters and outdoor cafes when food is present will be collected using one or both of two methods: 1) automation from Cellular Tracking Technologies, and/or 2) long focal follows (Altmann, 1974). For method 1, individuals will wear radio tags whose signals are detected (within approximately 75m) by nodes that are placed at approximately 200m intervals. For method 2, focal individuals will be followed for as long as possible until criterion is reached (see later in this paragraph). Reaching criterion may require conducting multiple focal follows on multiple days. Focal follows will be spaced apart by at least one day because grackles either have food sources that persist across the whole season (e.g., oranges in an orange grove) or for a few hours (e.g., french fries in a parking lot) (Logan, pers. obs.). Therefore, the foraging observations on temporary (on the order of a few hours) food sources should be independent. Before and after tracking/long focal follow data will be collected on a minimum of 20 individuals with a minimum of 4 separate visit events per individual pre-manipulation and a minimum of 4 separate visit events per individual post-manipulation. This data will allow us to detect pre- vs post-manipulation within-individual differences in visits to dumpsters and cafes to determine whether individuals change the proportion of time they spend at these locations when food is present.

For method 1, a determination will be made daily about whether food was present at a location and in which particular time period. We will visit the dumpsters each morning to record whether the lids were open and there was garbage inside or garbage on the ground around the dumpsters (1=food present) or whether the lids were closed and there was no garbage on the ground (0=food absent). We will initially visit cafes during lunch hours to determine when people generally eat outside and then set the default food presence period for each cafe to the widest time period we observed for that cafe (e.g., 12-2p = food present, before 12p and after 2p = food absent). For method 2, food presence/absence data is built in because the observer collects foraging, habitat, and location data at the same time.

We will track baseline behavior and changes after the flexibility manipulation in habitat use and foraging using the same long focal follows as above. We document all occurrences of microhabitats used and foraging of the focal individual (see Observational methods, above). These data will allow us to detect any pre- and post-manipulation within-individual differences in diversity of habitat use and foraging breadth.

Flexibility manipulation (Design 2 reversal learning)

When using automated feeders, we will set up 4 feeders (2 dark gray and 2 light gray) at one location in a particular spatial arrangement: one dark gray and one light gray feeder will be oriented in the same way 1-2m apart, and the other dark gray and light gray feeders will be facing them in a mirrored position, but 5-10m away. The feeders will be fitted with RFID readers to automatically record which individual (fitted with a PIT tag attached to a leg band) visits which feeder and when. When using remote controlled feeders, 2 feeders (1 dark gray and 1 light gray) will be set up at a location and we will attempt to minimize side bias by alternating which color is on which side each day as well as moving the feeders around so they are never in the exact location they were on a previous day. The feeders will be available for opening for throughout the day: we will find birds that need training, set up and run a training session, then move and repeat), 5 days per week. If the subject visits the feeder that has the rewarded color, the feeder will automatically, or via a remote control, deliver a small amount of food, and then close and reset more food in preparation for the next opening. If the subject visits the feeder that has the non-rewarded color, the presence data will be recorded, but the feeder will not open. All feeders will contain one type of high value food (e.g., goldfish crackers, cheetos, cheez-its, or peanuts).

Feeder habituation: All feeders will have food available and be open for 3 hours per day on 5 consecutive days or until at least 20 banded grackles have visited at least 1 feeder.

The experiment: Each time a subject lands on an automated feeder or approaches the remote controlled feeder, it is recorded as the color choice and counts as one trial. The experiment will begin with the dark gray feeder being the rewarded feeder for all individuals.

Jays

• Jay protocol and data sheet templates
• Protocol for applying radio tags and conducting GPS tracks from McCune KB et al. (2020)
• Jay processing protocol

Planned sample

We will catch up to 60 jays per year using walk in traps, mist nets, and bow nets; collect their biometric data, and a blood sample; apply colored leg bands; attach a radio tag using a leg loop harness; and release them at their point of capture. We will collect data on pre- and post-manipulation success measures and conduct the experiment within the non-breeding season to control for potential temporal differences in the environment and behavior.

To determine whether the flexibility manipulation has influenced the ability of jays to persist in human modified environments, we will catch half of the jays in areas with access to human-supplemented food (i.e. private property, a university campus, parks adjacent to neighborhoods with feeders) and the other half in natural areas (wildlife management areas, reserves).

Before and after manipulation success measures

We will track baseline behavior and changes after the flexibility manipulation via spatial movement tracking that lasts for 60 min, noting the GPS location and the jay’s behavior at 1 minute intervals. During tracks, we document all occurrences of microhabitats used within the territory (see Observational methods, above), foraging (see Observational methods, above), and breeding behaviors of the focal individual, if it is the breeding season (Altmann, 1974). The minimum sample size will be 20 individuals per species with a minimum of 4 tracks per individual (at least 2 per month) pre-manipulation and a minimum of 4 tracks per individual (at least 2 per month) post-manipulation (at least 320 tracks in total). These data will allow us to detect any pre- and post-manipulation within-individual differences in space use, diversity of habitat use, and foraging breadth.

Flexibility manipulation (Design 2 reversal learning)

We will set up feeding stations at a minimum of 4 study sites, each containing multiple jay territories, spaced at least 2km apart: 2 sites in natural habitat, 2 in human-modified habitat. If individuals are dispersed across multiple areas, we will attempt to add an equal number of sites in natural and human-modified habitats. Gravity feeders will be set up in territories in each of the sites in a similar spatial arrangement as in the grackle experiment (above).

Feeder habituation

All feeders at all locations will have food and be open for 2-hour sessions between 8a-3p on a minimum of 4 consecutive days or until at least 1 banded jay per territory per site (natural/human modified) has visited at least 3 feeders.

The experiment

Prior to the flexibility manipulation, we will collect the 4 minimum tracking sessions to determine the baseline values for space use, nest success (if appropriate), microhabitat use and foraging breadth. Afterwards, we will set up feeders to initiate the flexibility manipulation phase. Once jays are habituated to the feeders we will manipulate behavioral flexibility using serial reversals of the open feeder location or color. Only one feeder will be opened at a consistent location (or that is a consistent color) within territories across days and the manipulation treatment will consist of 30-min sessions per day per territory, up to 4 days per week, where each visit by the focal jay to a feeder is considered a trial. Jays pass a given reversal when they correctly choose the rewarded feeder in at least 10 trials out of the most recent 12 trials. Serial reversals will continue until jays pass two consecutive reversals in X trials or less (see Serial reversal passing criterion above). At this criterion, the jays will be considered to have increased their behavioral flexibility. After the manipulation is complete in each territory, we will again conduct the tracking sessions to measure space use, nest success (if appropriate), microhabitat use and foraging breadth.

Toutouwai

• Toutouwai protocol and data sheet templates

Planned sample

In the Zealandia toutouwai population the breeding season runs from October to February. Pairs typically produce 1-3 fledglings per nesting attempt and can nest up to 3 times per season. Each season, around 30 pairs nest in the long term study area (Figure 9), so we will aim to catch up to 30 fledgling toutouwai a year (one per pair, 60 in total) using a drop trap. We will collect their biometric data and a feather sample for DNA sexing. On each bird we will apply 2 colored leg bands for individual ID (one to each leg), a uniquely numbered metal BP sized band (supplied by the New Zealand National Bird Banding Scheme) and a single RFID tag attached to a leg band (on the opposite leg to the metal band). We will also attach a radio tag using a leg loop harness that degrades over time. Birds will be released at their point of capture. We will conduct the experiment in the breeding season and collect data on post-manipulation success measures in the breeding season as well as the non-breeding season. In this case, we do not need to control for potential temporal differences in the environment and behavior because we are measuring juvenile dispersal behavior.

 

Figure 9. The toutouwai study area shown in white, is situated centrally within Zealandia Ecosanctuary, outlined in purple.

 

Flexibility manipulation (Design 2 reversal learning)

Juvenile toutouwai typically begin to forage independently on their caregiver parent’s territory from 3-4 weeks post-fledging. We will set up two experimental feeders on the caregiver parent’s territory when the juvenile toutouwai is at least 3 weeks post-fledging and has been captured, radio tagged and fitted with an RFID tag. One juvenile per territory will be randomly selected to participate in the experiment. Half of all juveniles will be assigned to a manipulated condition and half will be assigned to a control condition. The condition (control or manipulation) will be randomly assigned to each bird using random.org. On each territory the two RFID activated feeders will be spaced a maximum of 4 m apart and within sight of each other, with one feeder mounted within 20 cm of the ground, and the other mounted at 2 m height. Feeders on the ground will be light gray in color and feeders mounted at height will be dark gray, to make them both spatially and visually distinct. To ensure that the juvenile belonging to the territory is the only bird that can access the feeders, each feeder will only open when the individual fitted with the RFID tag coded to the feeder approaches.

Feeder habituation

Both feeders on a territory will be available to the RFID tagged individual(s) from 8 am - 3 pm for 5 consecutive days, or until the juvenile has visited both feeders. When a juvenile lands on the perch of an available feeder, the RFID tag will be read and the feeder will automatically open to dispense a live mealworm reward.

The experiment

Juveniles in both the manipulation and control conditions will initially undergo one reversal learning procedure consisting of a discrimination learning phase and a reversal learning phase. During the initial discrimination learning phase, only one feeder will be available (hereafter called the rewarded feeder) and the location of this rewarded feeder (high or low) will be assigned randomly, with half of the birds receiving the low and half receiving the high feeder as their rewarded feeder. The rewarded feeder will be available for 2 hours each day, or until a maximum of 100 trials have been completed (whichever occurs first). The rewarded feeder will automatically open when the target bird (fitted with the correct RFID tag) lands on the feeder perch. However, when the target bird lands on the perch of either feeder (rewarded or unrewarded), the visit will be logged and counted as a single trial. To pass the discrimination phase, the bird must visit the rewarded feeder in 10 out of 12 consecutive trials. Once this criteria is reached, the juvenile toutouwai will be given a reversal phase, where the previously unrewarded feeder is now rewarded. To pass this reversal phase, the bird must again achieve the criterion of 10 out of 12 consecutive trials visiting the correct feeder. Birds in the manipulation group will then receive serial reversals until the point at which they are switching feeder preferences and pass the serial reversal criterion (see Methods). For juveniles in the control group, after the initial single reversal both feeders will remain available and rewarded for two hours each day (or until food is depleted) for a minimum of 8 days.

Post-manipulation success measures

We will radio track fledglings for 12 weeks following the end of the flexibility manipulation (i.e., until they are 112 days post-fledging). Tag signals will either be detected with a hand-held antenna, or by nodes that are placed at approximately 200m intervals across the 25 ha robin study area and around the sanctuary perimeter. Each fledgling will be located once per week over this time period and their location will be GPS marked to reconstruct dispersal tracks. By the end of 12 weeks, fledglings will be approximately 4 months post-fledging, at which point they are likely to be attempting to establish their own territories. The measures used to investigate the effect of the manipulation versus control treatment will include the final dispersal location (inside or outside the sanctuary), the total dispersal distance between the natal territory and final location, the age at which dispersal is first detected (defined as when the bird has left the caregiver parent’s territory for a minimum of 3 days), and their survival status at 16 weeks post-fledging (alive/dead).

ANALYSIS PLAN

We run analyses in R [current version 4.3.2; R Core Team (2017)] using the following R packages: rethinking (McElreath, 2020a), rstan (Stan Development Team, 2020), cmdstanr (Gabry & Češnovar, 2022), knitr (Xie, 2018), and irr (Gamer et al., 2012).

Can flexibility be increased to help individuals succeed in human modified environments?

DESIGN 1 - Reversal learning experiment in visual isolation

Can flexibility be increased? If most individuals in the flexibility manipulation (serial reversals) passed the passing criterion, then the answer is yes.

Do the flexibility manipulated (serial reversal) individuals have better success in human modified environments?

Response variable: success variable (e.g., predictability of movement behavior, number of different food items taken, etc.)

Explanatory variable:

• Condition (control, manipulated)

OR

• Condition (pre-manipulation, post-manipulation)

Random variable: ID (when response variable has multiple data points per individual)

DESIGN 2 - Reversal learning experiment in a group context

Same questions, response/explanatory/random variables as in Design 1.

Great-tailed grackles

G.Q1 Do flexibility manipulated individuals differ in the proportion of time spent at cafes and garbage dumpsters when food is present?

The model

Bayesian model for a binomial distribution:

\(R_i\) ~ Binomial(\(t_i\),\(p_i\))

\(R_i\) is the duration spent at cafes and dumpsters when food rewards were present and \(t_i\) is the total duration spent at cafes and dumpsters either before or after (i) the flexibility manipulation. A binomial distribution was used because the response variable is a proportion (McElreath, 2020b).

logit(\(p_i\)) = \(\gamma\)[condition] + \(\alpha\)[ind],[condition]

\(\gamma\)[condition] is the average log-odds for each condition (before/after) and \(\alpha\)[ind],[condition] is the effect for each individual in each condition. The Bayesian model was developed using McElreath (2020b) as a guide.

Power analysis

We estimated our power to detect differences between conditions at different sample sizes and with different mean changes in the proportion of time spent in the presence of a reward in the before vs. after conditions (Figure 10). We simulated the proportions of times that different sample sizes of individuals would spend in the presence of the reward before and after the flexibility manipulation. We analyzed these simulated data with the model we will use to analyze the actual data, estimating the change in the proportion of time spent in the presence of the reward between the before and after conditions. From the posterior estimates of the model, we extracted both the mean change as well as the ratio of the posterior estimates that were below zero.

If the mean ratio of estimates below zero is close to 0.5, the model assumes that the change in proportion of time spent in the presence of a reward before the flexibility manipulation is similar to after. If the ratio is close to zero, the model assumes individuals have changed their behavior. For changes smaller than 0.1, models are likely to assume that no changes occurred even with large sample sizes. If the change in the proportion of time an individual spends at a location when food is present before the flexibility manipulation vs. after is 0.1, on average 93% of the posterior of the model based on a sample size of 20 individuals will be larger than zero. This means that the model is quite certain there is a difference that is larger than zero. In addition, none of the models for a sample size of 20 at the mean change of 0.1 have a ratio larger than 0.3, meaning that the risk of having a false negative is unlikely.

In general, with sample sizes at or above 20 and mean changes in the proportion of time spent in the presence of a reward is 0.1 or larger, then it is highly likely that the model will indicate that individuals have changed their behavior. Mean changes below 0.1 can still be detected, however there is a higher risk that there will be a false negative. If the change in the proportion of time an individual spends at a location when food is present before the flexibility manipulation vs. after is 0.025, on average 61% of the posterior of the model based on a sample size of 20 individuals will be larger than zero. In addition, only 20% of the models for a sample size of 20 at the mean change of 0.025 have a ratio larger than 0.3, meaning that the risk of having a false negative is still low.

With small mean changes in the response variable, some individuals might not increase or even decrease their response after the manipulation because there is variation around the mean change in individual responses. With small sample sizes, there is a risk that only individuals who did not clearly increase their response will be studied, whereas larger sample sizes are more likely to include a wider spectrum of individuals.

To estimate the risk of detecting false positives, we set the mean change in the proportion of time spent in the presence of the reward to zero so there was no change between the before and after conditions. As expected, the average ratio of estimates below zero is close to 0.5 and independent of sample size. With a sample size of 20, 43% have a ratio smaller than 0.3, meaning that the risk of having a false positive is high. The risk would be lower if the variation among individuals was lower than what we assumed (across all models, the standard deviation of the mean change in proportion was 0.1, which is a conservative estimate).

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
rstan_options(auto_write = TRUE)
#rebuild_cmdstan() #run this if you get an error about a PCH file https://discourse.mc-stan.org/t/struggling-to-get-cmdstan-working-on-mac-os-big-sur/19774/4

### SIMULATE the population
# use the cafe example for normal distribution p.437, the primate pulling example p.447 for the binomial distribution, and the grad admissions example for how to deal with a response variable that is a proportion p.341 (McElreath 2020). Replace the 1 in the primate pulling model with the total duration; the response variable is duration when reward is present. So the model makes the proportion for you - feed it the raw data.
N_ind <- 20 #number of individuals tested

#choosing the mean proportion change in the amount of time spent at a location when a reward was present and standard deviations - visualize to see whether it would be possible to tell the difference between before (black) and after (red). An SD of 0.1 (when the mean before is 0.6) encompasses all of the range of proportions we expect to find (0.3-0.9) so it would be the largest SD we would expect and thus the most conservative sd to use in the simulations
prob_before<-0.6
changeprop<-0.025
plot(density(rnorm(10000,prob_before,sd=0.1)),lwd=2,xlim=c(0,1.2))
lines(density(rnorm(10000,prob_before+(1-prob_before)*(changeprop*2),sd=0.1)),lwd=2,col="red")

#before
durationtotalb <- round(rnorm(N_ind,mean=10000,sd=500),0) #observed for 10000 seconds before manipulation, 0=no decimal points
propbefore <- round(rnorm(N_ind,mean=0.5,sd=0.1),2) #proportion of duration rewarded:total duration, 2 means 2 decimal points
durationrewardb <- round(durationtotalb*propbefore,0) #duration at location when reward was present
changeprop <- rnorm(N_ind, mean=(1-propbefore)/5,sd=0.1) #if had a large proportion before, have a smaller change relative to the after condition. Divide by 2 = assume that on average they will improve by 0.25 (the before mean of 0.5 divided by 2)

#after
propafter <- round(propbefore+changeprop,2)
propafter[propafter>1]<-1 #values larger than 1 mess up the model so restrict it
durationtotala <- round(rnorm(N_ind,mean=10000,sd=500),0) #10000 seconds, after manipulation
durationrewarda <- round(durationtotala*propafter,0) 

#check the plot to see whether individuals are increasing their proportions
#plot(propbefore,propafter)
#abline(0,1)

# power analysis: run this simulation with different N_ind, changeprop means and sd, to find the boundaries of where you can detect a difference (run the model and use contrasts) between the two conditions (before/after). Automate the process like Dieter did in xpop figs 5 (line 1133) and 6 (line 1452)?


### POWER ANALYSIS: different sample sizes and different mean changes in proportions between before and after conditions
# We now set our range of likely sample sizes, from a minimum of 9 individuals per population to a maximum of 60 individuals per population. We draw repeated samples of individuals of that size from each of the 5 populations, during each repetition we draw two samples from each population (so we can infer the false negative rate of wrongly estimating that two samples come from a different population even though they were taken from the sample population). This means we have 10 repetitions for each of the 5 sample sizes and each of the 6 changeprops. So 10*6 changeprops (=60) for each of the 5 sample sizes, resulting in a total length of samplesizes of 300
samplesizes<-c(10,20,30,40,60) #c(10,20,30,40,60)

# Set the means you want to compare
meanchangeprop1<-0.4 
meanchangeprop2<-0.3
meanchangeprop3<-0.2 #e.g., a bird spends 0.6 of its time being rewarded at cafes and dumpsters before the manipulation and then increases this proportion to 0.8 after the manipulation, showing that the manipulation improved their foraging success
meanchangeprop4<-0.1
meanchangeprop5<-0.05
meanchangeprop6<-0.025
changepropvariants<-c(meanchangeprop1,meanchangeprop2,meanchangeprop3,meanchangeprop4,meanchangeprop5,meanchangeprop6) #for false negatives
#changepropvariants<-c(0) #for false positives
#changepropvariants<-c(meanchangeprop6) #for showing differences

#compare different baseline proportions for how much time these individuals spend when the reward is present before the manipulation
prop_before<-c(0.5,0.6,0.7) #c(0.5,0.6,0.7)


#set up data frame where results will be stored
simresults <- matrix(NA,nrow=900,ncol=7)
simresults <- data.frame(simresults)
colnames(simresults) <- c("n","changeprop_mean","changeprop_sd","repetition","proportion_before","proportion_estimates_below_zero","count_estimates_below_zero")
counter<-1
 
# run simulations. There are 5 different sample sizes, and for each we have 6 different mean change in proportions that we want to examine, and each will repeat 10 times for a total of 300 samples.
for (samplesize in 1:length(samplesizes)) {
        N_ind <- samplesizes[samplesize] #number of grackles per sample size
        
        #now run through each of the 6 proportion variations
        for (propvariant in 1:length(changepropvariants)) {
        
        #pick the current proportion variant from the list changepropvariants
        currentchangeprop <- changepropvariants[propvariant]
        
        for (baselineprop in 1:3) {
        #pick the current proportion variant from the list changepropvariants
        currentprop_before <- prop_before[baselineprop]
        
        for (repetition in 1:10) {
        #create data frame to store for each repetition of the simulated data to be analyzed by the model
        sites <- matrix(nrow=n,ncol=4) 
        colnames(sites)<-c("ID","condition","duration_rewarded","total_duration")
        
        #BEFORE manipulation
        #observed for 10000 seconds before manipulation, 0=no decimal points
        durationtotalb <- round(rnorm(N_ind,mean=10000,sd=500),0) 
        #proportion of duration rewarded:total duration, 2 means 2 decimal points. Assume baseline proportion is 0.5, 0.6, or 0.7
        propbefore <- round(rnorm(N_ind,mean=currentprop_before,sd=0.1),2)
        propbefore[propbefore>1]<-1
        #duration at location when reward was present
        durationrewardb <- round(durationtotalb*propbefore,0) 
        #choose the next proportion change for each individual. If had a large proportion before, have a smaller change relative to the after condition. Multiply by 0.2 = assume that on average they will improve by 0.2. Modifying the proportion after relative to the proportion before accounts for those individuals who were already successful before the manipulation and thus won't have much of a change in prop due to the manipulation. Using the relative change also ensures that an individual will never have an after proportion larger than 1. Multiply currentchangeprop by 2 bc want the relative change to be in about the same order of magnitude as the proportion of change for each individual as the absolute change. If the proportion before is 0.5, the relative change will be the same as the absolute change. If the proportion before >0.5, then the absolute change will be smaller. E.g., if proportion before = 0.7, then a relative change of 0.2 leads to an absolute change of 0.14. If it is <0.5, then the absolute change will be larger than 0.2.
        changeprop <- rnorm(N_ind, mean=(1-propbefore)*(currentchangeprop*2),sd=0.1)  

        #AFTER manipulation
        propafter <- round(propbefore+changeprop,2)
        propafter[propafter>1]<-1 #values larger than 1 mess up the model so restrict it
        durationtotala <- round(rnorm(N_ind,mean=10000,sd=500),0) 
        durationrewarda <- round(durationtotala*propafter,0) 
  
        ### run a STAN model to see whether the before and after proportions are estimated to be different from each other
        #sets up the data sheet with the simulated data that the model will run on
       dat <- list(
    response = c(durationrewardb,durationrewarda),
    total = c(durationtotalb,durationtotala),
    tid = c(rep(1,N_ind),rep(2,N_ind)), #1 refers to durationrewardb and 2 to durationrewarda
    actor = rep(1:N_ind,2) #go through all individuals twice because there are observations from before and after for each
)
       
        # Our MODEL, which will estimate a different mean and variance of phi for each site 
       #(note: had to add force_recompile bc kept giving me an error and wouldn't finish running. See https://github.com/rmcelreath/rethinking/issues/338)
s1 <- ulam(
    alist(
        response ~ dbinom(total,p),
        logit(p) <- g[tid] + alpha[actor,tid],
        # adaptive priors - non-centered which is a parameterization that improves sampling for the MCMC model fit by helping with complex varying effects (see p.453)
        transpars> matrix[actor,2]:alpha <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
        matrix[2,actor]:z_actor ~ normal( 0 , 1 ),
        # fixed priors
        g[tid] ~ normal(0,1),
        vector[2]:sigma_actor ~ dexp(1),
        cholesky_factor_corr[2]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
        # compute ordinary correlation matrixes from Cholesky factors
        gq> matrix[2,2]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
    ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))
      
        #run a CONTRAST to determine whether there is a difference between before (g1) and after (g2) (subtract g1 before from g2 after). Want probability scale because that is the scale we used in the simulation
post <- extract.samples(s1)
diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020
diff_p <- inv_logit(post$g[,2]) - inv_logit(post$g[,1]) #on the probability scale p.341 McElreath 2020
#precis( list( diff_g=diff_g , diff_p=diff_p ) )
    
    # enter all of the results into the data sheet
    simresults[counter,]$n <- N_ind
    simresults[counter,]$changeprop_mean <- currentchangeprop
    simresults[counter,]$changeprop_sd <- 0.1
    simresults[counter,]$repetition <- repetition
    simresults[counter,]$proportion_before <- currentprop_before
    simresults[counter,]$proportion_estimates_below_zero <- sum(diff_p<0)/length(diff_p)
    simresults[counter,]$count_estimates_below_zero <- sum(diff_p<0)
    counter<-counter+1 
    print(c(counter,"out of",10*length(samplesizes)*length(changepropvariants)*length(prop_before)))
        }
        }
        }
}

#write.csv(simresults,file="simresults20plus.csv") #put the outputs into a csv file so don't have to run the model again...it takes hours

## Prep a figure to show how sample size and mean change prop makes a difference in the results. Get posterior values for particular sample sizes and change props. The model kept stopping in the middle so I got the simulated data by running one sample size at a time. To do this, set samplesize (20 or 60), changepropvariants (0.1 or 0.025), and prop_before (always 0.6) to just one number and run the model. Generate the output, write to csv. Repeat for other model variants
#write.csv(diff_p,file="diff_p20changeprop0.1.csv")
#write.csv(diff_p,file="diff_p60changeprop0.1.csv")

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE)  #makes it so ulam always runs with cmdstan

# load the data sheet that was made from the simulation in
# the previous R chunk
d <- read.csv("https://raw.githubusercontent.com/ManyIndividuals/ManyIndividuals/main/Files/rrs/mi1simresultsgtgr.csv",
    header = TRUE, sep = ",", stringsAsFactors = FALSE)
# head(d)

# Set up the data sheet for the model
dat <- list(proportion_estimates_below_zero = as.numeric(d$proportion_estimates_below_zero),
    count_estimates_below_zero = as.numeric(d$count_estimates_below_zero),
    changeprop_mean = as.numeric(sqrt(d$changeprop_mean)), samplesize = as.integer(as.factor(d$n)))

