In oliviergimenez/R2ucare: Goodness-of-Fit Tests for Capture-Recapture Models
Introduction
Capture–recapture (CR) models have become a central tool in population ecology for estimating demographic parameters under imperfect detection of individuals (Lebreton et al. 1992; 2009). These methods rely on the longitudinal monitoring of individuals that are marked (or identifiable) and then captured or sighted alive over time.
Single-state CR models, and the Cormack-Jolly-Seber (CJS) model in particular (Lebreton et al. 1992), have been used to assess the effect of climate change (e.g. Guéry et al. 2017) or study senescence (e.g. Péron et al. 2016). The extension of single-state models to situations where individuals are detected in several geographical sites or equivalently states (e.g. breeding/non-breeding or sane/ill) are called multi-state CR models (Lebreton et al. 2009). Multistate CR models, and the Arnason-Schwarz model in particular (Lebreton et al. 2009), are appealing for addressing various biological questions such as metapopulation dynamics (e.g. Spendelow et al. 2016) or life-history trade–offs (e.g. Supp et al. 2015).
A necessary step for correct inference about demographic parameters is to assess the fit of single- and multi-state models to CR data, regardless of whether a Bayesian or a frequentist framework is adopted.
Two families of methods exist to perform goodness-of-fit (GOF) tests for CR models. First, an omnibus test of the null hypothesis that a given model fits the data adequately can be conducted using resampling methods and the deviance as a metric (White 2002). However when the null hypothesis is rejected, this omnibus approach does not inform about an alternative model that could be fitted. Second, specialized tests have been built to address biologically meaningful causes of departure from the null hypothesis. A global test for single- and multi-state CR models is decomposed into several interpretable components based on contingency tables, for example the presence of transients (Pradel et al., 1997; Pradel et al. 2003) or that for trap–dependence (Pradel, 1993; Pradel et al. 2003). These GOF tests are implemented in the Windows application U-CARE (Choquet et al. 2009).
Here, we introduce the R (R Development Core Team 2014) package R2ucare to perform GOF tests for single- and multi-state CR models. R2ucare also includes various functions to help manipulate CR data. As a package in the CRAN database, R2ucare provides full advantage of R’s many features (e.g. simulations, model fitting), while being multi-platform. We go through the theory first, then illustrate the use of R2ucare with wolves in France for single-state models and geese in the U.S. for multi-state models.
Theory
Once a model has been specified, GOF testing is the procedure that controls model assumptions. GOF testing and model fitting are two complementary procedures that share and compete for the information contained in the data. More liberal models require more information to be fitted (there are more parameters to estimate) but also fewer assumptions need to be verified. For instance, the time-dependent CJS model is merely content with the numbers of individuals captured at each occasion and the numbers never seen again from those released at each occasion when it comes to estimating its parameters. These summary statistics leave much of the details of the capture histories available to test its assumptions.
There are several ways in which this remaining information may be exploited to test the assumptions. The implementation retained in R2ucare builds on the optimal approach originally devised by Pollock et al. (1985) and later modified by Pradel (1993). It is based on contingency tables and aims at testing with chi-squared tests (and Fisher's exact tests when needed) for transients and trap-dependence. These aspects are examined specifically in two independent component tests called respectively Test 3.SR and Test 2.CT. The component tests directed at transients and trap-dependence actually address features of the data that are consequences of respectively the presence of transients and trap-dependence, so that these features may also be caused by other, completely different phenomena. They do verify respectively that:
-
Newly encountered individuals have the same chance to be later reobserved as recaptured (previously encountered) individuals; this is the null hypothesis of Test 3.SR.
-
Missed individuals have the same chance to be recaptured at the next occasion as currently captured individuals; this is the null hypothesis of Test 2.CT.
Although these components are often called 'test of transience' and 'test of trap-dependence', when it comes to interpretation, one should keep in mind that transience and trap-dependence are just two specific reasons why the tests respectively called 3.SR and 2.CT might be significant.
Interestingly, these two components provide formal tests for comparing the CJS model with more general models, namely a model with an interaction between age (2 classes) and time in the survival probability for Test 3.SR (Pradel et al. 1997) and a model allowing for a different recapture probability of individuals just released for Test 2.CT (Pradel 1993).
Beyond these two oriented components, the remaining information is distributed and structured into two additional components: Test 3.Sm and Test 2.CL. Those examine long-term features of the data:
-
Among those individuals seen again, when they were seen does not differ among previously and newly marked individuals; this is the null hypothesis of Test 3.Sm.
-
There is no difference in the timing of reencounters between the individuals encountered and not encountered at occasion $i$, conditional on presence at both occasions $i$ and $i+2$; this is the null hypothesis of Test 2.CL.
Data are generally sparse for these components and scattered over many occasions. Despite the implementation of some automatic pooling (see Choquet et al. 2005 for more details about the pooling rules), they are rarely significant alone.
Although many situations can lead to similar test results, we propose here a decision tree (Figure \ref{fig:dtcjs}) that should lead to reasonable solutions in most cases.
The theory for the GOF test of the multistate Arnason-Schwarz model was developed along similar lines as for the CJS model (Pradel et al. 2003). This test has yet more components and some components have a more complex structure (hence our non attempt to build a decision tree as for the CJS model), but for all that concerns us, the reasoning remains very similar. The test implemented in R2ucare is actually a test of the Jolly-Move model, a slightly more general model than the Arnason-Schwarz model in that it allows detection parameters to depend on the previous state occupied. This is biologically irrelevant in most common situations (Pradel et al. 2003), so that we may reason as if we were examining the Arnason-Schwarz model. Components here have been designed to detect transients, trap-dependence, and the memory of past states. This last point means that the component examines whether transitions to a new state depend on previous states beyond the current one. The corresponding components are respectively Test 3.GSR, Test M.ITEC, and Test WBWA. Like for the CJS case, they actually examine features of the data, namely that:
-
Newly encountered individuals have the same chance to be later reobserved as recaptured (i.e. previously encountered) individuals; this is the null hypothesis of Test 3.GSR which is the exact equivalent of 3.SR.
-
Individuals currently in the same state, whether captured or missed, have the same chance to be recaptured in each state at the next occasion; this is the null hypothesis of Test M.ITEC.
-
Individuals currently captured in the same state have the same chance to be next reobserved in the different states independently of their observed state at the most recent capture; this is the null hypothesis of Test WBWA.
These interpretable components are complemented by two composite components with no clearly identified interpretation, Test 3.GSm and Test M.LTEC. We do not attempt to give a description of these; let it suffice to say that Test 3.GSm is concerned with comparing newly and previously encountered, while Test M.LTEC contrasts missed and encountered individuals. Fortunately, these components play a secondary role as they are usually not significant alone.
For more details about the theory of GOF testing for CR models, we strongly encourage users to read Pradel et al. (2005) and Cooch and White (2006).
The R2ucare package
The R2ucare package contains R functions to perform GOF tests for CR models as well as various functions to manipulate CR data (see Table \ref{tab:functions} and the vignette of the package named vignette_R2ucare). It ensures reproducibility which was not possible with the U-CARE (Choquet et al. 2009) Windows standalone application. Besides, it can be used in combination with other R packages for fitting CR data like RMark (Laake 2013) or marked (Laake et al. 2013) or to carry out simulations to assess statistical power (e.g. Bromaghin et al. 2013; Fletcher et al. 2012).
Function | Description
:--------------- | :------------------------------------------------------------------------
marray | build a m-array for single-site/state capture-recapture data
multimarray | build a m-array for multi-site/state capture-recapture data
group_data | pool together individuals with the same encounter capture-recapture history
ungroup_data | split encounter capture-recapture histories into individual ones
read_inp | read MARK formated files
read_headed | read E-SURGE formated files
test3sr | implement Test 3.SR for single-site/state models (presence of transients)
test3sm | implement Test 3.Sm for single-site/state models
test2ct | implement Test 2.CT for single-site/state models (presence of trap-dependence)
test2cl | implement Test 2.CL for single-site/state models
test3Gsr | implement Test 3.GSR for multi-site/state models (presence of transients)
test3Gsm | implement Test 3.GSm for multi-site/state models
test3Gwbwa | implement Test WBWA for multi-site/state models (presence of memory)
testMitec | implement Test M.ITEC for multi-site/state models (presence of trap-dependence)
testMltec | implement Test M.LTEC for multi-site/state models
Table: The main functions of R2ucare and their description. See main text for more details.\label{tab:functions}
Goodness-of-fit tests for single-site/state models
We illustrate the use of R2ucare to assess the GOF of the CJS model to a dataset on wolves (Canis lupus) in France (e.g., Fletcher et al. 2012). Briefly, the data consist of capture histories for 160 individuals, partitioned into 35 3-month intervals (from spring 1995 to autumn 2003).