# We run the STAN model - it is a binomial model that
# assumes that the probability to estimate a difference
# depends on an interaction between the sample size and the
# difference between the conditions (before/after). These
# two models do the same thing with different versions of
# the response variable. In m_samplesize_diffs, the
# response variable is already a proportion. In
# m_samplesize_diffscount, the response variable is a count
# and the model turns it into a proportion. We ended up
# using the latter m_samplesize_diffs <- ulam( alist(
# proportion_estimates_below_zero ~ dnorm(mu,sigma), mu <-
# a[samplesize] + b[samplesize]*changeprop_mean,
# a[samplesize] ~ dnorm(a_bar,sigma_a), a_bar~dnorm(0,1.5),
# sigma_a~dexp(1), sigma~dexp(1),
# b[samplesize]~dnorm(0,1.5) ) , data=dat , chains=4 ,
# cores=4 , log_lik=TRUE , cmdstan = TRUE, control =
# list(adapt_delta = .95, force_recompile = TRUE))

m_samplesize_diffscount <- ulam(alist(count_estimates_below_zero ~
    dbinom(2000, p), logit(p) <- a[samplesize] - b[samplesize] *
    changeprop_mean, a[samplesize] ~ dnorm(a_bar, sigma_a), a_bar ~
    dnorm(0, 1.5), sigma_a ~ dexp(1), b[samplesize] ~ dnorm(0,
    1.5)), data = dat, log_lik = TRUE, messages = FALSE, cmdstan = T)

# Extract the posterior estimates from this STAN model
precis_output <- precis(m_samplesize_diffscount, depth = 2)

# Plot the predictions for the different sample sizes as
# separate lines = COUNT out of 2000
plot(NULL, xlim = c(-0.01, 0.41), ylim = c(0, 0.6), pch = 1,
    ann = F, frame.plot = F, cex = 1.5)

mu_10 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = sqrt(unique(d$changeprop_mean)),
    samplesize = rep(1, length(unique(d$changeprop_mean)))))
mu_10_mean <- apply(mu_10, 2, mean)
mu_10_ci <- apply(mu_10, 2, PI, prob = 0.97)
shade(mu_10_ci, unique(d$changeprop_mean), col = col.alpha("blue",
    0.7))
lines(x = c(-0.008, -0.008), y = c(min(d[d$n %in% 10 & d$changeprop_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprop_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.022, 0.022), y = c(min(d[d$n %in% 10 & d$changeprop_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    10 & d$changeprop_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.044, 0.044), y = c(min(d[d$n %in% 10 & d$changeprop_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    10 & d$changeprop_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.092, 0.092), y = c(min(d[d$n %in% 10 & d$changeprop_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprop_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.192, 0.192), y = c(min(d[d$n %in% 10 & d$changeprop_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprop_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.292, 0.292), y = c(min(d[d$n %in% 10 & d$changeprop_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprop_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.392, 0.392), y = c(min(d[d$n %in% 10 & d$changeprop_mean %in%
    0.4, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprop_mean %in% 0.4, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))

mu_20 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = sqrt(unique(d$changeprop_mean)),
    samplesize = rep(2, length(unique(d$changeprop_mean)))))
mu_20_mean <- apply(mu_20, 2, mean)
mu_20_ci <- apply(mu_20, 2, PI, prob = 0.97)
shade(mu_20_ci, unique(d$changeprop_mean), col = col.alpha("orange",
    0.7))
lines(x = c(-0.004, -0.004), y = c(min(d[d$n %in% 20 & d$changeprop_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprop_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.024, 0.024), y = c(min(d[d$n %in% 20 & d$changeprop_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    20 & d$changeprop_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.048, 0.048), y = c(min(d[d$n %in% 20 & d$changeprop_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    20 & d$changeprop_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.096, 0.096), y = c(min(d[d$n %in% 20 & d$changeprop_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprop_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.196, 0.196), y = c(min(d[d$n %in% 20 & d$changeprop_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprop_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.296, 0.296), y = c(min(d[d$n %in% 20 & d$changeprop_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprop_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.396, 0.396), y = c(min(d[d$n %in% 20 & d$changeprop_mean %in%
    0.4, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprop_mean %in% 0.4, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))

mu_30 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = sqrt(unique(d$changeprop_mean)),
    samplesize = rep(3, length(unique(d$changeprop_mean)))))
mu_30_mean <- apply(mu_30, 2, mean)
mu_30_ci <- apply(mu_30, 2, PI, prob = 0.97)
shade(mu_30_ci, unique(d$changeprop_mean), col = col.alpha("skyblue",
    0.7))
lines(x = c(0, 0), y = c(min(d[d$n %in% 30 & d$changeprop_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprop_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.028, 0.028), y = c(min(d[d$n %in% 30 & d$changeprop_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    30 & d$changeprop_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.052, 0.052), y = c(min(d[d$n %in% 30 & d$changeprop_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    30 & d$changeprop_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.1, 0.1), y = c(min(d[d$n %in% 30 & d$changeprop_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprop_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.2, 0.2), y = c(min(d[d$n %in% 30 & d$changeprop_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprop_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.3, 0.3), y = c(min(d[d$n %in% 30 & d$changeprop_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprop_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.4, 0.4), y = c(min(d[d$n %in% 30 & d$changeprop_mean %in%
    0.4, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprop_mean %in% 0.4, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))

mu_40 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = sqrt(unique(d$changeprop_mean)),
    samplesize = rep(4, length(unique(d$changeprop_mean)))))
mu_40_mean <- apply(mu_40, 2, mean)
mu_40_ci <- apply(mu_40, 2, PI, prob = 0.97)
shade(mu_40_ci, unique(d$changeprop_mean), col = col.alpha("#213940",
    0.7))
lines(x = c(0.004, 0.004), y = c(min(d[d$n %in% 40 & d$changeprop_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprop_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.032, 0.032), y = c(min(d[d$n %in% 40 & d$changeprop_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    40 & d$changeprop_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.056, 0.056), y = c(min(d[d$n %in% 40 & d$changeprop_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    40 & d$changeprop_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.104, 0.104), y = c(min(d[d$n %in% 40 & d$changeprop_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprop_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.204, 0.204), y = c(min(d[d$n %in% 40 & d$changeprop_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprop_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.304, 0.304), y = c(min(d[d$n %in% 40 & d$changeprop_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprop_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.404, 0.404), y = c(min(d[d$n %in% 40 & d$changeprop_mean %in%
    0.4, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprop_mean %in% 0.4, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))

mu_60 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = sqrt(unique(d$changeprop_mean)),
    samplesize = rep(5, length(unique(d$changeprop_mean)))))
mu_60_mean <- apply(mu_60, 2, mean)
mu_60_ci <- apply(mu_60, 2, PI, prob = 0.97)
shade(mu_60_ci, unique(d$changeprop_mean), col = col.alpha("red",
    0.7))
lines(x = c(0.008, 0.008), y = c(min(d[d$n %in% 60 & d$changeprop_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprop_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.036, 0.036), y = c(min(d[d$n %in% 60 & d$changeprop_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    60 & d$changeprop_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.06, 0.06), y = c(min(d[d$n %in% 60 & d$changeprop_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    60 & d$changeprop_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.108, 0.108), y = c(min(d[d$n %in% 60 & d$changeprop_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprop_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.208, 0.208), y = c(min(d[d$n %in% 60 & d$changeprop_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprop_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.308, 0.308), y = c(min(d[d$n %in% 60 & d$changeprop_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprop_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.408, 0.408), y = c(min(d[d$n %in% 60 & d$changeprop_mean %in%
    0.4, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprop_mean %in% 0.4, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))

legend(x = "topright", legend = c(lty2 = "10", lty1 = "20", lty3 = "30",
    lty2 = "40", lty3 = "60"), lty = c(1, 1, 1, 1, 1), lwd = c(5,
    5, 5, 5, 5), col = c(col.alpha("blue", 0.4), col.alpha("orange",
    0.4), col.alpha("skyblue", 0.4), col.alpha("#213940", 0.4),
    col.alpha("red", 0.4)), cex = 1, bty = "n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size", side = 3, line = -0.7, at = 0.4)
mtext("Power to detect differences between conditions", side = 3,
    line = 1, at = 0.2, cex = 1.5)
mtext("Mean change in proportion of time spent in reward presence before vs. after",
    side = 1, line = 3, cex = 1.2)
mtext("Ratio of estimates below zero", side = 2, line = 3, cex = 1.2)

# 93% of the posterior mass is above zero for a 0.1 or
# larger mean change in proportion between before and after
# when the sample size is 20. 96% for n=30, 99% for n=40,
# and 99.7% for n=60 mu_10_ci #82%. 3rd column from the
# right (column 4) for the 0.1 category on the x axis
# (ordered by unique(d$changeprop_mean)) mu_20_ci #93%
# mu_30_ci #96% mu_40_ci #99% mu_60_ci #99.7% sum(d[d$n
# %in% 20 & d$changeprop_mean %in%
# 0,]$proportion_estimates_below_zero<0.3) #13 values are
# <0.3 (out of 30) length(d[d$n %in% 20 & d$changeprop_mean
# %in% 0,]$proportion_estimates_below_zero) #30 values
# total

### extra figure to show that an increase in sample size
### reduces the SD, but does not shift the curve to the
### right d20 <-d[d$n==20,] #sample size of 20, sd=0.1 d60
### <-d[d$n==60,] #sample size of 60, sd=0.1
### plot(density(d20$proportion_estimates_below_zero),lwd=2,col='blue',xlim=c(-0.1,1),ylim=c(0,40),ylab='',ylab(''),xlab(''))
### lines(density(d60$proportion_estimates_below_zero),lwd=2,col='red')
### abline(v=0)
### legend(x='topright',legend=c(lty2='20',lty1='60'),lty=c(1,1),lwd=c(5,5),col=c(col.alpha('blue',0.4),col.alpha('red',0.4)),cex=1,bty='n')

# Axis labels (side 1=x axis, side 2=y axis) mtext('Sample
# size',side=3,line=0,at=1) mtext('Mean change in
# proportion of time spent in the presence of a reward
# between before and after conditions',side=1,line=3,
# cex=1.2) mtext('Density of the proportion of estimates
# below zero',side=2,line=3, cex=1.2)

 

Figure 10. Risk of false positives and false negatives depending on sample and effect sizes. Curves of the mean ratio of estimates below zero (vertical lines show the minimum and maximum ratios), which illustrates the power we have to detect differences between conditions at different sample sizes. The curves show the model estimates as the effect increases (larger changes in the mean proportion time spent after versus before on the x-axis) for different potential sample sizes (10-60, illustrated by different colors). When there is no change (x value of zero), estimates suggest that, as expected, half of the estimates are below zero because the before and after conditions are not different from each other. As the change increases, the estimates decrease because models are able to reliably tell that the before and after conditions differ from each other. Across all models, the standard deviation of the mean change in proportion was 0.1 (a conservative estimate).

# (not shown in the article, but useful if you want to
# learn more about this modeling process)

# a figure that shows that an increase in sample size
# usually only decreases the sd and doesn't shift the curve
# to the right where there would be less of a chance of
# crossing zero and thus being less likely that the before
# and after are different from each other load the data
# sheet that was made from the simulation
d <- read.csv("https://raw.githubusercontent.com/ManyIndividuals/ManyIndividuals/main/Files/rrs/mi1_posterior_diff_p.csv",
    header = TRUE, sep = ",", stringsAsFactors = FALSE)
# head(d)

# figure to show that an increase in sample size reduces
# the SD, but does not shift the curve to the right
plot(density(d$diff_p20changeprop0.1), lwd = 2, col = "blue",
    xlim = c(-0.1, 0.3), ylim = c(0, 25), ylab = "", ylab(""),
    xlab(""))
lines(density(d$diff_p60changeprop0.1), lwd = 2, col = "red")
lines(density(d$diff_p20changeprop0.025), lwd = 2, col = "black")
lines(density(d$diff_p60changeprop0.025), lwd = 2, col = "purple")
abline(v = 0)
legend(x = "topright", legend = c(lty2 = "n=20, change=0.1",
    lty1 = "n=60, change=0.1", lty1 = "n=20, change=0.025", lty1 = "n=60, change=0.025"),
    lty = c(1, 1, 1, 1), lwd = c(5, 5, 5, 5), col = c(col.alpha("blue",
        0.4), col.alpha("red", 0.4), col.alpha("black", 0.4),
        col.alpha("purple", 0.4)), cex = 1, bty = "n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size and mean proportion change", side = 3, line = 0,
    at = 1)
mtext("Mean change in proportion of time spent in the presence of a reward between before and after conditions",
    side = 1, line = 3, cex = 1.2)
mtext("Density", side = 2, line = 3, cex = 1.2)

Run this model on the actual data

Run the code below to determine whether there were differences between the before and after conditions in their proportion of time spent at locations when food was present.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan

# load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

# sort the data
beforepresent <- d[d$RewardPresent==1 & d$BeforeOrAfterManipulation=="Before",]
beforeabsent <- d[d$RewardPresent==0 & d$BeforeOrAfterManipulation=="Before",]
beforetotal <- c(beforepresent,beforeabsent)
afterpresent <- d[d$RewardPresent==1 & d$BeforeOrAfterManipulation=="After",]
afterabsent <- d[d$RewardPresent==0 & d$BeforeOrAfterManipulation=="After",]
aftertotal <- c(afterpresent,afterabsent)

# Set up the data sheet
dat <- list(
    response = c(beforepresent$Duration,afterpresent$Duration),
    total = c(beforetotal$Duration,aftertotal$Duration),
    tid = c(rep(1,nrow(beforepresent)),rep(2,nrow(afterpresent))), #for all rows with before data, assign a treatment=1, and for all rows with after data, assign a treatment=2
    actor = rep(unique(d$ID),2) #now each actor has 2 data points: one for before and one for after so multiply by 2
)

#simulated data to test the model
simulatedgtgrproportiondata<-matrix(nrow=40,ncol=4) #n=20*2 conditions
simulatedgtgrproportiondata<-as.data.frame(simulatedgtgrproportiondata)
colnames(simulatedgtgrproportiondata)<-c("id","condition","totalduration","presentduration")
simulatedgtgrproportiondata$id<-rep(c(1:20),2) #n=20 individuals in 2 conditions so need to repeat the ID twice
simulatedgtgrproportiondata$condition<-c(rep(1,20),rep(2,20)) #condition: first 20 rows = condition 1, second 20 rows = condition 2
simulatedgtgrproportiondata$totalduration<-rnorm(40,mean=3600,sd=600) #assume mean total duration = 3600 seconds for n=20*2 conditions
for(durationbefore in 1:20) {
  initialproportion<-runif(1,min=0.5,max=0.7) #proportion of time spent in the presence of a reward before the manipulation is between 0.5 and 0.7
simulatedgtgrproportiondata[durationbefore,]$presentduration<-simulatedgtgrproportiondata[durationbefore,]$totalduration*initialproportion #convert the proportion to the number of seconds by multiplying it by total duration before
simulatedgtgrproportiondata[durationbefore+20,]$presentduration<-simulatedgtgrproportiondata[durationbefore+20,]$totalduration*initialproportion*runif(1,1.1,1.3) #increase proportion by between 1.1 and 1.3 to get the proportion after the manipulation and convert it to seconds by multiplying it by duration after
}

dat2 <- list(
    response = round(simulatedgtgrproportiondata$presentduration,0),
    total = round(simulatedgtgrproportiondata$totalduration,0),
    tid = simulatedgtgrproportiondata$condition,
    actor = simulatedgtgrproportiondata$id 
)

# THE MODEL (p.441 for normal distribution, 448 for binomial distribution - McElreath 2020)
m1 <- ulam(
    alist(
        response ~ dbinom(total,p),
        logit(p) <- g[tid] + alpha[actor,tid],
        # adaptive priors - non-centered which is a parameterization that improves sampling for the MCMC model fit by helping with complex varying effects (see p.453)
        transpars> matrix[actor,2]:alpha <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
        matrix[2,actor]:z_actor ~ normal( 0 , 1 ),
        # fixed priors
        g[tid] ~ normal(0,1),
        vector[2]:sigma_actor ~ dexp(1),
        cholesky_factor_corr[2]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
        # compute ordinary correlation matrixes from Cholesky factors
        gq> matrix[2,2]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
    ) , data=dat2 , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE)

#model output: pay attention to before (g1) and after (g2)
#precis(m1,depth=2)

#run a CONTRAST to determine whether there is a difference between before (g1) and after (g2)
post <- extract.samples(m1)
diff_g <- post$g[,1] - post$g[,2] #log-odds difference p.341
diff_p <- inv_logit(post$g[,1]) - inv_logit(post$g[,2]) #on the probability scale p.341
precis( list( diff_g=diff_g , diff_p=diff_p ) )

G.Q2 More flexible = use more microhabitats?

The model

Bayesian model with a normal distribution:

habitatuse ~ \(\alpha\)[ind] + \(\beta\)[ind]*before

habitatuse is the response variable: the total number of different microhabitats used per individual. There will be one intercept, \(\alpha\), and one slope \(\beta\) per individual, which will be estimated for the two conditions, before (and after) the manipulation. ID is nested within condition as a random effect because there is more than one data point per individual: each individual has a data point in the before condition and in the after condition. A normal distribution was used because the response variable is a sum without an expected skew to the curve (see Figure 10.6 in McElreath, 2020b). The Bayesian model was developed using McElreath (2020b) as a guide.

Power analysis

See the jay Q3 microhabitat model for the power analysis.

Run this model on the actual data

Run the code below to determine whether there were differences between the before and after conditions in the proportion of microhabitats used. Only count that a microhabitat was used if the individual had at least 5% of their data points there. This prevents a microhabitat from being counted even if an individual was simply moving through it, and therefore not necessarily using it.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
                                                                                                                                  ## when the data do not yet exist, use simulated data to test the model
N_ind <- 20

dat <- list(
    habitatuse = rbinom(N_ind*2,size=10,prob=5/10), #20 individuals * 2 observations/individual, 10 max habitats; on average, individuals will use 5 of the 10 habitats more than 5% of the time
    tid = c(rep(1,N_ind),rep(2,N_ind)), #1 refers to before and 2 to after
    actor = rep(1:N_ind,2) #go through all individuals twice because there are observations from before and after for each
)


## When you have the actual data, load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

# add 4 microhabitats for all individuals in the natural habitat because suburban has 4 more microhabitats and this will equalize natural and suburban in the model
d[d$Habitat=="Natural",]$habitats <- d[d$Habitat=="Natural",]$habitats+3

# Set up the data sheet
dat <- list(
    habitatuse = d$habitats,
    before = d$BeforeOrAfterManipulation,
    ind = d$ID
)

# THE MODEL (p.447 for binomial distribution with advanced varying slopes - McElreath 2020). 
mj <- ulam(
    alist(
        habitatuse ~ dbinom(10,p), #max 10 habitats
        logit(p) <- g[tid] + alpha[actor,tid],
        # adaptive priors - non-centered
        transpars> matrix[actor,2]:alpha <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
        matrix[2,actor]:z_actor ~ normal( 0 , 1 ),
        # fixed priors
        g[tid] ~ normal(0,1),
        vector[2]:sigma_actor ~ dexp(1),
        cholesky_factor_corr[2]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
        # compute ordinary correlation matrixes from Cholesky factors
        gq> matrix[2,2]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
    ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))

precis(mj,depth=2) #pay attention to g (g1 is before, g2 is after)
boxplot(dat$habitatuse~dat$tid) #see if there is a difference visually

 # run a contrast to determine whether there is a difference between the before and after conditions (there is a difference if CI is on one side of zero)
post <- extract.samples(mj)
diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020
diff_p <- inv_logit(post$g[,2]) - inv_logit(post$g[,1]) #on the probability scale p.341 McElreath 2020
#precis( list( diff_g=diff_g , diff_p=diff_p ) )

G.Q3 More flexible = more food types?

The model

Bayesian model with a normal distribution:

y ~ \(\alpha\)[ind] + \(\beta\)[ind]*before

y is the response variable: the total number of different food types taken per individual. There will be one intercept, \(\alpha\), and one slope \(\beta\) per individual, which will be estimated for the two conditions, before (and after) the manipulation. ID is nested within condition as a random effect because there is more than one data point per individual: each individual has a data point in the before condition and in the after condition. A normal distribution was used because the response variable is a sum without an expected skew to the curve (see Figure 10.6 in McElreath, 2020b). The Bayesian model was developed using McElreath (2020b) as a guide.

Power analysis

See the jay Q4 foraging model for the power analysis.

Run this model on the actual data

Run the code below to determine whether there were differences between the before and after conditions in the number of food types taken.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
                                                                                                                                  ## when the data do not yet exist, use simulated data to test the model
N_ind <- 20

dat <- list(
    foods = standardize(rnorm(40,mean=6.5,sd=2)), #estimate 20 total individuals (*2 conditions) with a mean of 6.5 foods/individual (halfway between the max 13 possible food types)
    before = c(rep(1,N_ind),rep(2,N_ind)), #1 refers to before and 2 to after
    ind = rep(1:N_ind,2) #go through all individuals twice because there are observations from before and after for each
)

## load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

# Set up the data sheet
dat <- list(
    habitatuse = d$habitats,
    before = d$BeforeOrAfterManipulation,
    ind = d$ID
)

# THE MODEL (p.441 for normal distribution - McElreath 2020). Note that it is MULTIPLIED by condition (before), which means that you don't need to run a contrast because each individual value already accounts for the difference. So only look at beta to see if it is above zero or not for each individual or summed across individuals
mjf <- ulam(
    alist(
        foods ~ normal( mu , sigma ),
        mu <- a_ind[ind] + b_ind[ind]*before,
        c(a_ind,b_ind)[ind] ~ multi_normal( c(alpha,beta) , Rho , sigma_ind ),
        alpha ~ normal(6.5,2), #6.5=mean number of foods taken, 2=food types for the sd
        beta ~ normal(0,6), #0.5=mean slope, 6=sd which is like the max difference between before and after
        sigma_ind ~ exponential(1),
        sigma ~ exponential(1),
        Rho ~ lkj_corr(2)
    ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE)

#precis(mjf,depth=2) #model output: if the beta confidence interval are on one side of zero, then there was a difference between the before and after conditions. If the beta confidence interval crosses zero, then there was no difference between before and after.
#boxplot(dat$foods~dat$before) #see if there is a difference visually

Jays

J.Q1 Do jay populations in human modified areas differ in baseline behavioral flexibility compared to populations in natural areas?

The model

We used the reversal learning Bayesian model in Logan CJ et al. (2020) to simulate and analyze population differences in reversal learning, and calculate our ability to detect differences between populations. The model “accounts for every choice made in the reversal learning experiment and updates the probability of choosing either option after the choice was made depending on whether that choice contained a food reward or not. It does this by updating three main components for each choice: an attraction score, a learning rate (\(\phi\)), and a rate of deviating from learned attractions (\(\lambda\))” (Logan CJ et al., 2020).

Equation 1 (attraction and \(\phi\)): \(A_{i,j, t+1} = (1-\phi_j) A_{i,j,t} + \phi_j \pi_{i,j,t}\)

Equation 1 “tells us how attractions to different behavioral options \(A_{i,j, t+1}\) (i.e., how preferable option \(i\) is to the bird \(j\) at time \(t+1\)) change over time as a function of previous attractions \(A_{i ,j, t}\) and recently experienced payoffs \(\pi_{i,j,t}\) (i.e., whether they received a reward in a given trial or not). Attraction scores thus reflect the accumulated learning history up to this point. The (bird-specific) parameter \(\phi_j\) describes the weight of recent experience. The higher the value of \(\phi_j\), the faster the bird updates their attraction. It thus can be interpreted as the learning or updating rate of an individual. A value of \(\phi_j = 0.04\), for example, means that receiving a single reward for one of the two options will shift preferences by 0.02 from initial 0.5-0.5 attractions, a value of \(\phi_j = 0.06\) will shift preferences by 0.03 and so on” (Blaisdell et al., 2021).

Equation 2 (\(\lambda\)): \(P(i)_{t+1} = \frac{\exp(\lambda_j A_{i, j, t})}{\sum\limits_{m=1}^{2}\exp(\lambda_j A_{m, j, t})}.\)

Equation 2 “expresses the probability an individual \(j\) chooses option \(i\) in the next round, \(t+1\), based on the latent attractions. The parameter \(\lambda_j\) represents the rate of deviating from learned attractions of an individual (also called inverse temperature). It controls how sensitive choices are to differences in attraction scores. As \(\lambda_j\) gets larger, choices become more deterministic, as it gets smaller, choices become more exploratory (random choice if \(\lambda_j = 0\)). For instance, if an individual has a 0.6-0.4 preference for option A, a value of \(\lambda_j = 3\) means they choose A 65% of the time, a value of \(\lambda_j = 10\) means they choose A 88% of the time and a value of \(\lambda_j = 0.5\) means they choose A only 53% of the time” (Blaisdell et al., 2021).

We used the \(\phi_j\) and \(\lambda_j\) values as the response variable in the Bayesian model to examine whether there were differences in flexibility between the habitats:

y ~ \(\alpha\)[habitat]

y is the response variable (\(\phi_j\) and \(\lambda_j\), which are extracted from the correct and incorrect choices in the serial reversals). There is one intercept, \(\alpha\), per habitat (suburban or natural) and we will estimate the habitat’s average and standard deviation of the response variable.

Power analysis

Simulations using bespoke Bayesian models in Logan CJ et al. (2020) (the same model structure we use here) showed a high likelihood of detecting differences with a minimum sample size of 15 when mean differences in phi were at least 0.01 and mean differences in lambda at least 3.

Run this model on the actual data

Run the code below to determine whether there were differences between the two habitats in their phi and lambda flexibility measures.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE)  #makes it so ulam always runs with cmdstan

# load data sheet
dflex <- read.
v(url(""), header = T, sep = ",", stringsAsFactors = F)

# sort the data: only want data from the initial
# discrimination and first reversal to use as the baseline
# flexibility so we can compare between habitats
reduceddata <- matrix(ncol = ncol(dflex), nrow = 0)
reduceddata <- data.frame(reduceddata)
for (i in 1:length(unique(dflex$ID))) {
    thisbird <- unique(dflex$ID)[i]
    thisbirddata <- dflex[dflex$ID == thisbird, ]
    thisbirdslastreversal <- thisbirddata[thisbirddata$Reversal %in%
        c(0, 1), ]
    reduceddata <- rbind(reduceddata, thisbirdslastreversal)
}
dflex_beginning <- reduceddata


# Now we run a STAN model with all simulated habitats,
# estimating both means of phi and lambda for each habitat
# as well as each individuals' phi and lambda

# We set up the data for the model
dat_full <- as.list(simulatedreversaldata_attractionscores)
dat_full$N <- nrow(simulatedreversaldata_attractionscores)
dat_full$N_id <- length(unique(simulatedreversaldata_attractionscores$id))
dat_full$N_exp <- length(unique(simulatedreversaldata_attractionscores$Habitat))
dat_full$Choice <- as.numeric(as.factor(dat_full$Choice))

# This STAN model, in addition to estimating phi and lambda
# for each individual, also estimates means for each
# habitat It again starts with attractions set to 0.1 and
# assumes that individuals only learn about the option they
# chose.

reinforcement_model_id_site_nonzeroattraction <- "
data{
   int N;
   int N_id;
   int N_exp;
   int id[N];
   int Habitat[N];
   int Trial[N];
   int Choice[N];
   int Correct[N];
}

parameters{
  real logit_phi;
  real log_L;

  // Varying effects clustered on individual
  matrix[2,N_id] z_ID;
  vector<lower=0>[2] sigma_ID;       //SD of parameters among individuals
  cholesky_factor_corr[2] Rho_ID;

  // Varying effects clustered on experimenter
  matrix[2,N_exp] z_EXP;
  vector<lower=0>[2] sigma_EXP;
  cholesky_factor_corr[2] Rho_EXP;
}

transformed parameters{
matrix[N_id,2] v_ID; // varying effects on individuals
matrix[N_exp,2] v_EXP; // varying effects on experimenter

v_ID = ( diag_pre_multiply( sigma_ID , Rho_ID ) * z_ID )';
v_EXP = ( diag_pre_multiply( sigma_EXP , Rho_EXP ) * z_EXP )';
}

model{
matrix[N_id,2] A; // attraction matrix

logit_phi ~  normal(0,1);
log_L ~  normal(0,1);

// varying effects
to_vector(z_ID) ~ normal(0,1);
sigma_ID ~ exponential(1);
Rho_ID ~ lkj_corr_cholesky(4);

to_vector(z_EXP) ~ normal(0,1);
sigma_EXP ~ exponential(1);
Rho_EXP ~ lkj_corr_cholesky(4);

// initialize attraction scores
for ( i in 1:N_id ) {
A[i,1] = 0.1; A[i,2] = 0.1';
}

// loop over Choices
for ( i in 1:N ) {
vector[2] pay;
vector[2] p;
real L;
real phi;

// first, what is log-prob of observed choice
L =  exp(log_L + v_ID[id[i],1]+ v_EXP[Habitat[i],1]);
p = softmax(L*A[id[i],1:2]' );
Choice[i] ~ categorical( p );

// second, update attractions conditional on observed choice
phi =  inv_logit(logit_phi + v_ID[id[i],2]+ v_EXP[Habitat[i],2]);
pay[1:2] = rep_vector(0,2);
pay[ Choice[i] ] = Correct[i];
A[ id[i] , Choice[i] ] = ( (1-phi)*(A[ id[i] , Choice[i] ]) + phi*pay[Choice[i]])';
}//i
}
"

# Now we run the STAN model on the full data. This will
# take a few hours.
m_simulated_full <- stan(model_code = reinforcement_model_id_site_nonzeroattraction,
    data = dat_full, iter = 5000, cores = 4, chains = 4, control = list(adapt_delta = 0.9,
        max_treedepth = 12))

# We extract the posterior from the STAN model
s <- extract.samples(m_simulated_full)

# We calculate the mean phi and lambda for each habitat as
# estimated by the STAN model
lambda <- sapply(1:dat_full$N_exp, function(x) exp(mean(s$log_L) +
    mean(s$v_EXP[, x, 1])))
phi <- sapply(1:dat_full$N_exp, function(x) inv_logit(mean(s$logit_phi) +
    mean(s$v_EXP[, x, 2])))


# We want to know whether the STAN model estimated two
# habitats to be different or not. As criterion for this,
# we check whether the samples in the posterior for pairs
# of habitat overlap or not. That means that, when
# calculating the difference between the samples in the
# posterior for a pair of habitat, they are assumed to be
# different if the differences are all on one side of zero.

# We first check our power to estimate difference in mean
# phi between sites We calculate the pairwise differences
# between sites for the estimated phi means
pairwisedifferences_phiposteriors <- list(sapply(1:(dat_full$N_exp *
    (dat_full$N_exp - 1)/2), function(x) (s$v_EXP[, combn(1:dat_full$N_exp,
    m = 2)[1, x], 2] - (s$v_EXP[, combn(1:dat_full$N_exp, m = 2)[2,
    x], 2]))))

# We convert it into a dataframe for easier manipulation
estimates_pairwisedifferences_phiposteriors <- as.data.frame(precis(pairwisedifferences_phiposteriors,
    depth = 2))

# We add a column that classifies as true or false whether
# the estimates crosses zero (as opposed to all values
# being either smaller or larger than zero)
estimates_pairwisedifferences_phiposteriors$crosseszero <- ifelse(estimates_pairwisedifferences_phiposteriors$mean >
    0, estimates_pairwisedifferences_phiposteriors$`5.5%` < 0,
    estimates_pairwisedifferences_phiposteriors$`94.5%` > 0)

J.Q2 Are disturbance-resiliant jays more flexible than disturbance-resistant jays?

The model

Same as in J.Q1 above.

Power analysis

Same as in J.Q1 above.

Run this model on the actual data

Run the code below to determine whether there were differences between the species in their phi and lambda flexibility measures.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE)  #makes it so ulam always runs with cmdstan

# load data sheet
dflex <- read.csv(url(""), header = T, sep = ",", stringsAsFactors = F)

# sort the data: only want data from the initial
# discrimination and first reversal to use as the baseline
# flexibility so we can compare between species
reduceddata <- matrix(ncol = ncol(dflex), nrow = 0)
reduceddata <- data.frame(reduceddata)
for (i in 1:length(unique(dflex$ID))) {
    thisbird <- unique(dflex$ID)[i]
    thisbirddata <- dflex[dflex$ID == thisbird, ]
    thisbirdslastreversal <- thisbirddata[thisbirddata$Reversal %in%
        c(0, 1), ]
    reduceddata <- rbind(reduceddata, thisbirdslastreversal)
}
dflex_beginning <- reduceddata


# Now we run a STAN model with all simulated species,
# estimating both means of phi and lambda for each species
# as well as each individuals' phi and lambda

# We set up the data for the model
dat_full <- as.list(simulatedreversaldata_attractionscores)
dat_full$N <- nrow(simulatedreversaldata_attractionscores)
dat_full$N_id <- length(unique(simulatedreversaldata_attractionscores$id))
dat_full$N_exp <- length(unique(simulatedreversaldata_attractionscores$Habitat))
dat_full$Choice <- as.numeric(as.factor(dat_full$Choice))

# This STAN model, in addition to estimating phi and lambda
# for each individual, also estimates means for each
# habitat It again starts with attractions set to 0.1 and
# assumes that individuals only learn about the option they
# chose.

reinforcement_model_id_site_nonzeroattraction <- "
data{
   int N;
   int N_id;
   int N_exp;
   int id[N];
   int Species[N];
   int Trial[N];
   int Choice[N];
   int Correct[N];
}

parameters{
  real logit_phi;
  real log_L;

  // Varying effects clustered on individual
  matrix[2,N_id] z_ID;
  vector<lower=0>[2] sigma_ID;       //SD of parameters among individuals
  cholesky_factor_corr[2] Rho_ID;

  // Varying effects clustered on experimenter
  matrix[2,N_exp] z_EXP;
  vector<lower=0>[2] sigma_EXP;
  cholesky_factor_corr[2] Rho_EXP;
}

transformed parameters{
matrix[N_id,2] v_ID; // varying effects on individuals
matrix[N_exp,2] v_EXP; // varying effects on experimenter

v_ID = ( diag_pre_multiply( sigma_ID , Rho_ID ) * z_ID )';
v_EXP = ( diag_pre_multiply( sigma_EXP , Rho_EXP ) * z_EXP )';
}

model{
matrix[N_id,2] A; // attraction matrix

logit_phi ~  normal(0,1);
log_L ~  normal(0,1);

// varying effects
to_vector(z_ID) ~ normal(0,1);
sigma_ID ~ exponential(1);
Rho_ID ~ lkj_corr_cholesky(4);

to_vector(z_EXP) ~ normal(0,1);
sigma_EXP ~ exponential(1);
Rho_EXP ~ lkj_corr_cholesky(4);

// initialize attraction scores
for ( i in 1:N_id ) {
A[i,1] = 0.1; A[i,2] = 0.1';
}

// loop over Choices
for ( i in 1:N ) {
vector[2] pay;
vector[2] p;
real L;
real phi;

// first, what is log-prob of observed choice
L =  exp(log_L + v_ID[id[i],1]+ v_EXP[Species[i],1]);
p = softmax(L*A[id[i],1:2]' );
Choice[i] ~ categorical( p );

// second, update attractions conditional on observed choice
phi =  inv_logit(logit_phi + v_ID[id[i],2]+ v_EXP[Species[i],2]);
pay[1:2] = rep_vector(0,2);
pay[ Choice[i] ] = Correct[i];
A[ id[i] , Choice[i] ] = ( (1-phi)*(A[ id[i] , Choice[i] ]) + phi*pay[Choice[i]])';
}//i
}
"

# Now we run the STAN model on the full data. This will
# take a few hours.
m_simulated_full <- stan(model_code = reinforcement_model_id_site_nonzeroattraction,
    data = dat_full, iter = 5000, cores = 4, chains = 4, control = list(adapt_delta = 0.9,
        max_treedepth = 12))

# We extract the posterior from the STAN model
s <- extract.samples(m_simulated_full)

# We calculate the mean phi and lambda for each Species as
# estimated by the STAN model
lambda <- sapply(1:dat_full$N_exp, function(x) exp(mean(s$log_L) +
    mean(s$v_EXP[, x, 1])))
phi <- sapply(1:dat_full$N_exp, function(x) inv_logit(mean(s$logit_phi) +
    mean(s$v_EXP[, x, 2])))


# We want to know whether the STAN model estimated two
# habitats to be different or not. As criterion for this,
# we check whether the samples in the posterior for pairs
# of Species overlap or not. That means that, when
# calculating the difference between the samples in the
# posterior for a pair of Species, they are assumed to be
# different if the differences are all on one side of zero.

# We first check our power to estimate difference in mean
# phi between sites We calculate the pairwise differences
# between sites for the estimated phi means
pairwisedifferences_phiposteriors <- list(sapply(1:(dat_full$N_exp *
    (dat_full$N_exp - 1)/2), function(x) (s$v_EXP[, combn(1:dat_full$N_exp,
    m = 2)[1, x], 2] - (s$v_EXP[, combn(1:dat_full$N_exp, m = 2)[2,
    x], 2]))))

# We convert it into a dataframe for easier manipulation
estimates_pairwisedifferences_phiposteriors <- as.data.frame(precis(pairwisedifferences_phiposteriors,
    depth = 2))

# We add a column that classifies as true or false whether
# the estimates crosses zero (as opposed to all values
# being either smaller or larger than zero)
estimates_pairwisedifferences_phiposteriors$crosseszero <- ifelse(estimates_pairwisedifferences_phiposteriors$mean >
    0, estimates_pairwisedifferences_phiposteriors$`5.5%` < 0,
    estimates_pairwisedifferences_phiposteriors$`94.5%` > 0)

J.Q3 More flexible = use more microhabitats?

The model

Bayesian model with a normal distribution:

habitatuse ~ \(\alpha\)[ind] + \(\beta\)[ind]*before

habitatuse is the response variable: the total number of different microhabitats used per individual. There will be one intercept, \(\alpha\), and one slope \(\beta\) per individual, which will be estimated for the two conditions, before (and after) the manipulation. ID is nested within condition as a random effect because there is more than one data point per individual: each individual has a data point in the before condition and in the after condition. A normal distribution was used because the response variable is a sum without an expected skew to the curve (see Figure 10.6 in McElreath, 2020b). The Bayesian model was developed using McElreath (2020b) as a guide.

Power analysis

We estimated our power to detect differences between conditions at different sample sizes and with different mean changes in the proportion of different microhabitats used per individual in the before vs. after conditions (Figure 11). We simulated the proportion of habitats used for different sample sizes of individuals before and after the flexibility manipulation. We analyzed these simulated data with the model we will use to analyze the actual data, estimating the change in the proportion of habitats used between the before and after conditions. From the posterior estimates of the model, we extracted both the mean change as well as the ratio of the posterior estimates that were below zero.

If the mean ratio of estimates below zero is close to 0.5, the model assumes that the change in the proportion of habitats used before the flexibility manipulation is similar to after. If the ratio is close to zero, the model assumes individuals have changed their behavior. For changes in the proportion of habitats used smaller than 0.15 (standard deviation=0.2) or 0.1 (SD=0.1), models are likely to assume that no changes occurred even with large sample sizes. If the change in the proportion of habitats used before the flexibility manipulation vs. after is 0.15 with a standard deviation of 0.2, on average 94% of the posterior of the model based on a sample size of 20 individuals will be larger than zero (93% with a standard deviation of 0.1). This means that the model is quite certain there is a difference that is larger than zero. In addition, only four of the 30 models for a sample size of 20 at the mean change of 0.15 have a ratio larger than 0.3, meaning that the risk of having a false negative is not very likely.

In general, with sample sizes at or above 20 and mean changes in the proportion of habitats used at 0.1 or larger, it is highly likely that the model will indicate that individuals have changed their behavior. Mean changes below 0.1 can still be detected, however there is a higher risk that there will be a false negative and this risk is independent of sample size.

With small mean changes in the response variable, some individuals might not increase or even decrease their response after the manipulation because there is variation around the mean change in individual responses. With small sample sizes, there is a risk that only individuals who did not clearly increase their response will be studied, whereas larger sample sizes are more likely to include a wider spectrum of individuals.

To estimate the risk of detecting false positives, we set the mean change in the proportion of habitats used to zero so there was no change between the before and after conditions. As expected, the average ratio of estimates below zero is close to but below 0.5 and independent of sample size. The estimates went generally below 0.5 because the maximum number of habitats used was set to 10 and we had a condition where individuals before the manipulation used a mean of 7 habitats. Accordingly, if individuals randomly either increase or decrease their number of habitats used, decreases will be more severe because individuals can only increase by 3 habitats, but potentially decrease by 6 habitats. With a sample size of 20, 27% have a ratio smaller than 0.3, meaning that the risk of having a false positive is high. The risk would be lower if the variation among individuals was lower than what we assumed (the standard deviation of the mean change in number of habitats was 0.2, which is a conservative estimate).

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
rstan_options(auto_write = TRUE)

### SIMULATE the population

#exploration of the number of habitats jays might use (have to have at least 5% of the points in a given habitat for that habitat to count). Binomial distribution restricts it to the known maximum of 10 microhabitats in our study and makes it a positive number. Can likely distinguish between differences of 15% or more
plot(density(rbinom(10000,size=10,prob=5/10))) #10 max habitats; on average, individuals will use 5 of the 10 habitats more than 5% of the time
lines(density(rbinom(10000,size=10,prob=3.5/10)),col="red") #on average, individuals will use 3.5 of the 10 habitats more than 5% of the time
lines(density(rbinom(10000,size=10,prob=7/10)),col="blue") #on average, individuals will use 7 of the 10 habitats more than 5% of the time

#choose the mean proportion of habitats used before the manipulation
number_before<-5/10 #number of habitats used per bird in the before condition. 5 = 10/2, the halfway point of the max 10 habitats in the suburban habitat
changenumber<-1.5 #the difference in the number habitats used between the before and after conditions
N_ind <- 20 #number of individuals tested

numberbefore <- rbinom(N_ind,size=10,prob=number_before) #proportion of habitats used before
numberafter <- as.vector(rep(0,N_ind) )
for (i in 1:N_ind) {
  numberafter[i] <- round( numberbefore[i] + rnorm(n=1, mean=(10-numberbefore[i])*(changenumber/numberbefore[i]),sd=1),0 )
  if(numberafter[i]>10){numberafter[i] <- 10}
}

#change the number of habitats increased (e.g., 3.5) in changenumber in numberafter to determine which changenumber to run the simulations at. Visualize using the plot below to see when the distributions have more separation (and see whether it might be possible to tell the difference between before (black) and after (red))
plot(density(numberbefore),lwd=2,xlim=c(0,18),ylim=c(0,0.3))
lines(density(numberafter),lwd=2,col="red") #the after condition. Should be able to tell the difference when changenumber is at least 1.5 above number_before (and number_before is 5/10)

#check the plot to see whether individuals are increasing their numbers = most are above the 1:1 line. Those that were smaller on the x axis are farther from the line, which is what we wanted
plot(numberbefore,numberafter,xlim=c(0,10),ylim=c(0,10))
abline(0,1)


### POWER ANALYSIS: different sample sizes and different mean changes in proportions between before and after conditions
# We now set our range of likely sample sizes, from a minimum of 10 individuals per population to a maximum of 60 individuals per population. We draw repeated samples of individuals of that size from each of the 5 populations, during each repetition we draw two samples from each population (so we can infer the false negative rate of wrongly estimating that two samples come from a different population even though they were taken from the sample population). This means we have 10 repetitions for each of the 5 sample sizes and each of the 6 changemeans. So 10*6 changemeans. (=60) for each of the 5 sample sizes, resulting in a total length of samplesizes of 300
samplesizes<-c(10,20,30,40,60) #c(10,20,30,40,60)

# Set the means you want to compare
meanchangenumber1<-3/10 #Likely going to use 3 more out of the 10 habitats
meanchangenumber2<-2.5/10 #Likely going to use 2.5 more out of the 10 habitats
meanchangenumber3<-2/10 
meanchangenumber4<-1.5/10
meanchangenumber5<-1/10
meanchangenumber6<-0/10
changenumbervariants<-c(meanchangenumber1,meanchangenumber2,meanchangenumber3,meanchangenumber4,meanchangenumber5,meanchangenumber6) #for false negatives

#compare different baseline numbers for how many habitats these individuals are likely to use before the manipulation
average_number_before<-c(3/10, 5/10, 7/10)

#set up data frame where results will be stored
simresults <- matrix(NA,nrow=900,ncol=7)
simresults <- data.frame(simresults)
colnames(simresults) <- c("n","changenumber_mean","repetition","changeprop_sd","number_before","proportion_estimates_below_zero","count_estimates_below_zero")
counter<-1

# run simulations. There are 5 different sample sizes, and for each we have 6 different mean change in proportions that we want to examine, and each will repeat 10 times for a total of 300 samples.
for (samplesize in 1:length(samplesizes)) {
        N_ind <- samplesizes[samplesize] #number of grackles per sample size
        
        #now run through each of the 6 number variations
        for (numbervariant in 1:length(changenumbervariants)) {
        
        #pick the current proportion variant from the list changepropvariants
        currentchangenumber <- changenumbervariants[numbervariant]
        
        for (baselinenumber in 1:3) {
        #pick the current proportion variant from the list changepropvariants
        average_currentnumber_before <- average_number_before[baselinenumber]
        
        for (repetition in 1:10) {
       
        #BEFORE & AFTER manipulation
        individual_probs_before<-(rnorm(N_ind,mean=average_currentnumber_before,sd=0.05))  # Based on one of the three average baseline numbers before, each individual will be assigned a value close to this with some variation
        individual_probs_before[individual_probs_before<0]<-0
        individual_probs_before[individual_probs_before>1]<-1
        
        simulatedindividuals_number_before<-as.vector(rep(0,N_ind))
        individual_probs_after<-as.vector(rep(0,N_ind))
        simulatedindividuals_number_after<-as.vector(rep(0,N_ind))
        
        for (i in 1:N_ind) {
          simulatedindividuals_number_before[i] <- rbinom(1 ,size=10,prob=individual_probs_before[i]) # Based on their individual values for the proportion of habitats before, calculate proportion of habitats each individual used before
          simulatedindividuals_number_before[simulatedindividuals_number_before==0]<-1 # Individuals have to live in at least one habitat
          individual_probs_after[i]<-individual_probs_before[i]+rnorm(1,mean=currentchangenumber,sd=0.2) # We increase the individual baseline values by one of the six change numbers, with variation so that some individuals might also decrease their value from their baseline before.  #change this SD here and below in the simresults ouput to test different SDs
          individual_probs_after[individual_probs_after<0]<-0
          individual_probs_after[individual_probs_after>1]<-1
          simulatedindividuals_number_after[i] <- rbinom(1 ,size=10,prob=individual_probs_after[i]) # Based on their modified individual values for the proportion of habitats after, calculate how many habitats each individual used after
          simulatedindividuals_number_after[simulatedindividuals_number_after==0]<-1 # Individuals have to live in at least one habitat
          }

        ### run a STAN model to see whether the before and after proportions are estimated to be different from each other
        #sets up the data sheet with the simulated data that the model will run on
      dat <- list(
    habitatuse = c(simulatedindividuals_number_before,simulatedindividuals_number_after), #compare before and after
    tid = c(rep(1,N_ind),rep(2,N_ind)), #1 refers to before and 2 to after
    actor = rep(1:N_ind,2) #go through all individuals twice because there are observations from before and after for each
)

        # Our MODEL
sj <- ulam(
    alist(
        habitatuse ~ dbinom(10,p),
        logit(p) <- g[tid] + alpha[actor,tid],
        # adaptive priors - non-centered
        transpars> matrix[actor,2]:alpha <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
        matrix[2,actor]:z_actor ~ normal( 0 , 1 ),
        # fixed priors
        g[tid] ~ normal(0,1),
        vector[2]:sigma_actor ~ dexp(1),
        cholesky_factor_corr[2]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
        # compute ordinary correlation matrixes from Cholesky factors
        gq> matrix[2,2]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
    ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))
#precis(sj, depth=2) #check the output to make sure it looks as expected

#run contrasts
post <- extract.samples(sj)
diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020
diff_p <- inv_logit(post$g[,2]) - inv_logit(post$g[,1]) #on the probability scale p.341 McElreath 2020
#precis( list( diff_g=diff_g , diff_p=diff_p ) )

  # enter all of the results into the data sheet
    simresults[counter,]$n <- N_ind
    simresults[counter,]$changenumber_mean <- currentchangenumber #this will show the avg change in proportion of habitats
    simresults[counter,]$repetition <- repetition
    simresults[counter,]$changeprop_sd <- 0.2 #change this number and in individual_probs_after to test different SDs
    simresults[counter,]$number_before <- average_currentnumber_before
    simresults[counter,]$proportion_estimates_below_zero <- sum(diff_p<0)/length(diff_p)
    simresults[counter,]$count_estimates_below_zero <- sum(diff_p<0)
    counter<-counter+1 
    print(c(counter,"out of",10*length(samplesizes)*length(changenumbervariants)*length(average_number_before)))
        }
        }
        }
}

#write the output to csv, combine both SD data sets into one sheet, upload to github, and get the url for the next R chunk. FYI model takes 4hrs to run
#write.csv(simresults,file="simresultsjayhabitatsd0.1.csv")
#write.csv(simresults,file="simresultsjayhabitatsd0.2.csv")

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan

#load the data sheet that was made from the simulation in the previous R chunk
d <- read.csv("https://raw.githubusercontent.com/ManyIndividuals/ManyIndividuals/main/Files/rrs/mi1simresultsjayshabitats.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
#head(d)

#make a plot first for sd=2
d2 <- d[d$changeprop_sd==0.2,]

# Set up the data sheet for the model
dat2 <- list(
  proportion_estimates_below_zero = as.numeric(d2$proportion_estimates_below_zero),
  count_estimates_below_zero = as.numeric(d2$count_estimates_below_zero),
  changenumber_mean = as.factor(d2$changenumber_mean),
  samplesize = as.integer(as.factor(d2$n))
)

# We run the STAN model for sd=2 - it is a binomial model that assumes that the probability to estimate a difference depends on an interaction between the sample size and the difference between the conditions (before/after). The response variable is a count and the model turns it into a proportion
m_samplesize_diffscount2 <- ulam(
  alist(
    count_estimates_below_zero ~ dbinom(2000,p),
    logit(p) <- a[samplesize] + b[samplesize,changenumber_mean],
    a[samplesize]~dnorm(0,1),
    transpars> matrix[samplesize,6]:b <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
    matrix[6,samplesize]:z_actor ~ normal( 0 , 1 ),
    # fixed priors
    vector[6]:sigma_actor ~ dexp(1),
    cholesky_factor_corr[6]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
    gq> matrix[6,6]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
  ),data=dat2, log_lik=TRUE, messages=FALSE,cmdstan=T)

# Extract the posterior estimates from this STAN model
precis_output<-precis( m_samplesize_diffscount2 , depth=2 )


#now make a plot for sd=1
d1 <- d[d$changeprop_sd==0.1,]

# Set up the data sheet for the model
dat1 <- list(
  proportion_estimates_below_zero = as.numeric(d1$proportion_estimates_below_zero),
  count_estimates_below_zero = as.numeric(d1$count_estimates_below_zero),
  changenumber_mean = as.factor(d1$changenumber_mean),
  samplesize = as.integer(as.factor(d1$n))
)

# We run the STAN model
m_samplesize_diffscount <- ulam(
  alist(
    count_estimates_below_zero ~ dbinom(2000,p),
    logit(p) <- a[samplesize] + b[samplesize,changenumber_mean],
    a[samplesize]~dnorm(0,1),
    transpars> matrix[samplesize,6]:b <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
    matrix[6,samplesize]:z_actor ~ normal( 0 , 1 ),
    # fixed priors
    vector[6]:sigma_actor ~ dexp(1),
    cholesky_factor_corr[6]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
    gq> matrix[6,6]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
  ),data=dat1, log_lik=TRUE, messages=FALSE,cmdstan=T)

# Extract the posterior estimates from this STAN model
precis_output1<-precis( m_samplesize_diffscount , depth=2 )


# Plot the predictions for the different sample sizes as separate lines = COUNT out of 2000
# plot for sd=2
op <- par(mfrow=c(1,2), mar=c(5.9,4.9,2.3,0.9))
plot(NULL,xlim=c(-0.01,0.31),ylim=c(0.0,0.6),pch=1,ann=F,frame.plot=F,cex=1.5)

mu_10<- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(1,length(unique(d2$changenumber_mean))) ) )
mu_10_mean <- apply( mu_10 , 2 , mean )
mu_10_ci <- apply( mu_10 , 2 , PI , prob=0.97 )
shade( mu_10_ci , unique(d2$changenumber_mean) , col=col.alpha("blue",0.7) )
lines(x=c(-0.004,-0.004),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.096,0.096),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.146,0.146),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.196,0.196),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.246,0.246),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.296,0.296),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))

mu_20 <- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(2,length(unique(d2$changenumber_mean))) ) )
mu_20_mean <- apply( mu_20 , 2 , mean )
mu_20_ci <- apply( mu_20 , 2 , PI , prob=0.97 )
shade( mu_20_ci , unique(d2$changenumber_mean) , col=col.alpha("orange",0.7) )
lines(x=c(-0.002,-0.002),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.098,0.098),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.148,0.148),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.198,0.198),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.248,0.248),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.298,0.298),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))

mu_30 <- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(3,length(unique(d2$changenumber_mean))) ) )
mu_30_mean <- apply( mu_30 , 2 , mean )
mu_30_ci <- apply( mu_30 , 2 , PI , prob=0.97 )
shade( mu_30_ci ,unique(d2$changenumber_mean), col=col.alpha("skyblue",0.7) )
lines(x=c(0,0),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.1,0.1),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.15,0.15),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.2,0.2),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.25,0.25),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.3,0.3),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))

mu_40 <- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(4,length(unique(d2$changenumber_mean))) ) )
mu_40_mean <- apply( mu_40 , 2 , mean )
mu_40_ci <- apply( mu_40 , 2 , PI , prob=0.97 )
shade( mu_40_ci ,unique(d2$changenumber_mean) , col=col.alpha("#213940",0.7) )
lines(x=c(0.002,0.002),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.102,0.102),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.152,0.152),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.202,0.202),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.252,0.252),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.302,0.302),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))

mu_60 <- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(5,length(unique(d2$changenumber_mean))) ) )
mu_60_mean <- apply( mu_60 , 2 , mean )
mu_60_ci <- apply( mu_60 , 2 , PI , prob=0.97 )
shade( mu_60_ci , unique(d2$changenumber_mean), col=col.alpha("red",0.7) )
lines(x=c(0.004,0.004),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.104,0.104),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.154,0.154),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.204,0.204),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.254,0.254),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.304,0.304),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))

# Axis labels (side 1=x axis, side 2=y axis)
mtext("A. Power to detect differences between conditions: SD=2",side=3,line=1,at=0.13,cex=1.2)
mtext("Mean change in proortion of habitats used before vs after",side=1,line=3, cex=0.9)
mtext("Ratio of estimates below zero",side=2,line=3, cex=1.2)

#94% of the posterior mass is above zero for a 0.15 or larger mean change in number of habitats used between before and after when the sample size is 20
mu_10_ci #0.12. column 4 for the 0.15 habitat change category on the x axis (ordered by unique(d$changenumber_mean))
mu_20_ci #0.06
mu_30_ci #0.02
mu_40_ci #0.009
mu_60_ci #0.003
sum(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero<0.3) #mean change=0.15 then 26/30 values are <0.3, mean change=0.15 then 29/30 values are <0.3, mean change=0 then 8/30 values are <0.3
length(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero) #30 values total


# plot for sd=1
plot(NULL,xlim=c(-0.01,0.31),ylim=c(0.0,0.6),pch=1,ann=F,frame.plot=F,cex=1.5)

mu1_10<- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(1,length(unique(d1$changenumber_mean))) ) )
mu1_10_mean <- apply( mu1_10 , 2 , mean )
mu1_10_ci <- apply( mu1_10 , 2 , PI , prob=0.97 )
shade( mu1_10_ci , unique(d1$changenumber_mean) , col=col.alpha("blue",0.7) )
lines(x=c(-0.004,-0.004),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.096,0.096),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.146,0.146),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.196,0.196),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.246,0.246),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.296,0.296),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))

mu1_20 <- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(2,length(unique(d1$changenumber_mean))) ) )
mu1_20_mean <- apply( mu1_20 , 2 , mean )
mu1_20_ci <- apply( mu1_20 , 2 , PI , prob=0.97 )
shade( mu1_20_ci , unique(d1$changenumber_mean) , col=col.alpha("orange",0.7) )
lines(x=c(-0.002,-0.002),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.098,0.098),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.148,0.148),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.198,0.198),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.248,0.248),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.298,0.298),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))

mu1_30 <- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(3,length(unique(d1$changenumber_mean))) ) )
mu1_30_mean <- apply( mu1_30 , 2 , mean )
mu1_30_ci <- apply( mu1_30 , 2 , PI , prob=0.97 )
shade( mu1_30_ci ,unique(d1$changenumber_mean), col=col.alpha("skyblue",0.7) )
lines(x=c(0,0),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.1,0.1),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.15,0.15),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.2,0.2),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.25,0.25),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(0.3,0.3),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))

mu1_40 <- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(4,length(unique(d1$changenumber_mean))) ) )
mu1_40_mean <- apply( mu1_40 , 2 , mean )
mu1_40_ci <- apply( mu1_40 , 2 , PI , prob=0.97 )
shade( mu1_40_ci ,unique(d1$changenumber_mean) , col=col.alpha("#213940",0.7) )
lines(x=c(0.002,0.002),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.102,0.102),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.152,0.152),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.202,0.202),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.252,0.252),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(0.302,0.302),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))

mu1_60 <- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(5,length(unique(d1$changenumber_mean))) ) )
mu1_60_mean <- apply( mu1_60 , 2 , mean )
mu1_60_ci <- apply( mu1_60 , 2 , PI , prob=0.97 )
shade( mu1_60_ci , unique(d1$changenumber_mean), col=col.alpha("red",0.7) )
lines(x=c(0.004,0.004),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.104,0.104),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.154,0.154),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.15,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.204,0.204),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.2,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.254,0.254),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.25,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(0.304,0.304),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0.3,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))

legend(x="topright",legend=c(lty2="10",lty1="20",lty3="30",lty2="40",lty3="60"),lty=c(1,1,1,1,1),lwd=c(5,5,5,5,5),col=c(col.alpha("blue",0.4),col.alpha("orange",0.4),col.alpha("skyblue",0.4),col.alpha("#213940",0.4),col.alpha("red",0.4)),cex=1,bty="n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size",side=3,line=-0.6,at=0.29)
mtext("B. SD=1",side=3,line=1,at=0.01,cex=1.2)
mtext("Mean change in proportion of habitats used before vs after",side=1,line=3, cex=0.9)
par(op)

#93% of the posterior mass is above zero for a 0.1 or larger mean change in number of habitats used between before and after when the sample size is 20
mu1_10_ci #0.17. column 5 for the 0.1 habitat change category on the x axis (ordered by unique(d$changenumber_mean))
mu1_20_ci #0.07
mu1_30_ci #0.06
mu1_40_ci #0.03
mu1_60_ci #0.004
sum(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0.1,]$proportion_estimates_below_zero<0.3) #29/30 values are <0.3, 
#length(d1[d1$n %in% 20 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero) #30 values total

 

Figure 11. Risk of false positives and false negatives depending on sample and effect sizes. Curves of the mean ratio of estimates below zero (vertical lines show the minimum and maximum ratios), which illustrates the power we have to detect differences between conditions at different sample sizes. Across all models, the standard deviation of the mean change in proportion of habitats used was 0.2 (A) or 0.1 (B). A mean change in proportion of habitats of 0.3 is associated with a difference of 3 habitats (when the maximum number of habitats is 10). The curves show the model estimates as the effect increases (larger changes in the mean proportion time spent after versus before on the x-axis) for different potential sample sizes (10-60, illustrated by different colors). When there is no change (x value of zero), estimates suggest that, as expected, half of the estimates are below zero because the before and after conditions are not different from each other. As the change increases, the estimates decrease because models are able to reliably tell that the before and after conditions differ from each other.

 

Run this model on the actual data

Run the code below to determine whether there were differences between the before and after conditions in the proportion of microhabitats used. Only count that a microhabitat was used if the individual had at least 5% of their data points there. This prevents a microhabitat from being counted even if an individual was simply moving through it, and therefore not necessarily using it.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
                                                                                                                                  ## when the data do not yet exist, use simulated data to test the model
N_ind <- 20

dat <- list(
    habitatuse = rbinom(N_ind*2,size=10,prob=5/10), #20 individuals * 2 observations/individual, 10 max habitats; on average, individuals will use 5 of the 10 habitats more than 5% of the time
    tid = c(rep(1,N_ind),rep(2,N_ind)), #1 refers to before and 2 to after
    actor = rep(1:N_ind,2) #go through all individuals twice because there are observations from before and after for each
)


## When you have the actual data, load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

# add 3 microhabitats for all individuals in the natural habitat because suburban has 3 more microhabitats and this will equalize natural and suburban in the model
d[d$Habitat=="Natural",]$habitats <- d[d$Habitat=="Natural",]$habitats+3

# Set up the data sheet
dat <- list(
    habitatuse = d$habitats,
    before = d$BeforeOrAfterManipulation,
    ind = d$ID
)

# THE MODEL (p.447 for binomial distribution with advanced varying slopes - McElreath 2020). 
mj <- ulam(
    alist(
        habitatuse ~ dbinom(10,p), #max 10 habitats
        logit(p) <- g[tid] + alpha[actor,tid],
        # adaptive priors - non-centered
        transpars> matrix[actor,2]:alpha <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
        matrix[2,actor]:z_actor ~ normal( 0 , 1 ),
        # fixed priors
        g[tid] ~ normal(0,1),
        vector[2]:sigma_actor ~ dexp(1),
        cholesky_factor_corr[2]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
        # compute ordinary correlation matrixes from Cholesky factors
        gq> matrix[2,2]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
    ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))

precis(mj,depth=2) #pay attention to g (g1 is before, g2 is after)
boxplot(dat$habitatuse~dat$tid) #see if there is a difference visually

 # run a contrast to determine whether there is a difference between the before and after conditions (there is a difference if CI is on one side of zero)
post <- extract.samples(mj)
diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020
diff_p <- inv_logit(post$g[,2]) - inv_logit(post$g[,1]) #on the probability scale p.341 McElreath 2020
#precis( list( diff_g=diff_g , diff_p=diff_p ) )

J.Q4 More flexible = more food types?

The model

Bayesian model with a normal distribution:

y ~ \(\alpha\)[ind] + \(\beta\)[ind]*before

y is the response variable: the total number of different food types taken per individual. There will be one intercept, \(\alpha\), and one slope \(\beta\) per individual, which will be estimated for the two conditions, before (and after) the manipulation. ID is nested within condition as a random effect because there is more than one data point per individual: each individual has a data point in the before condition and in the after condition. A normal distribution was used because the response variable is a sum without an expected skew to the curve (see Figure 10.6 in McElreath, 2020b). The Bayesian model was developed using McElreath (2020b) as a guide.

Power analysis

We estimated our power to detect differences between conditions at different sample sizes and with different mean changes in the total number of different food types taken per individual in the before vs. after conditions (Figure 12). We simulated the number of food types taken for different sample sizes of individuals before and after the flexibility manipulation. We analyzed these simulated data with the model we will use to analyze the actual data, estimating the change in the number of food types taken between the before and after conditions. From the posterior estimates of the model, we extracted both the mean change as well as the ratio of the posterior estimates that were below zero.

If the mean ratio of estimates below zero is close to 0.5, the model assumes that the change in the number of food types taken before the flexibility manipulation is similar to after. If the ratio is close to zero, the model assumes individuals have changed their behavior. For changes in the number of food types taken smaller than 1, models are likely to assume that no changes occurred even with large sample sizes. If the change in the number of food types taken before the flexibility manipulation vs. after is 1 with a standard deviation of 2, on average 99.96% of the posterior of the model based on a sample size of 20 individuals will be larger than zero. This means that the model is quite certain there is a difference that is larger than zero. In addition, none of the 30 models for a sample size of 20 at the mean change of 1 have a ratio larger than 0.3, meaning that the risk of having a false negative is not very likely.

In general, with sample sizes at or above 20 and mean changes in the number of food types taken at 1 or larger, it is likely that the model will indicate that individuals have changed their behavior. Mean changes below 1 can still be detected, however there is a higher risk that there will be a false negative and this risk is independent of sample size. For example, 17% of the models for a sample size of 20 at the mean change of 0.5 have a ratio larger than 0.3, meaning there is a risk of having a false negative.

With small mean changes in the response variable, some individuals might not increase or even decrease their response after the manipulation because there is variation around the mean change in individual responses. With small sample sizes, there is a risk that only individuals who did not clearly increase their response will be studied, whereas larger sample sizes are more likely to include a wider spectrum of individuals.

To estimate the risk of detecting false positives, we set the mean change in the number of food types taken to zero so there was no change between the before and after conditions. As expected, the average ratio of estimates below zero is close to 0.5 and independent of sample size. With a sample size of 20, 30% have a ratio smaller than 0.3, meaning that the risk of having a false positive is high. The risk would be lower if the variation among individuals was lower than what we assumed (the standard deviation of the mean change in number of foods was 2, which is a conservative estimate).

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
rstan_options(auto_write = TRUE)

### SIMULATE the population (p.441 for normal distribution - McElreath 2020)
N_ind <- 20 #number of individuals tested

#choose the mean number of food types taken and standard deviations - visualize to see whether it might be possible to tell the difference between before (black) and after (red)
number_before<-6.5 #number of foods per bird in the before condition
changenumber<-1.3 #the difference in the number foods between the before and after conditions
plot(density(rnorm(10000,number_before,sd=2)),lwd=2,xlim=c(0,18),ylim=c(0,0.3)) #A sample size of 10k individuals. An SD of 1.3 (when the mean before is 6.5) encompasses all of the range of numbers of foods we expect to find (0-13) so it would be the largest SD we would expect and thus the most conservative sd to use in the simulations
lines(density(rnorm(10000,number_before*(changenumber*1),sd=2)),lwd=2,col="red") #the after condition. Should be able to tell the difference when changenumber is at least 1.3 and the SD is at least 2 (and number_before is 6.5)

#before
numberbefore <- round(rnorm(N_ind,mean=6.5,sd=2),2) #number of foods taken, 2 means 2 decimal points
numberbefore[numberbefore<1]<-0 #this needs to be a minimum of 0 bc an individual can never take negative food
changenumber <- rnorm(N_ind, mean=(13-numberbefore)*1,sd=2) #if had a large number before, have a smaller change relative to the after condition. Multiply by 1 to have a change of 6.5 food types (13-6.5 = 6.5*1 = 6.5)

#after
numberafter <- round(numberbefore+rnorm(N_ind,mean=changenumber,sd=2),2)
numberafter[numberafter<1]<-0

#change the number of foods increased (e.g., 4.5) in changenumber and the sd in numberafter to determine which sds to run the simulations at. Visualize using the plot below to see when the distributions have more separation. It looks like with a difference of 2 foods (4.5 in changenumber), which is probably the minimum we could detect differences with, the maximum numberafter sd would be 2 and an sd with larger detectable differences is 1. Run the simulation with an sd of 1 and 2 to explore the boundaries
plot(density(numberbefore),lwd=2,xlim=c(0,18),ylim=c(0,0.3)) #A sample size of 10k individuals. An SD of 2 (when the mean before is 6.5) encompasses all of the range of numbers of foods we expect to find (0-13) so it would be the largest SD we would expect and thus the most conservative sd to use in the simulations
lines(density(numberafter),lwd=2,col="red") #the after condition. Should be able to tell the difference when changenumber is at least 2 and the SD is at least 2 (and number_before is 6.5)

#check the plot to see whether individuals are increasing their numbers in general (as we are attempting with the changenumber) = 3 points are below the 1:1 line so it looks good
plot(numberbefore,numberafter)
abline(0,1)


### POWER ANALYSIS: different sample sizes and different mean changes in proportions between before and after conditions
# We now set our range of likely sample sizes, from a minimum of 9 individuals per population to a maximum of 60 individuals per population. We draw repeated samples of individuals of that size from each of the 5 populations, during each repetition we draw two samples from each population (so we can infer the false negative rate of wrongly estimating that two samples come from a different population even though they were taken from the sample population). This means we have 10 repetitions for each of the 5 sample sizes and each of the 6 changenumbers. So 10*6 changenumbers. (=60) for each of the 5 sample sizes, resulting in a total length of samplesizes of 300
samplesizes<-c(10,20,30,40,60) #c(10,20,30,40,60)

# Set the means you want to compare. To get a certain mean change in number of foods between before and after, set the multiplier in the changenumber mean: a large multiplier means a large change. Max (13) minus 1 (bc individuals will always live in at least one habitat) / 2 bc individuals who have low starting values will increase to the max, whereas individuals with high starting values will not increase at all
meanchangenumber1<-1 #individuals will on average take 6 more foods after the experiment ((13-numberbefore=6) 6*(6/6)=6*1 = 6. push everyone to 13 meaning that the lowest individuals will increase by 12 and the highest by 0 food items. Therefore, the average is 6 food items
meanchangenumber2<-0.5 # individuals will on average take 3 more foods after the experiment (6*(3/6) =6*0.5= 3)
meanchangenumber3<-0.333 #individuals will on average take 2 more foods after the experiment (6*(2/6) =6*0.33  = 2)
meanchangenumber4<-0.17 # individuals will on average take 1 more foods after the experiment (6*(1/6)=6*0.17  = 1)
meanchangenumber5<-0.08 # individuals will on average take 0.5 more foods after the experiment (6*(0.5/6)=6*0.08 = 0.5)
meanchangenumber6<-0 # individuals will on average take 0 more foods after the experiment (6*(0/6))=6*0 = 0). 0 for false positives
changenumbervariants<-c(meanchangenumber1,meanchangenumber2,meanchangenumber3,meanchangenumber4,meanchangenumber5,meanchangenumber6) #for false negatives

#compare different baseline numbers for how many foods these individuals take before the manipulation
number_before<-c(4, 6.5, 9)

#set up data frame where results will be stored
simresults <- matrix(NA,nrow=900,ncol=7)
simresults <- data.frame(simresults)
colnames(simresults) <- c("n","changenumber_mean","changenumber_sd","repetition","number_before","proportion_estimates_below_zero","count_estimates_below_zero")
counter<-1

# run simulations. There are 5 different sample sizes, and for each we have 6 different mean change in numbers that we want to examine, and each will repeat 10 times for a total of 300 samples.
for (samplesize in 1:length(samplesizes)) {
        N_ind <- samplesizes[samplesize] #number of grackles per sample size
        
        #now run through each of the 6 number variations
        for (numbervariant in 1:length(changenumbervariants)) {
        
        #pick the current proportion variant from the list changepropvariants
        currentchangenumber <- changenumbervariants[numbervariant]
        
        for (baselinenumber in 1:3) {
        #pick the current proportion variant from the list changepropvariants
        currentnumber_before <- number_before[baselinenumber]
        
        for (repetition in 1:10) {
        #create data frame to store for each repetition of the simulated data to be analyzed by the model
        sites <- matrix(nrow=n,ncol=3) 
        colnames(sites)<-c("ID","condition","number_habitats")
        
        #BEFORE manipulation
        numberbefore <- round(rnorm(N_ind,mean=currentnumber_before,sd=1.5),0) #number of foods, 2 means 2 decimal points
        numberbefore[numberbefore<1]<-1 #this needs to be a minimum of 1 bc an individual likely have been observed to take at least one piece of food
        changenumber <- rnorm(N_ind, mean=(13-numberbefore)*(currentchangenumber),sd=1) #change this SD to explore the boundaries of individual variation in their response (in terms of the success measure) to the flexibility manipulation. We explored SD=1 and SD=2

        #AFTER manipulation
        numberafter <- round(numberbefore+changenumber,0)
        numberafter[numberafter<1]<-1 #this needs to be a minimum of 1 bc an individual likely have been observed to take at least one piece of food
  
        ### run a STAN model to see whether the before and after proportions are estimated to be different from each other
        #sets up the data sheet with the simulated data that the model will run on
      dat <- list(
    foods = c(numberbefore,numberafter), #compare before and after
    before = c(rep(0,N_ind),rep(1,N_ind)), #1 refers to before and 2 to after
    ind = rep(1:N_ind,2) #go through all individuals twice because there are observations from before and after for each
)

        # Our MODEL, which will estimate a different mean and variance of phi for each site 
        # note: had to add force_recompile bc kept giving me an error and wouldn't finish running. See https://github.com/rmcelreath/rethinking/issues/338
sjf <- ulam(
    alist(
        foods ~ normal( mu , sigma ),
        mu <- a_ind[ind] + b_ind[ind]*before,
        c(a_ind,b_ind)[ind] ~ multi_normal( c(alpha,beta) , Rho , sigma_ind ),
        alpha ~ normal(6.5,2), #6.5=mean number of foods taken, 2=food types for the sd (don't change when exploring different SDs for changenumber)
        beta ~ normal(0,6), #0.5=mean slope, 6=sd which is like the max difference between before and after
        sigma_ind ~ exponential(1),
        sigma ~ exponential(1),
        Rho ~ lkj_corr(2)
    ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))

#precis(sjf, depth=2) #check the output to make sure it looks as expected
#Don't need to do the contrast because we already have each individual's number so we only need to look at their beta to see whether it is positive (more foods after) or negative (fewer foods after)
post <- extract.samples(sjf)

  # enter all of the results into the data sheet
    simresults[counter,]$n <- N_ind
    simresults[counter,]$changenumber_mean <- (13-currentnumber_before)*(currentchangenumber)
    simresults[counter,]$changenumber_sd <- 1 #change this to match the SD in changenumber (it only gets changed in these 2 places)
    simresults[counter,]$repetition <- repetition
    simresults[counter,]$number_before <- currentnumber_before
    simresults[counter,]$proportion_estimates_below_zero <- sum(post$beta<0)/2000
    simresults[counter,]$count_estimates_below_zero <- sum(post$beta<0)
    counter<-counter+1 
    print(c(counter,"out of",10*length(samplesizes)*length(changenumbervariants)*length(number_before)))
        }
        }
        }
}

#write.csv(simresults,file="simresultsjayforagingsd2.csv") #put the outputs into a csv file so don't have to run the model again (it takes 5-7 hours)
#write.csv(simresults,file="simresultsjayforagingsd1.csv")

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
#rebuild_cmdstan() #run this if you get an error about a PCH file https://discourse.mc-stan.org/t/struggling-to-get-cmdstan-working-on-mac-os-big-sur/19774/4

#load the data sheet that was made from the simulation in the previous R chunk
d <- read.csv("https://raw.githubusercontent.com/ManyIndividuals/ManyIndividuals/main/Files/rrs/mi1simresultsjaysforaging.csv", header = TRUE, sep = ",", stringsAsFactors = FALSE)
#head(d)

#make a plot first for sd=2
d2 <- d[d$changenumber_sd==2,]

# Set up the data sheet for the model
datsd2 <- list(
  proportion_estimates_below_zero = as.numeric(d2$proportion_estimates_below_zero),
  count_estimates_below_zero = as.numeric(d2$count_estimates_below_zero),
  changenumber_mean = as.factor(d2$changenumber_mean),
  samplesize = as.integer(as.factor(d2$n))
)

# We run the STAN model for sd=2 - it is a binomial model that assumes that the probability to estimate a difference depends on an interaction between the sample size and the difference between the conditions (before/after). The response variable is a count and the model turns it into a proportion
m_samplesize_diffscount2 <- ulam(
  alist(
    count_estimates_below_zero ~ dbinom(2000,p),
    logit(p) <- a[samplesize] + b[samplesize,changenumber_mean],
    a[samplesize]~dnorm(0,1),
    transpars> matrix[samplesize,6]:b <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
    matrix[6,samplesize]:z_actor ~ normal( 0 , 1 ),
    # fixed priors
    vector[6]:sigma_actor ~ dexp(1),
    cholesky_factor_corr[6]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
    gq> matrix[6,6]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
  ),data=datsd2, log_lik=TRUE, messages=FALSE,cmdstan=T)

# Extract the posterior estimates from this STAN model
precis_output<-precis( m_samplesize_diffscount2 , depth=2 )


#now make a plot for sd=1
d1 <- d[d$changenumber_sd==1,]

# Set up the data sheet for the model
datsd1 <- list(
  proportion_estimates_below_zero = as.numeric(d1$proportion_estimates_below_zero),
  count_estimates_below_zero = as.numeric(d1$count_estimates_below_zero),
  changenumber_mean = as.factor(d1$changenumber_mean),
  samplesize = as.integer(as.factor(d1$n))
)

# We run the STAN model for sd=1
m_samplesize_diffscount <- ulam(
  alist(
    count_estimates_below_zero ~ dbinom(2000,p),
    logit(p) <- a[samplesize] + b[samplesize,changenumber_mean],
    a[samplesize]~dnorm(0,1),
    transpars> matrix[samplesize,6]:b <- compose_noncentered( sigma_actor , L_Rho_actor , z_actor ),
    matrix[6,samplesize]:z_actor ~ normal( 0 , 1 ),
    vector[6]:sigma_actor ~ dexp(1),
    cholesky_factor_corr[6]:L_Rho_actor ~ lkj_corr_cholesky( 2 ),
    gq> matrix[6,6]:Rho_actor <<- Chol_to_Corr(L_Rho_actor)
  ),data=datsd1, log_lik=TRUE, messages=FALSE,cmdstan=T)

# Extract the posterior estimates from this STAN model
precis_output1<-precis( m_samplesize_diffscount , depth=2 )


# Plot the predictions for the different sample sizes as separate lines = COUNT out of 2000
# plot for sd=2
op <- par(mfrow=c(1,2), mar=c(5.9,4.9,2.3,0.9))
plot(NULL,xlim=c(-0.2,5),ylim=c(0.0,0.8),pch=1,ann=F,frame.plot=F,cex=1.5,xaxt="n")

mu_10<- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(1,length(unique(d2$changenumber_mean))) ) )
mu_10_mean <- apply( mu_10 , 2 , mean )
mu_10_ci <- apply( mu_10 , 2 , PI , prob=0.97 )
shade( mu_10_ci , unique(d2$changenumber_mean) , col=col.alpha("blue",0.7) )
lines(x=c(-0.1,-0.1),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.9,0.9),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(1.9,1.9),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(2.9,2.9),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(3.9,3.9),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(4.9,4.9),y=c(min(d2[d2$n %in% 10 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d2[d2$n %in% 10 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))

mu_20 <- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(2,length(unique(d2$changenumber_mean))) ) )
mu_20_mean <- apply( mu_20 , 2 , mean )
mu_20_ci <- apply( mu_20 , 2 , PI , prob=0.97 )
shade( mu_20_ci , unique(d2$changenumber_mean) , col=col.alpha("orange",0.7) )
lines(x=c(-0.05,-0.05),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.95,0.95),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(1.95,1.95),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(2.95,2.95),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(3.95,3.95),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(4.95,4.95),y=c(min(d2[d2$n %in% 20 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d2[d2$n %in% 20 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))

mu_30 <- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(3,length(unique(d2$changenumber_mean))) ) )
mu_30_mean <- apply( mu_30 , 2 , mean )
mu_30_ci <- apply( mu_30 , 2 , PI , prob=0.97 )
shade( mu_30_ci ,unique(d2$changenumber_mean), col=col.alpha("skyblue",0.7) )
lines(x=c(0,0),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(1,1),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(2,2),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(3,3),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(4,4),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(5,5),y=c(min(d2[d2$n %in% 30 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d2[d2$n %in% 30 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))

mu_40 <- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(4,length(unique(d2$changenumber_mean))) ) )
mu_40_mean <- apply( mu_40 , 2 , mean )
mu_40_ci <- apply( mu_40 , 2 , PI , prob=0.97 )
shade( mu_40_ci ,unique(d2$changenumber_mean) , col=col.alpha("#213940",0.7) )
lines(x=c(0.05,0.05),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(1.05,1.05),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(2.05,2.05),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(3.05,3.05),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(4.05,4.05),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(5.05,5.05),y=c(min(d2[d2$n %in% 40 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d2[d2$n %in% 40 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))

mu_60 <- link( m_samplesize_diffscount2 , data=data.frame( changenumber_mean=as.factor(unique(d2$changenumber_mean)),samplesize=rep(5,length(unique(d2$changenumber_mean))) ) )
mu_60_mean <- apply( mu_60 , 2 , mean )
mu_60_ci <- apply( mu_60 , 2 , PI , prob=0.97 )
shade( mu_60_ci , unique(d2$changenumber_mean), col=col.alpha("red",0.7) )
lines(x=c(0.1,0.1),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(1.1,1.1),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(2.1,2.1),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(3.1,3.1),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(4.1,4.1),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(5.1,5.1),y=c(min(d2[d2$n %in% 60 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d2[d2$n %in% 60 & d2$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))

# Axis labels (side 1=x axis, side 2=y axis)
mtext("A. Power to detect differences between conditions: SD=2",side=3,line=1,at=2.5,cex=1.2)
mtext("Mean change in number of food types taken before vs after",side=1,line=2, cex=1)
mtext("Ratio of estimates below zero",side=2,line=3, cex=1.2)
mtext("0",side=1,line=0, at=0, cex=1)
mtext("0.5",side=1,line=0, at=1, cex=1)
mtext("1",side=1,line=0, at=2, cex=1)
mtext("2",side=1,line=0, at=3, cex=1)
mtext("3",side=1,line=0, at=4, cex=1)
mtext("6",side=1,line=0, at=5, cex=1)

#99.96% of the posterior mass is above zero for a 1 or larger mean change in number of habitats used between before and after when the sample size is 20
#mu_10_ci #99.85%. column 4 for the 1 category on the x axis (ordered by unique(d$changenumber_mean))
#mu_20_ci #99.96%
#mu_30_ci #99.95%
#mu_40_ci #99.97%
#mu_60_ci #99.98%
#sum(d2[d2$n %in% 20 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero<0.3) #30 values are <0.3 (out of 30). 25/30 are <0.3 when the changenumber mean is 1 (equivalant to a 0.5 change in habitats), and 9/30 when changenumbermean=0
#length(d2[d2$n %in% 20 & d2$changenumber_mean %in% 2,]$proportion_estimates_below_zero) #30 values total


# plot for sd=1
plot(NULL,xlim=c(-0.2,5.2),ylim=c(0.0,0.8),pch=1,ann=F,frame.plot=F,cex=1.5,xaxt="n")

mu1_10<- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(1,length(unique(d1$changenumber_mean))) ) )
mu1_10_mean <- apply( mu1_10 , 2 , mean )
mu1_10_ci <- apply( mu1_10 , 2 , PI , prob=0.97 )
shade( mu1_10_ci , unique(d1$changenumber_mean) , col=col.alpha("blue",0.7) )
lines(x=c(-0.1,-0.1),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(0.9,0.9),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(1.9,1.9),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(2.9,2.9),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(3.9,3.9),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))
lines(x=c(4.9,4.9),y=c(min(d1[d1$n %in% 10 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d1[d1$n %in% 10 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("blue",0.5))

mu1_20 <- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(2,length(unique(d1$changenumber_mean))) ) )
mu1_20_mean <- apply( mu1_20 , 2 , mean )
mu1_20_ci <- apply( mu1_20 , 2 , PI , prob=0.97 )
shade( mu1_20_ci , unique(d1$changenumber_mean) , col=col.alpha("orange",0.7) )
lines(x=c(-0.05,-0.05),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(0.95,0.95),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(1.95,1.95),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(2.95,2.95),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(3.95,3.95),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))
lines(x=c(4.95,4.95),y=c(min(d1[d1$n %in% 20 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d1[d1$n %in% 20 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("orange",0.5))

mu1_30 <- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(3,length(unique(d1$changenumber_mean))) ) )
mu1_30_mean <- apply( mu1_30 , 2 , mean )
mu1_30_ci <- apply( mu1_30 , 2 , PI , prob=0.97 )
shade( mu1_30_ci ,unique(d1$changenumber_mean), col=col.alpha("skyblue",0.7) )
lines(x=c(0,0),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(1,1),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(2,2),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(3,3),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(4,4),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))
lines(x=c(5,5),y=c(min(d1[d1$n %in% 30 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d1[d1$n %in% 30 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("skyblue",0.5))

mu1_40 <- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(4,length(unique(d1$changenumber_mean))) ) )
mu1_40_mean <- apply( mu1_40 , 2 , mean )
mu1_40_ci <- apply( mu1_40 , 2 , PI , prob=0.97 )
shade( mu1_40_ci ,unique(d1$changenumber_mean) , col=col.alpha("#213940",0.7) )
lines(x=c(0.05,0.05),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(1.05,1.05),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(2.05,2.05),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(3.05,3.05),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(4.05,4.05),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))
lines(x=c(5.05,5.05),y=c(min(d1[d1$n %in% 40 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d1[d1$n %in% 40 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("#213940",0.5))

mu1_60 <- link( m_samplesize_diffscount , data=data.frame( changenumber_mean=as.factor(unique(d1$changenumber_mean)),samplesize=rep(5,length(unique(d1$changenumber_mean))) ) )
mu1_60_mean <- apply( mu1_60 , 2 , mean )
mu1_60_ci <- apply( mu1_60 , 2 , PI , prob=0.97 )
shade( mu1_60_ci , unique(d1$changenumber_mean), col=col.alpha("red",0.7) )
lines(x=c(0.1,0.1),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 0,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(1.1,1.1),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 1,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(2.1,2.1),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(3.1,3.1),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 3,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(4.1,4.1),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 4,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))
lines(x=c(5.1,5.1),y=c(min(d1[d1$n %in% 60 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero),max(d1[d1$n %in% 60 & d1$changenumber_mean %in% 5,]$proportion_estimates_below_zero)),col=col.alpha("red",0.5))

legend(x="topright",legend=c(lty2="10",lty1="20",lty3="30",lty2="40",lty3="60"),lty=c(1,1,1,1,1),lwd=c(5,5,5,5,5),col=c(col.alpha("blue",0.4),col.alpha("orange",0.4),col.alpha("skyblue",0.4),col.alpha("#213940",0.4),col.alpha("red",0.4)),cex=1,bty="n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size",side=3,line=-0.7,at=4.9)
mtext("0",side=1,line=0, at=0, cex=1)
mtext("0.5",side=1,line=0, at=1, cex=1)
mtext("1",side=1,line=0, at=2, cex=1)
mtext("2",side=1,line=0, at=3, cex=1)
mtext("3",side=1,line=0, at=4, cex=1)
mtext("6",side=1,line=0, at=5, cex=1)
mtext("B. SD=1",side=3,line=1,at=0.1,cex=1.2)
mtext("Mean change in number of foods taken before vs after",side=1,line=2, cex=1)
par(op)

#98% of the posterior mass is above zero for a 1 or larger mean change in number of habitats used between before and after when the sample size is 20
#mu1_10_ci #99.97%. column 4 for the 1 category on the x axis (ordered by unique(d$changenumber_mean))
#mu1_20_ci #99.99%
#mu1_30_ci #99.996%
#mu1_40_ci #99.998%
#mu1_60_ci #100%
#sum(d1[d1$n %in% 20 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero<0.3) #30 values are <0.3 (out of 30)
#length(d1[d1$n %in% 20 & d1$changenumber_mean %in% 2,]$proportion_estimates_below_zero) #30 values total

 

Figure 12. Risk of false positives and false negatives depending on sample and effect sizes. Curves of the mean ratio of estimates below zero (vertical lines show the minimum and maximum ratios), which illustrates the power we have to detect differences between conditions at different sample sizes. Across all models, the standard deviation of the mean change in number of food types taken was 2 (A) or 1 (B), and the number of food types taken before the flexibility manipulation was 6.5. A mean change in proportion of habitats of 0.3 is associated with a difference of 3 habitats (when the maximum number of habitats is 10). The curves show the model estimates as the effect increases (larger changes in the mean proportion time spent after versus before on the x-axis) for different potential sample sizes (10-60, illustrated by different colors). When there is no change (x value of zero), estimates suggest that, as expected, half of the estimates are below zero because the before and after conditions are not different from each other. As the change increases, the estimates decrease because models are able to reliably tell that the before and after conditions differ from each other.

 

Run this model on the actual data

Run the code below to determine whether there were differences between the before and after conditions in the number of food types taken.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
                                                                                                                                  ## when the data do not yet exist, use simulated data to test the model
N_ind <- 20

dat <- list(
    foods = standardize(rnorm(40,mean=4.5,sd=2)), #estimate 20 total individuals (*2 conditions) with a mean of 6.5 foods/individual (halfway between the max 13 possible food types)
    before = c(rep(1,N_ind),rep(2,N_ind)), #1 refers to before and 2 to after
    ind = rep(1:N_ind,2) #go through all individuals twice because there are observations from before and after for each
)

## load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

# Set up the data sheet
dat <- list(
    habitatuse = d$habitats,
    before = d$BeforeOrAfterManipulation,
    ind = d$ID
)

# THE MODEL (p.441 for normal distribution - McElreath 2020). Note that it is MULTIPLIED by condition (before), which means that you don't need to run a contrast because each individual value already accounts for the difference. So only look at beta 
mjf <- ulam(
    alist(
        foods ~ normal( mu , sigma ),
        mu <- a_ind[ind] + b_ind[ind]*before,
        c(a_ind,b_ind)[ind] ~ multi_normal( c(alpha,beta) , Rho , sigma_ind ),
        alpha ~ normal(6.5,2), #6.5=mean number of foods taken, 2=food types for the sd
        beta ~ normal(0,6), #0.5=mean slope, 6=sd which is like the max difference between before and after
        sigma_ind ~ exponential(1),
        sigma ~ exponential(1),
        Rho ~ lkj_corr(2)
    ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE)

#precis(mjf,depth=2)
#boxplot(dat$foods~dat$before) #see if there is a difference visually

Toutouwai

T.Q1 Does a flexibility manipulation alter dispersal timing (1.1) and distance (1.2), as well as the likelihood that juvenile toutouwai will disperse beyond the protection of Zealandia’s fence and attempt to establish in the adjacent urban area (1.3)?

The model: dispersal timing (1.1)

Bayesian model for a zero-inflated poisson distribution:

\(D_i\) ~ Zero-inflated Poisson(\(p_i\),\(lambda_i\))

\(D_i\) is how many days before the cut-off of observations at 112 days individuals dispersed (for example, if a bird dispersed 50 days after fledging, D_i is 112-50=72); the subtraction covers cases where the age at dispersal would be very late such that some birds might disperse after day 112 post-fledging. The latter individuals will be included with the non-dispersers as having a zero value. \(lambda_i\) is the mean of the Poisson distribution describing the average day (and variance) for individuals who have dispersed. Dispersal is defined as the time point at which the juvenile is first detected leaving the natal territory on three consecutive days. A zero inflated Poisson distribution was used because a small percentage of individuals do not disperse in their first season (zero values) and individuals who do disperse have a response variable that is a count number of days post-fledging do so within a relatively short period, which is captured by the Poisson distribution which counts the number of days (McElreath, 2020b). These patterns are described by (Richard, 2007), which shows that only a small minority of individuals does not disperse, and that dispersers were observed to leave their natal territory on average 47.5 days after fledging, with a range between 32-72 days. We assume that both the likelihood to disperse and the day at which dispersers leave might differ between individuals in the manipulated and the control conditions:

log(\(p_i\)) = \(\gamma_p\)[condition] + \(\alpha_p\)[parentid],[condition]

The two \(\gamma\)[condition] estimates reflect the difference between control and manipulated birds for the probability to not disperse (\(\gamma_p\)) and when to disperse (\(\gamma_l\)). The Bayesian model was developed using McElreath (2020b) as a guide.

Power analysis (1.1)

We estimated our power to detect differences between conditions at different sample sizes and with different mean changes in age at which individuals disperse after fledging (Figure 13). We simulated the age at which individuals would disperse in the control condition (with averages of 36, 48, or 72 days after fledging), and assumed that manipulated individuals might disperse sooner. We analyzed these simulated data with the model we will use to analyze the actual data, estimating the contrast in average time at which individuals disperse. The model also accounts for differences in the likelihood to not disperse at all between control and manipulated birds, but we did not examine this here in the simulation given that the number of individuals who will not disperse is likely to be very low. From the posterior estimates of the model, we extracted both the mean change as well as the ratio of the posterior estimates that were below zero, indicating that there would be a clear difference between control and manipulated birds.

If the mean ratio of estimates below zero is close to 0.5, the model assumes that the age at which individuals disperse is similar for control and manipulated birds. If the ratio is close to zero, the model assumes individuals have changed their behavior and disperse earlier when they were manipulated. We find that for changes smaller than 1 standard deviation (which equals roughly 7 days), models are likely to assume that there are no differences between control and manipulated individuals even with large sample sizes. If the contrast in time of dispersal of manipulated compared to control individuals is 1 standard deviation, on average 92% of the posterior of the model based on a sample size of 20 individuals will be larger than zero. This means that the model is quite certain there is a contrast that is larger than zero.

In general, with sample sizes at or above 20 and mean contrasts between manipulated and control individuals of 1 standard deviation or higher, it is highly likely that the model will indicate that manipulated individuals have changed their behavior. Mean contrasts of less than 1 standard deviation can still be detected, however there is a higher risk that there will be a false negative (27% risk of a false negative with a sample size of 20 when the contrast is 0.25 standard deviations = ~2 days earlier dispersal).

To estimate the risk of detecting false positives, we set the mean change in age of dispersal for manipulated individuals compared to control individuals to zero so there was no change. As expected, the average ratio of estimates below zero is close to 0.5 and independent of sample size. With a sample size of 20, <7% of the simulations have a ratio smaller than 0.1, meaning that the risk of having a false positive is low.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
rstan_options(auto_write = TRUE)
#rebuild_cmdstan() #run this if you get an error about a PCH file https://discourse.mc-stan.org/t/struggling-to-get-cmdstan-working-on-mac-os-big-sur/19774/4

### SIMULATE the population
### We first start with a single population to identify the parameters for the zero-inflated poisson distribution.
# We base this on the information on page 69 in this thesis: 
# https://mro.massey.ac.nz/bitstream/handle/10179/1596/02_whole.pdf?sequence=1
# It suggests that only a small minority of individuals do not disperse. Based on previous observations of our study population we estimate that ~10% of individuals do not disperse. The dispersers were observed to leave their natal territory on average 47.5 days after fledging, with a range between 32-72 days. We convert this to ask how many days before the end of the observation period birds dispersed, so subtracting these values from 112 (average = 64, range = 40-80). Accordingly, we set p=0.1 and lambda = 112-48 = 64
# plot the distribution
hist(rzipois(10000,p=0.1,lambda=64))

# We expect to have only very few non-dispersing individuals in our sample (10% with a sample size of 60 would be 6 non-dispersers, split between control and manipulated groups). Accordingly, we think that the uncertainty will be too high to say whether there is a difference in the probability to disperse between control and manipulated individuals. We will estimate the separate probabilities in the model, but not vary it in the simulation.
# For the timing of dispersal, we assume that there might be seasonal differences in when the optimal timing would be, so we set the average of the control group to disperse 40 days before the end of the observation (dispersed on average 72 days after fledging), 64 (48 days after fledging), and 76 (36 days after fledging). 
# For the expected change due to the manipulation, we assume that it will be within the natural range given that external conditions might set limits on when birds might be first able to disperse. Accordingly, we set the maximum expected improvement to the previous mean minus 2 standard deviations. The standard deviation for a poisson distribution is the square root of the mean. We take this for the days after fledging, not the subtracted means, because we expect potentially more variation when birds on average disperse later. 

# power analysis: run this simulation with different N_ind, change means, to find the boundaries of where you can detect a difference (run the model and use contrasts) between the two conditions (control/manipulated)


### POWER ANALYSIS: different sample sizes and different mean changes in dispersal distance between control and manipulated conditions
# We now set our range of likely sample sizes, from a minimum of 10 individuals to a maximum of 60 individuals (half in the control and the other half in the manipulated condition). We draw repeated samples of individuals of that size, with 10 repetitions for each of the 5 sample sizes and each of the 6 mean changes. So 10*6 mean change (=60) for each of the 5 sample sizes, resulting in a total length of samplesizes of 300
samplesizes<-c(10,20,30,40,60) #c(10,20,30,40,60)

# Set the means you want to compare (number of standard deviations of change)
meanchange1<-2 
meanchange2<-1.5
meanchange3<-1 
meanchange4<-0.5
meanchange5<-0.25
meanchange6<-0 #for false positives
changevariants<-c(meanchange1,meanchange2,meanchange3,meanchange4,meanchange5,meanchange6) #for false negatives

#compare different baseline means
mean_day_control<-c(40,64,76) #


#set up data frame where results will be stored
simresults <- matrix(NA,nrow=900,ncol=6)
simresults <- data.frame(simresults)
colnames(simresults) <- c("n","change_mean","repetition","timing_control","proportion_estimates_below_zero","count_estimates_below_zero")
counter<-1

# run simulations. There are 5 different sample sizes, and for each we have 6 different mean change in proportions that we want to examine, and each will repeat 10 times for a total of 300 samples.
for (samplesize in 1:length(samplesizes)) {
  N_ind <- samplesizes[samplesize] #number of grackles per sample size
  
  #now run through each of the 6 change variations
  for (propvariant in 1:length(changevariants)) {
    
    #pick the current change variant from the list changevariants
    currentchange <- changevariants[propvariant]
    
    for (baselineprop in 1:3) {
      #pick the baseline distance variant from the list 
      currentprop_before <- mean_day_control[baselineprop]
      
      for (repetition in 1:10) {
        #create data frame to store for each repetition of the simulated data to be analyzed by the model
   
        #CONTROL condition
        days_control <- rzipois(N_ind/2,p=0.1,lambda=currentprop_before)
        #MANIPULATED condition
        days_manipulated<- rzipois(N_ind/2,p=0.1,lambda=currentprop_before+currentchange*(sqrt(112-currentprop_before)))
        
        ### run a STAN model to see whether the before and after proportions are estimated to be different from each other
        #sets up the data sheet with the simulated data that the model will run on
        dat <- list(
          response = c(days_control,days_manipulated),
          condition = c(rep(1,N_ind/2),rep(2,N_ind/2)) #1 refers to control and 2 to manipulated
        )
        
        # Our MODEL
        s1 <- ulam(
          alist(
            response ~ dzipois(p,lambda),
            logit(p) <- g_p[condition] ,
            log(lambda) <- g[condition] ,
            # adaptive priors - non-centered which is a parameterization that improves sampling for the MCMC model fit by helping with complex varying effects (see p.453)
            g_p[condition] ~ normal(0,2),
            g[condition] ~ normal(0,2)
          ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))
        
        #run a CONTRAST to determine whether there is a difference between before (g1) and after (g2) (subtract g1 before from g2 after)
        post <- extract.samples(s1)
        diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020

        # enter all of the results into the data sheet
        simresults[counter,]$n <- N_ind
        simresults[counter,]$change_mean <- currentchange
        simresults[counter,]$repetition <- repetition
        simresults[counter,]$timing_control <- currentprop_before
        simresults[counter,]$proportion_estimates_below_zero <- sum(diff_g<0)/length(diff_g)
        simresults[counter,]$count_estimates_below_zero <- sum(diff_g<0)
        counter<-counter+1 
        print(c(counter,"out of",10*length(samplesizes)*length(changevariants)*length(distance_control)))
      }
    }
  }
}

#write.csv(m1simresults_toutouawidispersaltiming.csv") #put the outputs into a csv file so don't have to run the model again...it takes hours

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE)  #makes it so ulam always runs with cmdstan

# load the data sheet that was made from the simulation in
# the previous R chunk
d <- read.csv("https://raw.githubusercontent.com/ManyIndividuals/ManyIndividuals/main/Files/rrs/m1simresults_toutouawidispersaltiming.csv",
    header = TRUE, sep = ",", stringsAsFactors = FALSE)
# head(d)

# Set up the data sheet for the model
dat <- list(proportion_estimates_below_zero = as.numeric(d$proportion_estimates_below_zero),
    count_estimates_below_zero = as.numeric(d$count_estimates_below_zero),
    change_mean = as.numeric(sqrt(d$change_mean)), samplesize = as.integer(as.factor(d$n)))

# We run the STAN model - it is a binomial model that
# assumes that the probability to estimate a difference
# depends on an interaction between the sample size and the
# difference between the conditions (before/after)
m_samplesize_diffscount <- ulam(alist(count_estimates_below_zero ~
    dbinom(2000, p), logit(p) <- a[samplesize] - b[samplesize] *
    change_mean, a[samplesize] ~ dnorm(a_bar, sigma_a), a_bar ~
    dnorm(0, 1.5), sigma_a ~ dexp(1), b[samplesize] ~ dnorm(0,
    1.5)), data = dat, log_lik = TRUE, messages = FALSE, cmdstan = T)

# Extract the posterior estimates from this STAN model
precis_output <- precis(m_samplesize_diffscount, depth = 2)

# Plot the predictions for the different sample sizes as
# separate lines = COUNT out of 2000
plot(NULL, xlim = c(-0.01, 2.01), ylim = c(0, 0.7), pch = 1,
    ann = F, frame.plot = F, cex = 1.5)

mu_10 <- link(m_samplesize_diffscount, data = data.frame(change_mean = sqrt(unique(d$change_mean)),
    samplesize = rep(1, length(unique(d$change_mean)))))
mu_10_mean <- apply(mu_10, 2, mean)
mu_10_ci <- apply(mu_10, 2, PI, prob = 0.97)
shade(mu_10_ci, unique(d$change_mean), col = col.alpha("blue",
    0.7))
lines(x = c(-0.04, -0.04), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.21, 0.21), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    0.25, ]$proportion_estimates_below_zero), max(d[d$n %in%
    10 & d$change_mean %in% 0.25, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.46, 0.46), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    0.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 0.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.96, 0.96), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    1, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 1, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(1.46, 1.46), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    1.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 1.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(1.96, 1.96), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    2, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 2, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))

mu_20 <- link(m_samplesize_diffscount, data = data.frame(change_mean = sqrt(unique(d$change_mean)),
    samplesize = rep(2, length(unique(d$change_mean)))))
mu_20_mean <- apply(mu_20, 2, mean)
mu_20_ci <- apply(mu_20, 2, PI, prob = 0.97)
shade(mu_20_ci, unique(d$change_mean), col = col.alpha("orange",
    0.7))
lines(x = c(-0.02, -0.02), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.23, 0.23), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    0.25, ]$proportion_estimates_below_zero), max(d[d$n %in%
    20 & d$change_mean %in% 0.25, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.48, 0.48), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    0.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 0.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.98, 0.98), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    1, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 1, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(1.48, 1.48), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    1.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 1.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(1.98, 1.98), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    2, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 2, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))

mu_30 <- link(m_samplesize_diffscount, data = data.frame(change_mean = sqrt(unique(d$change_mean)),
    samplesize = rep(3, length(unique(d$change_mean)))))
mu_30_mean <- apply(mu_30, 2, mean)
mu_30_ci <- apply(mu_30, 2, PI, prob = 0.97)
shade(mu_30_ci, unique(d$change_mean), col = col.alpha("skyblue",
    0.7))
lines(x = c(0, 0), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.25, 0.25), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    0.25, ]$proportion_estimates_below_zero), max(d[d$n %in%
    30 & d$change_mean %in% 0.25, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.5, 0.5), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    0.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 0.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(1, 1), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    1, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 1, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(1.5, 1.5), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    1.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 1.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(2, 2), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    2, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 2, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))

mu_40 <- link(m_samplesize_diffscount, data = data.frame(change_mean = sqrt(unique(d$change_mean)),
    samplesize = rep(4, length(unique(d$change_mean)))))
mu_40_mean <- apply(mu_40, 2, mean)
mu_40_ci <- apply(mu_40, 2, PI, prob = 0.97)
shade(mu_40_ci, unique(d$change_mean), col = col.alpha("#213940",
    0.7))
lines(x = c(0.02, 0.02), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.27, 0.27), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    0.25, ]$proportion_estimates_below_zero), max(d[d$n %in%
    40 & d$change_mean %in% 0.25, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.52, 0.52), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    0.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 0.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(1.02, 1.02), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    1, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 1, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(1.52, 1.52), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    1.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 1.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(2.2, 2.2), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    2, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 2, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))

mu_60 <- link(m_samplesize_diffscount, data = data.frame(change_mean = sqrt(unique(d$change_mean)),
    samplesize = rep(5, length(unique(d$change_mean)))))
mu_60_mean <- apply(mu_60, 2, mean)
mu_60_ci <- apply(mu_60, 2, PI, prob = 0.97)
shade(mu_60_ci, unique(d$change_mean), col = col.alpha("red",
    0.7))
lines(x = c(0.04, 0.04), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.29, 0.29), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    0.25, ]$proportion_estimates_below_zero), max(d[d$n %in%
    60 & d$change_mean %in% 0.25, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.54, 0.54), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    0.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 0.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(1.04, 1.04), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    1, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 1, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(1.54, 1.54), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    1.5, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 1.5, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(2.2, 2.2), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    2, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 2, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))

legend(x = "topright", legend = c(lty2 = "10", lty1 = "20", lty3 = "30",
    lty2 = "40", lty3 = "60"), lty = c(1, 1, 1, 1, 1), lwd = c(5,
    5, 5, 5, 5), col = c(col.alpha("blue", 0.4), col.alpha("orange",
    0.4), col.alpha("skyblue", 0.4), col.alpha("#213940", 0.4),
    col.alpha("red", 0.4)), cex = 1, bty = "n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size", side = 3, line = -1.3, at = 1.6)
mtext("Power to detect differences between conditions", side = 3,
    line = 1, at = 1, cex = 1.5)
mtext("Weeks manipulated indvs disperse earlier than control indvs",
    side = 1, line = 3, cex = 1.2)
mtext("Ratio of estimates below zero", side = 2, line = 3, cex = 1.2)

# 92% of the posterior mass is above zero for a change of 1
# sd (1 week) between control and manipulated indviduals
# when the sample size is 20. It is 83% for a change of 0.5
# sd 1-mu_10_mean[3] #85%. 3rd column from the left 1 sd (1
# week) category on the x axis (ordered by
# unique(d$change_mean)) 1-mu_20_mean[3] #92%
# 1-mu_30_mean[3] #94% 1-mu_40_mean[3] #95% 1-mu_60_mean[3]
# #97%

# There is a <7% chance to get a false positive with a
# sample size of 20 sum(d[d$n %in% 20 & d$change_mean %in%
# 0,]$proportion_estimates_below_zero<0.3) #2 values are
# <0.1 (out of 30) length(d[d$n %in% 20 & d$changeprop_mean
# %in% 0,]$proportion_estimates_below_zero) #30 values
# total

 

Figure 13. Risk of false positives and false negatives depending on sample and effect sizes. Curves of the mean ratio of estimates below zero (vertical lines show the minimum and maximum ratios), which illustrates the power we have to detect differences between conditions at different sample sizes. The curves show the model estimates as the effect increases (larger changes in the mean proportion time spent after versus before on the x-axis) for different potential sample sizes (10-60, illustrated by different colors). When there is no change (x value of zero), estimates suggest that, as expected, half of the estimates are below zero because the before and after conditions are not different from each other. As the change increases, the estimates decrease because models are able to reliably tell that the before and after conditions differ from each other.

 

Run this model on the actual data (1.1)

Run the code below to determine whether the manipulated individuals differ from the control individuals in their age of dispersal.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan

# load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

#sets up the data sheet with the simulated data that the model will run on
        dat <- list(
          response = d$DispersalTiming, #for each individual the day at which they dispersed
          condition = d$Condition #1 refers to control and 2 to manipulated
        )
        
        # Our MODEL
        m1 <- ulam(
          alist(
            response ~ dzipois(p,lambda),
            logit(p) <- g_p[condition] ,
            log(lambda) <- g[condition] ,
            # adaptive priors - non-centered which is a parameterization that improves sampling for the MCMC model fit by helping with complex varying effects (see p.453)
            g_p[condition] ~ normal(0,2),
            g[condition] ~ normal(0,2)
          ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))
        
#run a CONTRAST to determine whether there is a difference between before (g1) and after (g2) (subtract g1 before from g2 after)
        post <- extract.samples(m1)
        diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020
        precis( diff_g )

The model: dispersal distance (1.2)

Bayesian model for a gamma-poisson distribution:

\(D_i\) ~ Gamma-Poisson(\(lambda_i\),\(phi_i\))

\(D_i\) is the distance in meters that individuals are found away from their parents’ territory on day 112 after they fledged (measured as a straight line between the center of the natal territory and final location the bird is detected at or the point of death/disappearance), \(lambda_i\) is the mean of the poisson describing the distance individuals who have dispersed are found on average away from their parents’ territory, and \(phi_i\) controls the variance. A gamma-poisson distribution was used because the dispersal distance is a non-negative count, almost all individuals disperse, but the distribution is skewed. Most individuals disperse not very far, but a small proportion can disperse for large distances. These patterns are described by (Richard, 2007), which shows that the average dispersal distance ~1,000m and the maximum dispersal distance is ~10,000m. We assume that distance dispersers move might differ between individuals in the manipulated and the control conditions:

log(\(lambda_i\)) = \(\gamma_l\)[condition]

The \(\gamma\)[condition] reflects the average distance individuals disperse for each condition (control/manipulated) to disperse. The Bayesian model was developed using McElreath (2020b) as a guide.

Power analysis (1.2)

We estimated our power to detect differences between conditions at different sample sizes and with different mean changes in distance individuals disperse (Figure 14). We simulated the distance individuals would disperse in the control condition (with averages of 500m, 1000m, or 1500m), and assumed that manipulated individuals might disperse up to 1000m farther. We analyzed these simulated data with the model we will use to analyze the actual data, estimating the contrast in average distance individuals disperse. From the posterior estimates of the model, we extracted both the mean change as well as the ratio of the posterior estimates that were below zero, indicating that there would be a clear difference between control and manipulated birds.

If the mean ratio of estimates below zero is close to 0.5, the model assumes that the distance individuals disperse is similar for control and manipulated birds. If the ratio is close to zero, the model assumes individuals have changed their behavior and disperse earlier when they were manipulated. We find that for changes smaller than 300m, models are likely to assume that there are no differences between control and manipulated individuals even with large sample sizes. Also for the larger increases in dispersal distance, we would need sample sizes of at least 40 (for a 1000m increase) or 60 individuals (for a 300m increase) to reliably detect differences between control and manipulated individuals. If the contrast in dispersal distance of manipulated compared to control individuals is 500m, on average 93% of the posterior of the model based on a sample size of 60 individuals will be larger than zero (81% with a sample size of 20). This means that the model is quite certain there is a contrast that is larger than zero.

In general, with sample sizes at or above 40 and mean contrasts between manipulated and control individuals of 300m or more, it is likely that the model will indicate that manipulated individuals have changed their behavior. Mean contrasts of less than 300m can still be detected, however there is a high risk that there will be a false negative (67% risk of a false negative with a sample size of 40 when the contrast is 200m).

To estimate the risk of detecting false positives, we set the mean change in age of dispersal for manipulated individuals compared to control individuals to zero so there was no change. As expected, the average ratio of estimates below zero is close to 0.5 and independent of sample size. With a sample size of 40, <7% of the simulations have a ratio smaller than 0.1, meaning that the risk of having a false positive is low.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
rstan_options(auto_write = TRUE)
#rebuild_cmdstan() #run this if you get an error about a PCH file https://discourse.mc-stan.org/t/struggling-to-get-cmdstan-working-on-mac-os-big-sur/19774/4

### SIMULATE the population
### We first start with a single population to identify the parameters for the Gamma-Poission distribution.
# We base this on the information on page 71ff in this thesis on toutouwai: 
# https://mro.massey.ac.nz/bitstream/handle/10179/1596/02_whole.pdf?sequence=1
# It suggests that i) almost all individuals disperse, ii) most individuals do not disperse far: average dispersal distance ~1,000m, and iii) the maximum dispersal distance is ~10,000m. This distribution resembles a gamma-poisson with a mean (lambda) of 1000 and a dispersion (phi) also of 1000
# plot the distribution
hist(rgampois(10000,1000,1000),breaks=100)
# percentage who disperse less than 100m (so are assumed to have remained on the natal territory): 10%
sum(rgampois(10000,1000,1000)<100)/10000

# If manipulated individuals disperse on average 500 meters more, they will have a mean of 1500m rather than 1000m, 2.5% rather than 10% will remain in the natal territory, and the maximum will go up from ~14,000 to ~16,000 meters. 

# Potentially the individuals in this population will not disperse as far or farther than the ones we based the information on. We will accordingly change the dispersal distance for the control population to 500m and to 1500m in the simulation. We will keep the variance parameter the same as the mean (so lambda and phi are both 500, both 1000, or both 1500).


### POWER ANALYSIS: different sample sizes and different mean changes in dispersal distance between control and manipulated conditions
# We now set our range of likely sample sizes, from a minimum of 10 individuals to a maximum of 60 individuals (half in the control and the other half in the manipulated condition). We draw repeated samples of individuals of that size, with 10 repetitions for each of the 5 sample sizes and each of the 6 mean changes. So 10*6 mean change (=60) for each of the 5 sample sizes, resulting in a total length of samplesizes of 300
samplesizes<-c(10,20,30,40,60) #c(10,20,30,40,60)

# Set the means you want to compare (mean change in meters of dispersal distance)
meanchange1<-1000 
meanchange2<-500
meanchange3<-300
meanchange4<-200
meanchange5<-100
meanchange6<-0 #for false positives
changevariants<-c(meanchange1,meanchange2,meanchange3,meanchange4,meanchange5,meanchange6) #for false negatives

#compare different baseline distances for control individuals
mean_distance_control<-c(500,1000,1500)

#set up data frame where results will be stored
simresults <- matrix(NA,nrow=900,ncol=6)
simresults <- data.frame(simresults)
colnames(simresults) <- c("n","change_mean","repetition","distance_control","proportion_estimates_below_zero","count_estimates_below_zero")
counter<-1

# run simulations. There are 5 different sample sizes, and for each we have 6 different mean change in proportions that we want to examine, and each will repeat 10 times for a total of 300 samples.
for (samplesize in 1:length(samplesizes)) {
  N_ind <- samplesizes[samplesize] #number of grackles per sample size
  
  #now run through each of the 6 change variations
  for (propvariant in 1:length(changevariants)) {
    
    #pick the current change variant from the list changevariants
    currentchangeprop <- changevariants[propvariant]
    
    for (baselineprop in 1:3) {
      #pick the baseline distance variant from the list 
      currentprop_before <- mean_distance_control[baselineprop]
      
      for (repetition in 1:10) {
        #create data frame to store for each repetition of the simulated data to be analyzed by the model
   
        #CONTROL condition
        distance_control <- round(rgampois(N_ind/2,currentprop_before,currentprop_before),0) 
        #MANIPULATED condition
        distance_manipulated<- round(rgampois(N_ind/2,currentprop_before+currentchangeprop,currentprop_before),0) 
       
        
        ### run a STAN model to see whether the distance differences are estimated to be different from each other
        #sets up the data sheet with the simulated data that the model will run on
        dat <- list(
          response = c(distance_control,distance_manipulated),
          condition = c(rep(1,N_ind/2),rep(2,N_ind/2)) #1 refers to control and 2 to manipulated
        )
        
        # Our MODEL
        s1 <- ulam(
          alist(
            response ~ dgampois(lambda,phi),
            lambda <- g[condition],
            # fixed priors
            g[condition] ~ normal(1000,1000),
            phi ~ normal(1000,1500)
          ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))
        
        #run a CONTRAST to determine whether there is a difference between before (g1) and after (g2) (subtract g1 before from g2 after)
        post <- extract.samples(s1)
        diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020

        # enter all of the results into the data sheet
        simresults[counter,]$n <- N_ind
        simresults[counter,]$change_mean <- currentchangeprop
        simresults[counter,]$repetition <- repetition
        simresults[counter,]$distance_control <- currentprop_before
        simresults[counter,]$proportion_estimates_below_zero <- sum(diff_g<0)/length(diff_g)
        simresults[counter,]$count_estimates_below_zero <- sum(diff_g<0)
        counter<-counter+1 
        print(c(counter,"out of",10*length(samplesizes)*length(changevariants)*length(distance_control)))
      }
    }
  }
}

#write.csv(simresults,file="simresults20plus.csv") #put the outputs into a csv file so don't have to run the model again...it takes hours

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE)  #makes it so ulam always runs with cmdstan

# load the data sheet that was made from the simulation in
# the previous R chunk
d <- read.csv("https://raw.githubusercontent.com/ManyIndividuals/ManyIndividuals/main/Files/rrs/m1simresults_toutouawidispersaldistance.csv",
    header = TRUE, sep = ",", stringsAsFactors = FALSE)
# head(d)

# Set up the data sheet for the model
dat <- list(proportion_estimates_below_zero = as.numeric(d$proportion_estimates_below_zero),
    count_estimates_below_zero = as.numeric(d$count_estimates_below_zero),
    change_mean = as.numeric(as.factor(d$change_mean)), samplesize = as.integer(as.factor(d$n)))

# We run the STAN model - it is a binomial model that
# assumes that the probability to estimate a difference
# depends on an interaction between the sample size and the
# difference between the conditions
m_samplesize_diffscount <- ulam(alist(count_estimates_below_zero ~
    dbinom(2000, p), logit(p) <- a[samplesize] - b[samplesize] *
    change_mean, a[samplesize] ~ dnorm(a_bar, sigma_a), a_bar ~
    dnorm(0, 1.5), sigma_a ~ dexp(1), b[samplesize] ~ dnorm(0,
    1.5)), data = dat, log_lik = TRUE, messages = FALSE, cmdstan = T)

# Extract the posterior estimates from this STAN model
precis_output <- precis(m_samplesize_diffscount, depth = 2)

# Plot the predictions for the different sample sizes as
# separate lines = COUNT out of 2000
plot(NULL, xlim = c(0.5, 6.5), ylim = c(0, 0.8), pch = 1, ann = F,
    frame.plot = F, cex = 1.5, xaxt = "n")
axis(side = 1, at = c(1, 2, 3, 4, 5, 6), labels = c(0, 100, 200,
    300, 500, 1000))

mu_10 <- link(m_samplesize_diffscount, data = data.frame(change_mean = as.numeric(as.factor(unique(d$change_mean))),
    samplesize = rep(1, length(unique(d$change_mean)))))
mu_10_mean <- apply(mu_10, 2, mean)
mu_10_ci <- apply(mu_10, 2, PI, prob = 0.97)
shade(mu_10_ci, as.numeric(as.factor(unique(d$change_mean))),
    col = col.alpha("blue", 0.7))
lines(x = c(0.96, 0.96), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(1.96, 1.96), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    100, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 100, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(2.96, 2.96), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    200, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(3.96, 3.96), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    300, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(4.96, 4.96), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    500, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$change_mean %in% 500, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(5.96, 5.96), y = c(min(d[d$n %in% 10 & d$change_mean %in%
    1000, ]$proportion_estimates_below_zero), max(d[d$n %in%
    10 & d$change_mean %in% 1000, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))

mu_20 <- link(m_samplesize_diffscount, data = data.frame(change_mean = as.numeric(as.factor(unique(d$change_mean))),
    samplesize = rep(2, length(unique(d$change_mean)))))
mu_20_mean <- apply(mu_20, 2, mean)
mu_20_ci <- apply(mu_20, 2, PI, prob = 0.97)
shade(mu_20_ci, as.numeric(as.factor(unique(d$change_mean))),
    col = col.alpha("orange", 0.7))
lines(x = c(0.98, 0.98), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(1.98, 1.98), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    100, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 100, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(2.98, 2.98), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    200, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(3.98, 3.98), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    300, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(4.98, 4.98), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    500, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$change_mean %in% 500, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(5.98, 5.98), y = c(min(d[d$n %in% 20 & d$change_mean %in%
    1000, ]$proportion_estimates_below_zero), max(d[d$n %in%
    20 & d$change_mean %in% 1000, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))

mu_30 <- link(m_samplesize_diffscount, data = data.frame(change_mean = as.numeric(as.factor(unique(d$change_mean))),
    samplesize = rep(3, length(unique(d$change_mean)))))
mu_30_mean <- apply(mu_30, 2, mean)
mu_30_ci <- apply(mu_30, 2, PI, prob = 0.97)
shade(mu_30_ci, as.numeric(as.factor(unique(d$change_mean))),
    col = col.alpha("skyblue", 0.7))
lines(x = c(1, 1), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(2, 2), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    100, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 100, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(3, 3), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    200, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(4, 4), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    300, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(5, 5), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    500, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$change_mean %in% 500, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(6, 6), y = c(min(d[d$n %in% 30 & d$change_mean %in%
    1000, ]$proportion_estimates_below_zero), max(d[d$n %in%
    30 & d$change_mean %in% 1000, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))

mu_40 <- link(m_samplesize_diffscount, data = data.frame(change_mean = as.numeric(as.factor(unique(d$change_mean))),
    samplesize = rep(4, length(unique(d$change_mean)))))
mu_40_mean <- apply(mu_40, 2, mean)
mu_40_ci <- apply(mu_40, 2, PI, prob = 0.97)
shade(mu_40_ci, as.numeric(as.factor(unique(d$change_mean))),
    col = col.alpha("#213940", 0.7))
lines(x = c(1.02, 1.02), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(2.02, 2.02), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    100, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 100, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(3.02, 3.02), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    200, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(4.02, 4.02), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    300, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(5.02, 5.02), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    500, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$change_mean %in% 500, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(6.02, 6.02), y = c(min(d[d$n %in% 40 & d$change_mean %in%
    1000, ]$proportion_estimates_below_zero), max(d[d$n %in%
    40 & d$change_mean %in% 1000, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))

mu_60 <- link(m_samplesize_diffscount, data = data.frame(change_mean = as.numeric(as.factor(unique(d$change_mean))),
    samplesize = rep(5, length(unique(d$change_mean)))))
mu_60_mean <- apply(mu_60, 2, mean)
mu_60_ci <- apply(mu_60, 2, PI, prob = 0.97)
shade(mu_60_ci, as.numeric(as.factor(unique(d$change_mean))),
    col = col.alpha("red", 0.7))
lines(x = c(1.04, 1.04), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(2.04, 2.04), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    100, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 100, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(3.04, 3.04), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    200, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(4.04, 4.04), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    300, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 300, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(5.04, 5.04), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    500, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$change_mean %in% 500, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(6.04, 6.04), y = c(min(d[d$n %in% 60 & d$change_mean %in%
    1000, ]$proportion_estimates_below_zero), max(d[d$n %in%
    60 & d$change_mean %in% 1000, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))

legend(x = "topright", legend = c(lty2 = "10", lty1 = "20", lty3 = "30",
    lty2 = "40", lty3 = "60"), lty = c(1, 1, 1, 1, 1), lwd = c(5,
    5, 5, 5, 5), col = c(col.alpha("blue", 0.4), col.alpha("orange",
    0.4), col.alpha("skyblue", 0.4), col.alpha("#213940", 0.4),
    col.alpha("red", 0.4)), cex = 1, bty = "n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size", side = 3, line = -1, at = 5.5)
mtext("Power to detect differences between conditions", side = 3,
    line = 1, at = 3.5, cex = 1.5)
mtext("Increase in dispersal distance (meters) of manipulated vs control individuals",
    side = 1, line = 3, cex = 1.2)
mtext("Ratio of estimates below zero", side = 2, line = 3, cex = 1.2)

# 93% of the posterior mass is above zero for a 500m or
# larger mean change between conditions when the sample
# size is 60, 86% for n=40, 83% for n=30, and 81% for n=20
# mu_10_ci #82%. 3rd column from the right (column 4) for
# the 0.1 category on the x axis (ordered by
# unique(d$changeprop_mean)) 1-mu_60_mean[2]
# 1-mu_30_mean[2]

# sum(d[d$n %in% 40 & d$change_mean %in%
# 0,]$proportion_estimates_below_zero<0.1) #2 values are
# <0.1 (out of 30) length(d[d$n %in% 40 & d$change_mean
# %in% 0,]$proportion_estimates_below_zero) #30 values
# total

 

Figure 14 Risk of false positives and false negatives depending on sample and effect sizes. Curves of the mean ratio of estimates below zero (vertical lines show the minimum and maximum ratios), which illustrates the power we have to detect differences between conditions at different sample sizes. The curves show the model estimates as the effect increases (larger changes in the mean proportion time spent after versus before on the x-axis) for different potential sample sizes (10-60, illustrated by different colors). When there is no change (x value of zero), estimates suggest that, as expected, half of the estimates are below zero because the before and after conditions are not different from each other. As the change increases, the estimates decrease because models are able to reliably tell that the before and after conditions differ from each other.

 

Run this model on the actual data (1.2)

Run the code below to determine whether there were differences between the control and manipulated conditions in the distance of dispersal.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan

# load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

# Set up the data for the analysis. This assumes you have a variable "dispersaldistance" that reports for each individual how many meters they dispersed, and a variable "condition" that indicates whether the individual is part of the control or the manipulated group 
dat <- list(
          response = as.numeric(d$dispersaldistance),
          condition = as.numeric(as.factor(d$condition)) #1 refers to control and 2 to manipulated
        )
        
# Our MODEL
        s1 <- ulam(
          alist(
            response ~ dgampois(lambda,phi),
            lambda <- g[condition],
            # fixed priors
            g[condition] ~ normal(1000,1000),
            phi ~ normal(1000,1500)
          ) , data=dat , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))

#model output: pay attention to control (g1) and manipulated (g2)
#precis(m1,depth=2)

#run a CONTRAST to determine whether there is a difference between control (g1) and manipulated (g2)
post <- extract.samples(m1)
diff_g <- post$g[,1] - post$g[,2] #log-odds difference p.341 McElreath 2020
precis( diff_g )

The model: dispersal location (1.3)

location ~ dbinom(1,p)

logit(p) <- g[condition]

The response variable is dispersal location (0=inside Zealandia, 1=outside Zealandia) on day 112 post-fledging. g[condition] is the average log odds for each condition (control or flexibility manipulated). A binomial distribution was used because the response variable is binomial (see Figure 10.6 in McElreath, 2020b). The Bayesian model was developed using McElreath (2020b) as a guide.

Power analysis (1.3)

We estimated our power to detect differences between conditions at different sample sizes and with different mean changes in the dispersal location in the control vs. manipulated conditions (Figure 15). We analyzed simulated data with the model we will use to analyze the actual data, estimating the change in dispersal location between the conditions. From the posterior estimates of the model, we extracted both the mean change as well as the ratio of the posterior estimates that were below zero.

If the mean ratio of estimates below zero is close to 0.5, the model assumes that the change in the number of food types taken before the flexibility manipulation is similar to after. If the ratio is close to zero, the model assumes individuals have changed their behavior. For differences in the dispersal location at or larger than 0.3, models are likely to assume that no changes occurred even with large sample sizes. If the difference in dispersal location between control and manipulated conditions is 0.3, on average 92% of the posterior of the model based on a sample size of 40 individuals will be larger than zero. This means that the model is quite certain there is a difference that is larger than zero.

In general, with sample sizes at or above 40 and mean differences in the dispersal location of 0.3 or larger, it is likely that the model will indicate that the flexibility experiment influenced their behavior in a way that affected their dispersal location. Mean changes below 0.3 can still be detected, however there is a higher risk that there will be a false negative. For example, 17% of the models for a sample size of 40 at the mean change of 0.2 have a ratio larger than 0.3, meaning there is a large risk of having a false negative.

With small mean changes in the response variable, some individuals might not increase or even decrease their response after the manipulation because there is variation around the mean change in individual responses. With small sample sizes, there is a risk that only individuals who did not clearly increase their response will be studied, whereas larger sample sizes are more likely to include a wider spectrum of individuals.

To estimate the risk of detecting false positives, we set the mean change in survival to zero so there was no change between the conditions. As expected, the average ratio of estimates below zero is close to 0.5 and independent of sample size. With a sample size of 40, 60% have a ratio smaller than 0.3, meaning that the risk of having a false positive is high.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE)  #makes it so ulam always runs with cmdstan
rstan_options(auto_write = TRUE)
# rebuild_cmdstan() #run this if you get an error about a
# PCH file
# https://discourse.mc-stan.org/t/struggling-to-get-cmdstan-working-on-mac-os-big-sur/19774/4

# SIMULATE the population
N_ind <- 30  #number of individuals tested

# choosing the mean change in location between control and
# manipulated groups and standard deviations - visualize to
# see whether it would be possible to tell the difference
# between control (black) and manipulated (red)
prob_control <- 0.6
changemanip <- 0.1
plot(density(rnorm(10000, prob_control, sd = 0.1)), lwd = 2,
    xlim = c(0, 1.2))
lines(density(rnorm(10000, prob_control + (1 - prob_control) *
    (changemanip * 2), sd = 0.1)), lwd = 2, col = "red")

# control
probcontrol <- round(rbinom(n = N_ind, size = 1, prob = 0.6),
    2)  #for 20 data points, randomly assign a 0 (inside) or 1 (outside) location such that each appears 60% of the time
changemanip <- rnorm(N_ind, mean = (1 - probcontrol)/5, sd = 0.1)  #if had a large proportion for control, have a smaller change relative to the manipulated condition. Divide by 2 = assume that on average manipulated will disperse outside 0.25 (the control mean of 0.5 divided by 2)

# manip
probmanip <- round(probcontrol + changemanip, 2)
probmanip[probmanip > 1] <- 1  #values larger than 1 mess up the model so restrict it

# check the plot to see whether manip are more likely to
# disperse outside plot(probcontrol,probmanip) abline(0,1)


### POWER ANALYSIS: different sample sizes and different
### mean changes in proportions between before and after
### conditions We now set our range of likely sample sizes
### from 10 to 60 individuals per population. We draw
### repeated samples of individuals of that size from each
### of the 5 populations, during each repetition we draw
### two samples from each population (so we can infer the
### false negative rate of wrongly estimating that two
### samples come from a different population even though
### they were taken from the sample population). This means
### we have 10 repetitions for each of the 5 sample sizes
### and each of the 6 changeprops. So 10*6 changeprops
### (=60) for each of the 5 sample sizes, resulting in a
### total length of samplesizes of 300
samplesizes <- c(10, 20, 30, 40, 60)  #c(10,20,30,40,60)

# Set the means you want to compare
meanchangeprop1 <- 0.3
meanchangeprop2 <- 0.2
meanchangeprop3 <- 0.1
meanchangeprop4 <- 0.05
meanchangeprop5 <- 0.025
meanchangeprop6 <- 0  #for false positives
changeprobvariants <- c(meanchangeprop1, meanchangeprop2, meanchangeprop3,
    meanchangeprop4, meanchangeprop5, meanchangeprop6)  #for false negatives

# compare different control individual location
# probabilities
prob_control <- c(0.6, 0.7, 0.8)  #expected baseline location probability of being inside Zealandia

# set up data frame where results will be stored
simresults <- matrix(NA, nrow = 900, ncol = 6)
simresults <- data.frame(simresults)
colnames(simresults) <- c("n", "changeprob_mean", "repetition",
    "proportion_inside", "proportion_estimates_below_zero", "count_estimates_below_zero")
counter <- 1

# run simulations. There are 5 different sample sizes, and
# for each we have 6 different mean change in proportions
# that we want to examine, and each will repeat 10 times
# for a total of 300 samples.
for (samplesize in 1:length(samplesizes)) {
    N_ind <- samplesizes[samplesize]  #number of grackles per sample size

    # now run through each of the 6 proportion variations
    for (probvariant in 1:length(changeprobvariants)) {

        # pick the current proportion variant from the list
        # changepropvariants
        currentchangeprob <- changeprobvariants[probvariant]

        for (baselineprob in 1:3) {
            # pick the current proportion variant from the
            # list changepropvariants
            currentprob_control <- prob_control[baselineprob]

            for (repetition in 1:10) {
                # CONTROL & MANIP groups
                probcontrol <- round(rbinom(n = N_ind/2, size = 1,
                  prob = currentprob_control), 2)  #for half the data points, randomly assign a 0 (inside Zealandia) or 1 (outside Zealandia) according to the baseline location probabilities (0.5,0.6,0.7)
                probmanip <- round(rbinom(n = N_ind/2, size = 1,
                  prob = (currentprob_control - currentchangeprob)),
                  2)  #for the other half of the data points, starting from the probability that the control individuals stay inside Zealandia (0.5, 0.6, and 0.7), we are decreasing the probability that they remain in Zealandia (add zero) for the manipulated individuals (add meanchangeprop 1 through 6)

                ### run a STAN model to see whether the
                ### conditions are estimated to be
                ### different from each other sets up the
                ### data sheet with the simulated data that
                ### the model will run on
                dat2 <- list(location = c(probcontrol, probmanip),
                  condition = c(rep(1, N_ind/2), rep(2, N_ind/2))  #1=inside zealandia, 2=outside zealandia
)

                # The MODEL
                s1 <- ulam(alist(location ~ dbinom(1, p), logit(p) <- g[condition],
                  g[condition] ~ normal(0, 1)), data = dat2,
                  chains = 4, cores = 4, log_lik = TRUE, cmdstan = TRUE,
                  control = list(adapt_delta = 0.95, force_recompile = TRUE))

                # run a CONTRAST to determine whether there
                # is a difference between before (g1) and
                # after (g2) (subtract g1 before from g2
                # after)
                post <- extract.samples(s1)
                diff_g <- post$g[, 2] - post$g[, 1]  #log-odds difference p.341 McElreath 2020
                diff_p <- inv_logit(post$g[, 2]) - inv_logit(post$g[,
                  1])  #on the probability scale p.341 McElreath 2020
                # precis( list( diff_g=diff_g ,
                # diff_p=diff_p ) )

                # enter all of the results into the data
                # sheet
                simresults[counter, ]$n <- N_ind
                simresults[counter, ]$changeprob_mean <- currentchangeprob
                simresults[counter, ]$repetition <- repetition
                simresults[counter, ]$proportion_inside <- currentprob_control
                simresults[counter, ]$proportion_estimates_below_zero <- sum(diff_p <
                  0)/length(diff_p)
                simresults[counter, ]$count_estimates_below_zero <- sum(diff_p <
                  0)
                counter <- counter + 1
                print(c(counter, "out of", 10 * length(samplesizes) *
                  length(changeprobvariants) * length(prob_control)))
            }
        }
    }
}
# write.csv(simresults,file='simresultsTouLocation.csv')
# #put the outputs into a csv file so don't have to run
# model again - it takes about 2.5 hours

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE)  #makes it so ulam always runs with cmdstan

# load the data sheet that was made from the simulation in
# the previous R chunk
d <- read.csv("https://raw.githubusercontent.com/ManyIndividuals/ManyIndividuals/main/Files/rrs/mi1simresultstoulocation.csv",
    header = TRUE, sep = ",", stringsAsFactors = FALSE)
# head(d)

# make the data comparable with the other graphs by
# subtracting from the total
d$proportion_estimates_below_zero <- 1 - d$proportion_estimates_below_zero
d$count_estimates_below_zero <- 2000 - as.numeric(d$count_estimates_below_zero)

# Set up the data sheet for the model
dat <- list(proportion_estimates_below_zero = as.numeric(d$proportion_estimates_below_zero),
    count_estimates_below_zero = as.numeric(d$count_estimates_below_zero),
    changeprop_mean = as.numeric(d$changeprob_mean), samplesize = as.integer(as.factor(d$n)))

# We run the STAN model - it is a binomial model that
# assumes that the probability to estimate a difference
# depends on an interaction between the sample size and the
# difference between the conditions Condition
m_samplesize_diffscount <- ulam(alist(count_estimates_below_zero ~
    dbinom(2000, p), logit(p) <- a[samplesize] + b[samplesize] *
    changeprop_mean, a[samplesize] ~ dnorm(a_bar, sigma_a), a_bar ~
    dnorm(0, 1.5), sigma_a ~ dexp(1), b[samplesize] ~ dnorm(0,
    1.5)), data = dat, log_lik = TRUE, messages = FALSE, cmdstan = T)

# Extract the posterior estimates from this STAN model
precis_output <- precis(m_samplesize_diffscount, depth = 2)  #b gets more negative with increasing sample size as expected

# Plot the predictions for the different sample sizes as
# separate lines = COUNT out of 2000
plot(NULL, xlim = c(-0.01, 0.31), ylim = c(0, 0.6), pch = 1,
    ann = F, frame.plot = F, cex = 1.5)

mu_10 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(1, length(unique(d$changeprob_mean)))))
mu_10_mean <- apply(mu_10, 2, mean)
mu_10_ci <- apply(mu_10, 2, PI, prob = 0.97)
shade(mu_10_ci, unique(d$changeprob_mean), col = col.alpha("blue",
    0.7))
lines(x = c(-0.008, -0.008), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.022, 0.022), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.044, 0.044), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.092, 0.092), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.192, 0.192), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.292, 0.292), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 10 &
    d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("blue", 0.5))

mu_20 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(2, length(unique(d$changeprob_mean)))))
mu_20_mean <- apply(mu_20, 2, mean)
mu_20_ci <- apply(mu_20, 2, PI, prob = 0.97)
shade(mu_20_ci, unique(d$changeprob_mean), col = col.alpha("orange",
    0.7))
lines(x = c(-0.004, -0.004), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.024, 0.024), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.048, 0.048), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.096, 0.096), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.196, 0.196), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.296, 0.296), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 20 &
    d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("orange", 0.5))

mu_30 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(3, length(unique(d$changeprob_mean)))))
mu_30_mean <- apply(mu_30, 2, mean)
mu_30_ci <- apply(mu_30, 2, PI, prob = 0.97)
shade(mu_30_ci, unique(d$changeprob_mean), col = col.alpha("skyblue",
    0.7))
lines(x = c(0, 0), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.028, 0.028), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.052, 0.052), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.1, 0.1), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.2, 0.2), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.3, 0.3), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 30 &
    d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("skyblue", 0.5))

mu_40 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(4, length(unique(d$changeprob_mean)))))
mu_40_mean <- apply(mu_40, 2, mean)
mu_40_ci <- apply(mu_40, 2, PI, prob = 0.97)
shade(mu_40_ci, unique(d$changeprob_mean), col = col.alpha("#213940",
    0.7))
lines(x = c(0.004, 0.004), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.032, 0.032), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.056, 0.056), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.104, 0.104), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.204, 0.204), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.304, 0.304), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 40 &
    d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("#213940", 0.5))

mu_60 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(5, length(unique(d$changeprob_mean)))))
mu_60_mean <- apply(mu_60, 2, mean)
mu_60_ci <- apply(mu_60, 2, PI, prob = 0.97)
shade(mu_60_ci, unique(d$changeprob_mean), col = col.alpha("red",
    0.7))
lines(x = c(0.008, 0.008), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.036, 0.036), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.06, 0.06), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.108, 0.108), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.208, 0.208), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))
lines(x = c(0.308, 0.308), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero), max(d[d$n %in% 60 &
    d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero)),
    col = col.alpha("red", 0.5))

legend(x = "topright", legend = c(lty2 = "10", lty1 = "20", lty3 = "30",
    lty2 = "40", lty3 = "60"), lty = c(1, 1, 1, 1, 1), lwd = c(5,
    5, 5, 5, 5), col = c(col.alpha("blue", 0.4), col.alpha("orange",
    0.4), col.alpha("skyblue", 0.4), col.alpha("#213940", 0.4),
    col.alpha("red", 0.4)), cex = 1, bty = "n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size", side = 3, line = -0.7, at = 0.31)
mtext("Power to detect differences between conditions", side = 3,
    line = 1, at = 0.15, cex = 1.5)
mtext("Mean difference in dispersal location (control vs. manipulated)",
    side = 1, line = 3, cex = 1.2)
mtext("Ratio of estimates below zero", side = 2, line = 3, cex = 1.2)

# 95% of the posterior mass is above zero for a 0.3 or
# larger mean difference between conditions when the sample
# size is 60
mu_10_ci  #0.28. Column 1 for the 0.3 category on the x axis (ordered by unique(d$changeprob_mean))
mu_20_ci  #0.20
mu_30_ci  #0.14
mu_40_ci  #0.08
mu_60_ci  #0.06
sum(d[d$n %in% 40 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero <
    0.3)  #30/30 values are <0.3 when mean change=0.3; 18/30 when mean change = 0, 25/30 when mean change = 0.2
length(d[d$n %in% 40 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero)  #30 values total

 

Figure 15. Risk of false positives and false negatives depending on sample and effect sizes. Curves of the mean ratio of estimates below zero (vertical lines show the minimum and maximum ratios), which illustrates the power we have to detect differences between conditions at different sample sizes. The curves show the model estimates as the effect increases (larger changes in the mean proportion time spent after versus before on the x-axis) for different potential sample sizes (10-60, illustrated by different colors). When there is no change (x value of zero), estimates suggest that, as expected, half of the estimates are below zero because the before and after conditions are not different from each other. As the change increases, the estimates decrease because models are able to reliably tell that the before and after conditions differ from each other.

 

Run this model on the actual data (1.3)

Run the code below to determine whether there were differences between the control and manipulated conditions in their survival status (alive or dead when individuals are or would be 112 days old).

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
rstan_options("auto_write" = TRUE)
#rebuild_cmdstan() #run this if you get an error about a PCH file https://discourse.mc-stan.org/t/struggling-to-get-cmdstan-working-on-mac-os-big-sur/19774/4

# load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

# Set up the data sheet
dat <- list(
    status = d$status,
    condition = d$condition
)

#simulated data to test the model
dat2 <- list(
    location = rbinom(40,1,0.5), #for 40 data points, randomly assign a 0 (inside) or 1 (outside) dispersal location such that each appears 50% of the time
    condition = c(rep(1,20),rep(2,20)) #for all rows with control data, assign a condition=1, and for all rows with manipulated data, assign a condition=2, repeat each 20 times (for a total sample size of 40)
)

# THE MODEL (p.448 for binomial distribution - McElreath 2020)
#g[condition] = avg log odds for each condition
mtl <- ulam(
    alist(
        location ~ dbinom(1,p),
        logit(p) <- g[condition],
        g[condition] ~ normal(0,1)
    ) , data=dat2 , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))

#model output: pay attention to before (g1) and after (g2)
#precis(mtl,depth=2)

#run a CONTRAST to determine whether there is a difference in survival status between control (g1) and manipulated (g2)
post <- extract.samples(mtl)
diff_g <- post$g[,1] - post$g[,2] #log-odds difference p.341
diff_p <- inv_logit(post$g[,1]) - inv_logit(post$g[,2]) #on the probability scale p.341
precis( list( diff_g=diff_g , diff_p=diff_p ) )

T.Q2 Compared to control individuals, are flexibility manipulated individuals more likely to survive their first 16 weeks post-fledging, particularly if they disperse into the urban reserves outside the sanctuary fence?

The model

Bayesian models with a binomial distribution:

Examining the difference between conditions:

status ~ dbinom(1,p)

logit(p) <- g[condition]

The response variable is survival status (0=dead, 1=alive) on day 112 post-fledging. g[condition] is the average log odds for each condition (control or flexibility manipulated). A binomial distribution was used because the response variable is binomial (see Figure 10.6 in McElreath, 2020b). The Bayesian model was developed using McElreath (2020b) as a guide.

Examining the difference between conditions in each of the two areas:

status ~ dbinom(1,p)

logit(p) <- g[habitat] + \(\beta\)[habitat,condition]

The response variable is survival status (0=dead, 1=alive) on day 112 post-fledging. g[habitat] is the average log odds for each area (inside Zealandia, which is a natural area, or outside Zealandia, which can include natural/suburban/and urban areas), and \(\beta\)[habitat,condition] is an effect for each area (inside Zealandia or outside Zealandia) in each condition (control or flexibility manipulated).

Power analysis

We estimated our power to detect differences between conditions at different sample sizes and with different mean changes in the survival status in the control vs. manipulated conditions (Figure 16). We analyzed simulated data with the model we will use to analyze the actual data, estimating the change in survival status between the conditions. From the posterior estimates of the model, we extracted both the mean change as well as the ratio of the posterior estimates that were below zero.

If the mean ratio of estimates below zero is close to 0.5, the model assumes that the change in the number of food types taken before the flexibility manipulation is similar to after. If the ratio is close to zero, the model assumes individuals have changed their behavior. For differences in the survival status at or larger than 0.3, models are likely to assume that no changes occurred even with large sample sizes. If the difference in survival status between control and manipulated conditions is 0.3, on average 88% of the posterior of the model based on a sample size of 30 individuals will be larger than zero. This means that the model is quite certain there is a difference that is larger than zero.

In general, with sample sizes at or above 30 and mean differences in the survival status of 0.3 or larger, it is likely that the model will indicate that the flexibility experiment influenced their behavior in a way that affected their survival. Mean changes below 0.3 can still be detected, however there is a higher risk that there will be a false negative. For example, 17% of the models for a sample size of 30 at the mean change of 0.2 have a ratio larger than 0.3, meaning there is a large risk of having a false negative.

With small mean changes in the response variable, some individuals might not increase or even decrease their response after the manipulation because there is variation around the mean change in individual responses. With small sample sizes, there is a risk that only individuals who did not clearly increase their response will be studied, whereas larger sample sizes are more likely to include a wider spectrum of individuals.

To estimate the risk of detecting false positives, we set the mean change in survival to zero so there was no change between the conditions. As expected, the average ratio of estimates below zero is close to 0.5 and independent of sample size. With a sample size of 30, 17% have a ratio smaller than 0.3, meaning that the risk of having a false positive is high.

We also estimated the power to detect differences between conditions in different habitats (inside or outside Zealandia) at different sample sizes and with different mean changes in the survival status (Figure 17). We found that mean changes in survival status of 0.3 are likely to be detected with a sample size of at least 40 when considering the individuals inside Zealandia, but we will not be able to detect differences outside of Zealandia because the subset sample size will likely be too small.

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
rstan_options("auto_write" = TRUE)
#rebuild_cmdstan() #run this if you get an error about a PCH file https://discourse.mc-stan.org/t/struggling-to-get-cmdstan-working-on-mac-os-big-sur/19774/4

#SIMULATE the population
N_ind <- 20 #number of individuals tested

#choosing the mean change in survival between control and manipulated groups and standard deviations - visualize to see whether it might be possible to tell the difference between control (black) and manipulated (red)
prob_control<-0.6
changemanip<-0.1
plot(density(rnorm(10000,prob_control,sd=0.1)),lwd=2,xlim=c(0,1.2))
lines(density(rnorm(10000,prob_control+(1-prob_control)*(changemanip*2),sd=0.1)),lwd=2,col="red")

#control
probcontrol <- round(rbinom(n=N_ind,size=1,prob=0.6),2) #for 20 data points, randomly assign a 0 (dead) or 1 (alive) survival status such that each appears 60% of the time
changemanip <- rnorm(N_ind, mean=(1-probcontrol)/5,sd=0.1) #if had a large proportion before, have a smaller change relative to the control condition. Divide by 2 = assume that on average manipulated is more likely to survive by 0.25 (the control mean of 0.5 divided by 2)

#manip
probmanip <- round(probcontrol+changemanip,2)
probmanip[probmanip>1]<-1 #values larger than 1 mess up the model so restrict it

#check the plot to see whether manip increases prob of survival
#plot(probcontrol,probmanip)
#abline(0,1)


### POWER ANALYSIS: different sample sizes and different mean changes in conditions
# We now set our range of likely sample sizes, from a minimum of 10 individuals per population to a maximum of 60 individuals per population. We draw repeated samples of individuals of that size from each of the 5 populations, during each repetition we draw two samples from each population (so we can infer the false negative rate of wrongly estimating that two samples come from a different population even though they were taken from the sample population). This means we have 10 repetitions for each of the 5 sample sizes and each of the 6 changeprops. So 10*6 changeprops (=60) for each of the 5 sample sizes, resulting in a total length of samplesizes of 300
samplesizes<-c(10,20,30,40,60)

# Set the means you want to compare
meanchangeprop1<-0.3 
meanchangeprop2<-0.2
meanchangeprop3<-0.1 
meanchangeprop4<-0.05
meanchangeprop5<-0.025
meanchangeprop6<-0 #for false positives
changeprobvariants<-c(meanchangeprop1,meanchangeprop2,meanchangeprop3,meanchangeprop4,meanchangeprop5,meanchangeprop6) #for false negatives

#compare different control individual survival probabilities
prob_control<-c(0.5,0.6,0.7) #expected baseline survival in the natural area. We assume that it will be half of this in the suburban areas

#set up data frame where results will be stored
simresults <- matrix(NA,nrow=900,ncol=9)
simresults <- data.frame(simresults)
colnames(simresults) <- c("n","changeprob_mean","n_suburban","repetition","survival_control","proportion_estimates_below_zero_condition","count_estimates_below_zero_condition","proportion_estimates_below_zero_habitatnatural","proportion_estimates_below_zero_habitatsuburban")
counter<-1
 
# run simulations. There are 5 different sample sizes, and for each we have 6 different mean change in proportions that we want to examine, and each will repeat 10 times for a total of 300 samples.
for (samplesize in 1:length(samplesizes)) {
        N_ind <- samplesizes[samplesize] #number of grackles per sample size
        
        #now run through each of the 6 proportion variations
        for (probvariant in 1:length(changeprobvariants)) {
        
        #pick the current proportion variant from the list changepropvariants
        currentchangeprob <- changeprobvariants[probvariant]
        
        for (baselineprob in 1:3) {
        #pick the current proportion variant from the list changepropvariants
        currentprob_control <- prob_control[baselineprob]
        
        for (repetition in 1:10) {
        #CONTROL & MANIP groups
        habitatcontrol <- rbinom(N_ind/2,1,0.25)+1  
        probcontrol <- round(rbinom(n=N_ind/2,size=1,prob=currentprob_control/habitatcontrol),2) #for half the data points, randomly assign a 0 (dead) or 1 (alive) survival status according to the baseline survival probabilities (0.5,0.6,0.7)
        
        habitatmanip <- rbinom(N_ind/2,1,0.25)+1  
        probmanip <- round(rbinom(n=N_ind/2,size=1,prob=(currentprob_control/habitatmanip)+currentchangeprob),2) #for the other half the data points, randomly assign a 0 (dead) or 1 (alive) survival status according to the baseline survival probabilities (0.5,0.6,0.7)

        ### run a STAN model to see whether the condition proportions are estimated to be different from each other
        #sets up the data sheet with the simulated data that the model will run on
  dat2 <- list(
    status = c(probcontrol,probmanip),
    habitat = c(habitatcontrol,habitatmanip), #75% of individuals in the reserve, 25% in suburban randomly assigned from control and manip groups. 1=reserve, 2=suburban
    condition = c(rep(1,N_ind/2),rep(2,N_ind/2))
)
       
        # Our MODEL
s1 <- ulam(
    alist(
        status ~ dbinom(1,p),
        logit(p) <- g[condition],
        g[condition] ~ normal(0,1)
       ) , data=dat2 , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))

        #run a CONTRAST to determine whether there is a difference between before (g1) and after (g2) (subtract g1 before from g2 after)
post <- extract.samples(s1)
diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020
diff_p <- inv_logit(post$g[,2]) - inv_logit(post$g[,1]) #on the probability scale p.341 McElreath 2020
#precis( list( diff_g=diff_g , diff_p=diff_p ) )      

s2 <- ulam(
    alist(
        status ~ dbinom(1,p),
        logit(p) <- g[habitat] + beta[condition,habitat],
        # adaptive priors - non-centered which is a parameterization that improves sampling for the MCMC model fit by helping with complex varying effects (see p.453)
        transpars> matrix[condition,2]:beta <- compose_noncentered( sigma_habitat , L_Rho_habitat , z_habitat ),
        matrix[2,condition]:z_habitat ~ normal( 0 , 1 ),
        # fixed priors
        g[condition] ~ normal(0,1),
        vector[2]:sigma_habitat ~ dexp(1),
        cholesky_factor_corr[2]:L_Rho_habitat ~ lkj_corr_cholesky( 2 ),
        # compute ordinary correlation matrixes from Cholesky factors
        gq> matrix[2,2]:Rho_habitat <<- Chol_to_Corr(L_Rho_habitat)
    ) , data=dat2 , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))

      #run a CONTRAST to determine whether there is a difference in survival status between control (g1) and manipulated (g2) and habitat (reserve vs suburban)
post2 <- extract.samples(s2)
if(length(unique(dat2$habitat))>1){
  diff_betanatural <- inv_logit(post2$beta[,2,1]) - inv_logit(post2$beta[,1,1]) # Comparison for habitat 1 (natural): take the inverse logit of the sum (posterior control habitat 1 + posterior control) minus the inverse logit of the sum (posterior manipulated habitat 1 + posterior manipulated). For the beta values, the first entry in the square brackets refers to the rows (we want all of them so we leave this empty), the second entry refers to control (1) versus manipulated (2), and the third entry refers to the habitat (here habitat 1)
diff_betasuburban <- inv_logit(post2$beta[,2,2]) - inv_logit(post2$beta[,1,2]) # Comparison for habitat 2 (suburban): take the inverse logit of the sum (posterior control habitat 2 + posterior control) minus the inverse logit of the sum (posterior manipulated habitat 2 + posterior manipulated). For the beta values, the first entry in the square brackets refers to the rows (we want all of them so we leave this empty), the second entry refers to control (1) versus manipulated (2), and the third entry refers to the habitat (here habitat 2)
}
    # enter all of the results into the data sheet
    simresults[counter,]$n <- N_ind
    simresults[counter,]$changeprob_mean <- currentchangeprob
    simresults[counter,]$n_suburban <- sum(dat2$habitat==2)
    simresults[counter,]$repetition <- repetition
    simresults[counter,]$survival_control <- currentprob_control
    simresults[counter,]$proportion_estimates_below_zero_condition <- sum(diff_p<0)/length(diff_p)
    simresults[counter,]$count_estimates_below_zero_condition <- sum(diff_p<0)
     if(length(unique(dat2$habitat))>1){simresults[counter,]$proportion_estimates_below_zero_habitatnatural <-sum(diff_betanatural<0)/length(diff_betanatural)}else{simresults[counter,]$proportion_estimates_below_zero_habitatnatural <-"NA"}
     if(length(unique(dat2$habitat))>1){simresults[counter,]$proportion_estimates_below_zero_habitatsuburban <-sum(diff_betasuburban<0)/length(diff_betasuburban)}else{simresults[counter,]$proportion_estimates_below_zero_habitatsuburban <-"NA"}
    counter<-counter+1 
    print(c(counter,"out of",10*length(samplesizes)*length(changeprobvariants)*length(prob_control)))
        }
        }
        }
}
#write.csv(simresults,file="simresultsTouSurvival.csv") #put the outputs into a csv file so don't have to run model again - it takes about 9 hours

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE)  #makes it so ulam always runs with cmdstan

# load the data sheet that was made from the simulation in
# the previous R chunk
d <- read.csv("https://raw.githubusercontent.com/ManyIndividuals/ManyIndividuals/main/Files/rrs/mi1simresultstousurvival.csv",
    header = TRUE, sep = ",", stringsAsFactors = FALSE)
# head(d)
d_noNA <- d[complete.cases(d$proportion_estimates_below_zero_habitatnatural),
    ]

# Set up the data sheet for the model condition
dat <- list(proportion_estimates_below_zero = as.numeric(d$proportion_estimates_below_zero_condition),
    count_estimates_below_zero = as.numeric(d$count_estimates_below_zero_condition),
    changeprop_mean = as.numeric(d$changeprob_mean), samplesize = as.integer(as.factor(d$n)))

# habitat: natural
datn <- list(proportion_estimates_below_zero_n = as.numeric(d_noNA$proportion_estimates_below_zero_habitatnatural),
    changeprop_mean_n = as.numeric(d_noNA$changeprob_mean), samplesize_n = as.integer(as.factor(d_noNA$n)))

# habitat: suburban
dats <- list(proportion_estimates_below_zero_s = as.numeric(d_noNA$proportion_estimates_below_zero_habitatsuburban),
    changeprop_mean_s = as.numeric(d_noNA$changeprob_mean), samplesize_s = as.integer(as.factor(d_noNA$n)))

# We run the STAN model - it is a binomial model that
# assumes that the probability to estimate a difference
# depends on an interaction between the sample size and the
# difference between the conditions Condition
m_samplesize_diffscount <- ulam(alist(count_estimates_below_zero ~
    dbinom(2000, p), logit(p) <- a[samplesize] + b[samplesize] *
    changeprop_mean, a[samplesize] ~ dnorm(a_bar, sigma_a), a_bar ~
    dnorm(0, 1.5), sigma_a ~ dexp(1), b[samplesize] ~ dnorm(0,
    1.5)), data = dat, log_lik = TRUE, messages = FALSE, cmdstan = T)

# Habitat: natural
m_samplesize_diffsnatural <- ulam(alist(proportion_estimates_below_zero_n ~
    dnorm(mu, sigma), mu <- a[samplesize_n] + b[samplesize_n] *
    changeprop_mean_n, a[samplesize_n] ~ dnorm(a_bar, sigma_a),
    a_bar ~ dnorm(0, 1.5), sigma_a ~ dexp(1), sigma ~ dexp(1),
    b[samplesize_n] ~ dnorm(0, 1.5)), data = datn, chains = 4,
    cores = 4, cmdstan = TRUE, control = list(adapt_delta = 0.95,
        force_recompile = TRUE))

# Habitat: suburban
m_samplesize_diffssuburban <- ulam(alist(proportion_estimates_below_zero_s ~
    dnorm(mu, sigma), mu <- a[samplesize_s] + b[samplesize_s] *
    changeprop_mean_s, a[samplesize_s] ~ dnorm(a_bar, sigma_a),
    a_bar ~ dnorm(0, 1.5), sigma_a ~ dexp(1), sigma ~ dexp(1),
    b[samplesize_s] ~ dnorm(0, 1.5)), data = dats, chains = 4,
    cores = 4, cmdstan = TRUE, control = list(adapt_delta = 0.95,
        force_recompile = TRUE))

# Extract the posterior estimates from this STAN model
precis_output <- precis(m_samplesize_diffscount, depth = 2)
precis_outputn <- precis(m_samplesize_diffsnatural, depth = 2)
precis_outputs <- precis(m_samplesize_diffssuburban, depth = 2)

# Plot the predictions for the different sample sizes as
# separate lines = COUNT out of 2000

# CONDITION
plot(NULL, xlim = c(-0.01, 0.31), ylim = c(0, 0.6), pch = 1,
    ann = F, frame.plot = F, cex = 1.5)

mu_10 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(1, length(unique(d$changeprob_mean)))))
mu_10_mean <- apply(mu_10, 2, mean)
mu_10_ci <- apply(mu_10, 2, PI, prob = 0.97)
shade(mu_10_ci, unique(d$changeprob_mean), col = col.alpha("blue",
    0.7))
lines(x = c(-0.008, -0.008), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.022, 0.022), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.044, 0.044), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.092, 0.092), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.192, 0.192), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.292, 0.292), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("blue", 0.5))

mu_20 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(2, length(unique(d$changeprob_mean)))))
mu_20_mean <- apply(mu_20, 2, mean)
mu_20_ci <- apply(mu_20, 2, PI, prob = 0.97)
shade(mu_20_ci, unique(d$changeprob_mean), col = col.alpha("orange",
    0.7))
lines(x = c(-0.004, -0.004), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.024, 0.024), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.048, 0.048), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.096, 0.096), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.196, 0.196), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.296, 0.296), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("orange", 0.5))

mu_30 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(3, length(unique(d$changeprob_mean)))))
mu_30_mean <- apply(mu_30, 2, mean)
mu_30_ci <- apply(mu_30, 2, PI, prob = 0.97)
shade(mu_30_ci, unique(d$changeprob_mean), col = col.alpha("skyblue",
    0.7))
lines(x = c(0, 0), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.028, 0.028), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.052, 0.052), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.1, 0.1), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.2, 0.2), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.3, 0.3), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("skyblue", 0.5))

mu_40 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(4, length(unique(d$changeprob_mean)))))
mu_40_mean <- apply(mu_40, 2, mean)
mu_40_ci <- apply(mu_40, 2, PI, prob = 0.97)
shade(mu_40_ci, unique(d$changeprob_mean), col = col.alpha("#213940",
    0.7))
lines(x = c(0.004, 0.004), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.032, 0.032), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.056, 0.056), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.104, 0.104), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.204, 0.204), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.304, 0.304), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("#213940", 0.5))

mu_60 <- link(m_samplesize_diffscount, data = data.frame(changeprop_mean = as.numeric(unique(d$changeprob_mean)),
    samplesize = rep(5, length(unique(d$changeprob_mean)))))
mu_60_mean <- apply(mu_60, 2, mean)
mu_60_ci <- apply(mu_60, 2, PI, prob = 0.97)
shade(mu_60_ci, unique(d$changeprob_mean), col = col.alpha("red",
    0.7))
lines(x = c(0.008, 0.008), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("red", 0.5))
lines(x = c(0.036, 0.036), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("red", 0.5))
lines(x = c(0.06, 0.06), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("red", 0.5))
lines(x = c(0.108, 0.108), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("red", 0.5))
lines(x = c(0.208, 0.208), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("red", 0.5))
lines(x = c(0.308, 0.308), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_condition), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_condition)),
    col = col.alpha("red", 0.5))

legend(x = "topright", legend = c(lty2 = "10", lty1 = "20", lty3 = "30",
    lty2 = "40", lty3 = "60"), lty = c(1, 1, 1, 1, 1), lwd = c(5,
    5, 5, 5, 5), col = c(col.alpha("blue", 0.4), col.alpha("orange",
    0.4), col.alpha("skyblue", 0.4), col.alpha("#213940", 0.4),
    col.alpha("red", 0.4)), cex = 1, bty = "n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size", side = 3, line = -0.7, at = 0.31)
mtext("Power to detect differences between conditions", side = 3,
    line = 1, at = 0.15, cex = 1.5)
mtext("Mean difference in chance of survival (control vs. manipulated)",
    side = 1, line = 3, cex = 1.2)
mtext("Ratio of estimates below zero", side = 2, line = 3, cex = 1.2)

# 93% of the posterior mass is above zero for a 0.3 or
# larger mean difference between conditions when the sample
# size is 30
mu_10_ci  #29%. Column 1 for the 0.3 category on the x axis (ordered by unique(d$changeprob_mean))
mu_20_ci  #18%
mu_30_ci  #12%
mu_40_ci  #11%
mu_60_ci  #8%
sum(d[d$n %in% 30 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_condition <
    0.3)  #28 values are <0.3 (out of 30) when mean change=0.3; 5/30 when mean change = 0, 25/30 when mean change = 0.2
length(d[d$n %in% 30 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_condition)  #30 values total


# NATURAL
op <- par(mfrow = c(1, 2), mar = c(5.9, 4.9, 2.3, 1.6))
plot(NULL, xlim = c(-0.01, 0.31), ylim = c(0, 0.6), pch = 1,
    ann = F, frame.plot = F, cex = 1.5)

mu_10n <- link(m_samplesize_diffsnatural, data = data.frame(changeprop_mean_n = as.numeric(unique(d$changeprob_mean)),
    samplesize_n = rep(1, length(unique(d$changeprob_mean)))))
mu_10n_mean <- apply(mu_10n, 2, mean)
mu_10n_ci <- apply(mu_10n, 2, PI, prob = 0.97)
shade(mu_10n_ci, unique(d$changeprob_mean), col = col.alpha("blue",
    0.7))
lines(x = c(-0.008, -0.008), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.022, 0.022), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 10 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.044, 0.044), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 10 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.092, 0.092), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.192, 0.192), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.292, 0.292), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("blue", 0.5))

mu_20n <- link(m_samplesize_diffsnatural, data = data.frame(changeprop_mean_n = as.numeric(unique(d$changeprob_mean)),
    samplesize_n = rep(2, length(unique(d$changeprob_mean)))))
mu_20n_mean <- apply(mu_20n, 2, mean)
mu_20n_ci <- apply(mu_20n, 2, PI, prob = 0.97)
shade(mu_20n_ci, unique(d$changeprob_mean), col = col.alpha("orange",
    0.7))
lines(x = c(-0.004, -0.004), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.024, 0.024), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 20 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.048, 0.048), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 20 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.096, 0.096), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.196, 0.196), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.296, 0.296), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("orange", 0.5))

mu_30n <- link(m_samplesize_diffsnatural, data = data.frame(changeprop_mean_n = as.numeric(unique(d$changeprob_mean)),
    samplesize_n = rep(3, length(unique(d$changeprob_mean)))))
mu_30n_mean <- apply(mu_30n, 2, mean)
mu_30n_ci <- apply(mu_30n, 2, PI, prob = 0.97)
shade(mu_30n_ci, unique(d$changeprob_mean), col = col.alpha("skyblue",
    0.7))
lines(x = c(0, 0), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.028, 0.028), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 30 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.052, 0.052), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 30 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.1, 0.1), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.2, 0.2), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.3, 0.3), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("skyblue", 0.5))

mu_40n <- link(m_samplesize_diffsnatural, data = data.frame(changeprop_mean_n = as.numeric(unique(d$changeprob_mean)),
    samplesize_n = rep(4, length(unique(d$changeprob_mean)))))
mu_40n_mean <- apply(mu_40n, 2, mean)
mu_40n_ci <- apply(mu_40n, 2, PI, prob = 0.97)
shade(mu_40n_ci, unique(d$changeprob_mean), col = col.alpha("#213940",
    0.7))
lines(x = c(0.004, 0.004), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.032, 0.032), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 40 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.056, 0.056), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 40 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.104, 0.104), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.204, 0.204), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.304, 0.304), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("#213940", 0.5))

mu_60n <- link(m_samplesize_diffsnatural, data = data.frame(changeprop_mean_n = as.numeric(unique(d$changeprob_mean)),
    samplesize_n = rep(5, length(unique(d$changeprob_mean)))))
mu_60n_mean <- apply(mu_60n, 2, mean)
mu_60n_ci <- apply(mu_60n, 2, PI, prob = 0.97)
shade(mu_60n_ci, unique(d$changeprob_mean), col = col.alpha("red",
    0.7))
lines(x = c(0.008, 0.008), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("red", 0.5))
lines(x = c(0.036, 0.036), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 60 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("red", 0.5))
lines(x = c(0.06, 0.06), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatnatural),
    max(d[d$n %in% 60 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("red", 0.5))
lines(x = c(0.108, 0.108), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("red", 0.5))
lines(x = c(0.208, 0.208), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("red", 0.5))
lines(x = c(0.308, 0.308), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatnatural), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatnatural)),
    col = col.alpha("red", 0.5))

legend(x = "topright", legend = c(lty2 = "10 / 7.5", lty1 = "20 / 15",
    lty3 = "30 / 22.5", lty2 = "40 / 30", lty3 = "60 / 45"),
    lty = c(1, 1, 1, 1, 1), lwd = c(5, 5, 5, 5, 5), col = c(col.alpha("blue",
        0.4), col.alpha("orange", 0.4), col.alpha("skyblue",
        0.4), col.alpha("#213940", 0.4), col.alpha("red", 0.4)),
    cex = 1, bty = "n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size", side = 3, line = -0.7, at = 0.26)
mtext("A. Habitat: inside Zealandia", side = 3, line = 1, at = 0.15,
    cex = 1.5)
mtext("Mean difference in survival (control/manipulated)", side = 1,
    line = 3, cex = 1.2)
mtext("Ratio of estimates below zero", side = 2, line = 3, cex = 1.2)

# SUBURBAN
plot(NULL, xlim = c(-0.01, 0.31), ylim = c(0, 0.6), pch = 1,
    ann = F, frame.plot = F, cex = 1.5)

mu_10s <- link(m_samplesize_diffssuburban, data = data.frame(changeprop_mean_s = as.numeric(unique(d$changeprob_mean)),
    samplesize_s = rep(1, length(unique(d$changeprob_mean)))))
mu_10s_mean <- apply(mu_10s, 2, mean)
mu_10s_ci <- apply(mu_10s, 2, PI, prob = 0.97)
shade(mu_10s_ci, unique(d$changeprob_mean), col = col.alpha("blue",
    0.7))
lines(x = c(-0.008, -0.008), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatsuburban), max(d[d$n %in%
    10 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.022, 0.022), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 10 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.044, 0.044), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 10 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.092, 0.092), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 10 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.192, 0.192), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 10 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("blue", 0.5))
lines(x = c(0.292, 0.292), y = c(min(d[d$n %in% 10 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 10 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("blue", 0.5))

mu_20s <- link(m_samplesize_diffssuburban, data = data.frame(changeprop_mean_s = as.numeric(unique(d$changeprob_mean)),
    samplesize_s = rep(2, length(unique(d$changeprob_mean)))))
mu_20s_mean <- apply(mu_20s, 2, mean)
mu_20s_ci <- apply(mu_20s, 2, PI, prob = 0.97)
shade(mu_20s_ci, unique(d$changeprob_mean), col = col.alpha("orange",
    0.7))
lines(x = c(-0.004, -0.004), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatsuburban), max(d[d$n %in%
    20 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.024, 0.024), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 20 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.048, 0.048), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 20 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.096, 0.096), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 20 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.196, 0.196), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 20 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("orange", 0.5))
lines(x = c(0.296, 0.296), y = c(min(d[d$n %in% 20 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 20 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("orange", 0.5))

mu_30s <- link(m_samplesize_diffssuburban, data = data.frame(changeprop_mean_s = as.numeric(unique(d$changeprob_mean)),
    samplesize_s = rep(3, length(unique(d$changeprob_mean)))))
mu_30s_mean <- apply(mu_30s, 2, mean)
mu_30s_ci <- apply(mu_30s, 2, PI, prob = 0.97)
shade(mu_30s_ci, unique(d$changeprob_mean), col = col.alpha("skyblue",
    0.7))
lines(x = c(0, 0), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatsuburban), max(d[d$n %in%
    30 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.028, 0.028), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 30 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.052, 0.052), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 30 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.1, 0.1), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 30 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.2, 0.2), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 30 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("skyblue", 0.5))
lines(x = c(0.3, 0.3), y = c(min(d[d$n %in% 30 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 30 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("skyblue", 0.5))

mu_40s <- link(m_samplesize_diffssuburban, data = data.frame(changeprop_mean_s = as.numeric(unique(d$changeprob_mean)),
    samplesize_s = rep(4, length(unique(d$changeprob_mean)))))
mu_40s_mean <- apply(mu_40s, 2, mean)
mu_40s_ci <- apply(mu_40s, 2, PI, prob = 0.97)
shade(mu_40s_ci, unique(d$changeprob_mean), col = col.alpha("#213940",
    0.7))
lines(x = c(0.004, 0.004), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatsuburban), max(d[d$n %in%
    40 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.032, 0.032), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 40 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.056, 0.056), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 40 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.104, 0.104), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 40 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.204, 0.204), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 40 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("#213940", 0.5))
lines(x = c(0.304, 0.304), y = c(min(d[d$n %in% 40 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 40 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("#213940", 0.5))

mu_60s <- link(m_samplesize_diffssuburban, data = data.frame(changeprop_mean_s = as.numeric(unique(d$changeprob_mean)),
    samplesize_s = rep(5, length(unique(d$changeprob_mean)))))
mu_60s_mean <- apply(mu_60s, 2, mean)
mu_60s_ci <- apply(mu_60s, 2, PI, prob = 0.97)
shade(mu_60s_ci, unique(d$changeprob_mean), col = col.alpha("red",
    0.7))
lines(x = c(0.008, 0.008), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0, ]$proportion_estimates_below_zero_habitatsuburban), max(d[d$n %in%
    60 & d$changeprob_mean %in% 0, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("red", 0.5))
lines(x = c(0.036, 0.036), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.025, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 60 & d$changeprob_mean %in% 0.025, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("red", 0.5))
lines(x = c(0.06, 0.06), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.05, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 60 & d$changeprob_mean %in% 0.05, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("red", 0.5))
lines(x = c(0.108, 0.108), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.1, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 60 & d$changeprob_mean %in% 0.1, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("red", 0.5))
lines(x = c(0.208, 0.208), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.2, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 60 & d$changeprob_mean %in% 0.2, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("red", 0.5))
lines(x = c(0.308, 0.308), y = c(min(d[d$n %in% 60 & d$changeprob_mean %in%
    0.3, ]$proportion_estimates_below_zero_habitatsuburban),
    max(d[d$n %in% 60 & d$changeprob_mean %in% 0.3, ]$proportion_estimates_below_zero_habitatsuburban)),
    col = col.alpha("red", 0.5))

legend(x = "topright", legend = c(lty2 = "10 / 2.5", lty1 = "20 / 5",
    lty3 = "30 / 7.5", lty2 = "40 / 10", lty3 = "60 / 15"), lty = c(1,
    1, 1, 1, 1), lwd = c(5, 5, 5, 5, 5), col = c(col.alpha("blue",
    0.4), col.alpha("orange", 0.4), col.alpha("skyblue", 0.4),
    col.alpha("#213940", 0.4), col.alpha("red", 0.4)), cex = 1,
    bty = "n")

# Axis labels (side 1=x axis, side 2=y axis)
mtext("Sample size", side = 3, line = -0.7, at = 0.26)
mtext("B. Habitat: outside Zealandia", side = 3, line = 1, at = 0.15,
    cex = 1.5)
mtext("Mean difference in survival (control/manipulated)", side = 1,
    line = 3, cex = 1.2)
mtext("Ratio of estimates below zero", side = 2, line = 3, cex = 1.2)
par(op)

 

Figure 16. Risk of false positives and false negatives depending on sample and effect sizes. Curves of the mean ratio of estimates below zero (vertical lines show the minimum and maximum ratios), which illustrates the power we have to detect differences between conditions at different sample sizes. The curves show the model estimates as the effect increases (larger changes in the mean proportion time spent after versus before on the x-axis) for different potential sample sizes (10-60, illustrated by different colors). When there is no change (x value of zero), estimates suggest that, as expected, half of the estimates are below zero because the before and after conditions are not different from each other. As the change increases, the estimates decrease because models are able to reliably tell that the before and after conditions differ from each other.

 

Figure 17. Risk of false positives and false negatives depending on sample and effect sizes. Curves of the mean ratio of estimates below zero (vertical lines show the minimum and maximum ratios), which illustrates the power we have to detect differences between conditions (control vs. manipulated) in different habitats (inside or outside Zealandia) at different sample sizes. The sample size was set at 75% inside Zealandia and 25% outside Zealandia and the associated numbers with these percentages are shown for each overall sample size in the legend. The curves show the model estimates as the effect increases (larger changes in the mean proportion time spent after versus before on the x-axis) for different potential sample sizes (10-60, illustrated by different colors). When there is no change (x value of zero), estimates suggest that, as expected, half of the estimates are below zero because the before and after conditions are not different from each other. As the change increases, the estimates decrease because models are able to reliably tell that the before and after conditions differ from each other.

 

Run this model on the actual data

Run the code below to determine whether there were differences between the control and manipulated conditions in their survival status (alive or dead when individuals are or would be 112 days old).

library(rethinking)
library(cmdstanr)
set_ulam_cmdstan(TRUE) #makes it so ulam always runs with cmdstan
rstan_options("auto_write" = TRUE)
#rebuild_cmdstan() #run this if you get an error about a PCH file https://discourse.mc-stan.org/t/struggling-to-get-cmdstan-working-on-mac-os-big-sur/19774/4

# load data sheet
d <- read.csv(url(""), header=T, sep=",", stringsAsFactors=F)

# Set up the data sheet
dat <- list(
    status = d$status,
    habitat = d$habitat,
    condition = d$condition
)

#simulated data to test the model
dat2 <- list(
    status = rbinom(40,1,0.5), #for 40 data points, randomly assign a 0 (dead) or 1 (alive) survival status such that each appears 50% of the time
    habitat = rbinom(40,1,0.5)+1, #for 40 data points, randomly assign a 0 (reserve) or 1 (suburban) habitat such that each appears 50% of the time. Add a 1 to each data point to make it 1 & 2 rather than 0 and 1 (if don't do this, rstan will throw a csv error and break)
    condition = c(rep(1,20),rep(2,20)) #for all rows with control data, assign a condition=1, and for all rows with manipulated data, assign a condition=2, repeat each 20 times (for a total sample size of 40)
)

# THE MODEL (p.448 for binomial distribution - McElreath 2020)
#g[condition] = avg log odds for each condition
mt <- ulam(
    alist(
        status ~ dbinom(1,p),
        logit(p) <- g[condition],
        g[condition] ~ normal(0,1)
       ) , data=dat2 , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))
#precis(mt, depth=2)

#run a CONTRAST to determine whether there is a difference between before (g1) and after (g2) (subtract g1 before from g2 after)
post <- extract.samples(mt)
diff_g <- post$g[,2] - post$g[,1] #log-odds difference p.341 McElreath 2020
diff_p <- inv_logit(post$g[,2]) - inv_logit(post$g[,1]) #on the probability scale p.341 McElreath 2020
#precis( list( diff_g=diff_g , diff_p=diff_p ) )      

# now run the model to determine whether there were differences between conditions within each habitat
mt2 <- ulam(
    alist(
        status ~ dbinom(1,p),
        logit(p) <- g[habitat] + beta[condition,habitat],
        # adaptive priors - non-centered which is a parameterization that improves sampling for the MCMC model fit by helping with complex varying effects (see p.453)
        transpars> matrix[condition,2]:beta <- compose_noncentered( sigma_habitat , L_Rho_habitat , z_habitat ),
        matrix[2,condition]:z_habitat ~ normal( 0 , 1 ),
        # fixed priors
        g[condition] ~ normal(0,1),
        vector[2]:sigma_habitat ~ dexp(1),
        cholesky_factor_corr[2]:L_Rho_habitat ~ lkj_corr_cholesky( 2 ),
        # compute ordinary correlation matrixes from Cholesky factors
        gq> matrix[2,2]:Rho_habitat <<- Chol_to_Corr(L_Rho_habitat)
    ) , data=dat2 , chains=4 , cores=4 , log_lik=TRUE , cmdstan = TRUE, control = list(adapt_delta = .95, force_recompile = TRUE))
#precis(mt2, depth=2)

#run a CONTRAST to determine whether there is a difference in survival status between control (g1) and manipulated (g2) and habitat (natural vs suburban)
post2 <- extract.samples(mt2)
#str(post2)
if(length(unique(dat2$habitat))>1){
  diff_betanatural <- inv_logit(post2$beta[,2,1]) - inv_logit(post2$beta[,1,1]) # Comparison for habitat 1 (natural): take the inverse logit of the sum (posterior control habitat 1 + posterior control) minus the inverse logit of the sum (posterior manipulated habitat 1 + posterior manipulated). For the beta values, the first entry in the square brackets refers to the rows (we want all of them so we leave this empty), the second entry refers to control (1) versus manipulated (2), and the third entry refers to the habitat (here habitat 1)
diff_betasuburban <- inv_logit(post2$beta[,2,2]) - inv_logit(post2$beta[,1,2]) # Comparison for habitat 2 (suburban): take the inverse logit of the sum (posterior control habitat 2 + posterior control) minus the inverse logit of the sum (posterior manipulated habitat 2 + posterior manipulated). For the beta values, the first entry in the square brackets refers to the rows (we want all of them so we leave this empty), the second entry refers to control (1) versus manipulated (2), and the third entry refers to the habitat (here habitat 2)
}
#precis( list( diff_betanatural=diff_betanatural , diff_betasuburban=diff_betasuburban ) )

ETHICS

This research is carried out in accordance with permits from the:

1. US Fish and Wildlife Service (scientific collecting permit number MBPER0039225)
2. US Geological Survey Bird Banding Laboratory (federal bird banding permit number 23872 for grackles and 24273 for jays)
3. Institutional Animal Care and Use Committee at the University of California Santa Barbara (protocol number 958)
4. California Department of Fish and Wildlife (scientific collecting permit [specific use] number S‐192100001‐19210‐001 for grackles and S-192100002-20329-001 for jays)
5. Oregon Department of Fish and Wildlife (scientific collecting permit 125-22)
6. The New Zealand Department of Conservation (permit number 97554-FAU)

AUTHOR CONTRIBUTIONS

Logan: Hypothesis development, data collection, data analysis and interpretation, write up, revising/editing, materials/funding.

Shaw: Hypothesis development, data collection, data analysis and interpretation, write up, revising/editing, materials/funding.

Lukas: Analysis design and code, sample size modeling, validated reversal passing criteria, write up, revising/editing

McCune: Hypothesis development, data collection, data analysis and interpretation, write up, revising/editing.

FUNDING

This research is funded by the Department of Human Behavior, Ecology and Culture at the Max Planck Institute for Evolutionary Anthropology (to Logan) and the Rutherford Discovery Fellowship from the Royal Society Te Apārangi (to Shaw).

CONFLICT OF INTEREST DISCLOSURE

We, the authors, declare that we have no financial conflicts of interest with the content of this article. CJ Logan is a co-founder of and on the Managing Board at PCI Registered Reports.

ACKNOWLEDGEMENTS

We are grateful to our funders: Richard McElreath at the Max Planck Institute for Evolutionary Anthropology (whole project) and the Rutherford Discovery Fellowship from the Royal Society Te Apārangi (Shaw).

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