We first load the R2ucare package:
library(R2ucare)
Then we read in the wolf data that is provided with the package. To do so, R2ucare contains two functions that accomodate the most frequent CR formats: read_inp deals with the MARK format (Cooch and White 2006) while read_headed deals with the E-SURGE format (Choquet et al. 2009). The wolf dataset has the MARK format, therefore:
wolf = system.file("extdata", "wolf.inp", package = "R2ucare")
wolf = read_inp(wolf)
We then get the matrix and number of CR encounter histories:
ch = wolf$encounter_histories
n = wolf$sample_size
Following the procedure described in Figure \ref{fig:dtcjs}, we first assess the overall fit of the CJS model by using the function overall_CJS:
overall_CJS(ch,n)
Clearly, the CJS model does not fit the data well ($\chi^2_{115}=180.73$, $P < 0.01$). We then test for an effect of trap-dependence:
test2ct(ch,n,verbose = FALSE)
Test 2.CT is significant ($\chi^2_{31}=64.45$, $P < 0.01$). We also provide the signed square root (sign_test) of the Pearson chi–square statistic as a directional test of the null hypothesis (Pradel et al. 2005), which is negative when there is an excess of individuals encountered at a given occasion among the individuals encountered at the previous occasion.
Note that, by default, the GOF functions in R2ucare returns all the contingency tables that compose the test under scrutiny, which might not be of immediate use and rather cumbersome on screen, hence the use of verbose=FALSE in the call to the test2ct function above. Now we ask whether there is a transient effect:
test3sr(ch,n,verbose = FALSE)
Test 3.SR is also significant ($\chi^2_{29}=65.41$, $P < 0.01$). We also provide the signed square root (sign_test) of the Pearson chi–square statistic (Pradel et al. 2005), which is positive when there is an excess of never seen again among the newly marked.
Navigating through the decision tree in Figure \ref{fig:dtcjs} suggests we should perform the two remaining tests:
test3sm(ch,n,verbose = FALSE)
test2cl(ch,n,verbose = FALSE)
Neither Test 3.Sm ($\chi^2_{25}=22.98$, $P = 0.58$) nor Test 2.CL ($\chi^2_{30}=27.89$, $P = 0.58$) is significant, therefore we recommend fitting a CJS model incorporating both a transience effect and a trap-dependence effect and start the analysis from there. In passing, it is possible to calculate a GOF test for this new model by removing the two components Test 3.SR and Test 2.CT to the overall GOF test (Pradel et al. 2005):
# substract the components 3SR and 2CT to the CJS test statistic
stat_new = overall_CJS(ch,n)$chi2 - (test3sr(ch, n)$test3sr[[1]]
+ test2ct(ch, n)$test2ct[[1]])
# calculate degree of freedom associated with the new test statistic
df_new = overall_CJS(ch,n)$degree_of_freedom -
(test3sr(ch, n)$test3sr[[2]] + test2ct(ch, n)$test2ct[[2]])
# compute p-value
1 - pchisq(stat_new, df_new)
This new model incorporating transient and trap-dependence effects fits the wolf data well ($\chi^2_{55}=50.87$, $P = 0.63$).
To date, no GOF test exists for models with individual covariates (unless we discretize them and use groups), individual time-varying covariates (unless we treat them as states) or temporal covariates; therefore, these covariates should be removed from the dataset before using R2ucare. For groups, we recommend treating the groups separately (see e.g. the example in the help file for overall_CJS).
Goodness-of-fit tests for the Arnason-Schwarz model
We now wish to assess the GOF of the Arnason-Schwarz model to a dataset on Canada Geese (Branta canadensis) (Pradel et al. 2005). Briefly, the data consist of capture histories for 28,849 individuals marked and re–observed at wintering locations in the US between 1984 and 1986.
We first read in the geese data that are provided with the package:
geese = system.file("extdata", "geese.inp", package = "R2ucare")
geese = read_inp(geese)
We then get the matrix and number of CR encounter histories:
ch = geese$encounter_histories
n = geese$sample_size
Then we assess the quality of fit of the Arnason-Schwarz model to the geese CR data with the overall_JMV function. Beware that it takes a minute or so to run the test because an iterative optimization procedure is involved to perform Test M.ITEC and Test M.LTEC (Pradel et al. 2003) that is repeated several times to try and avoid local minima.
overall_JMV(ch,n)
The null hypothesis that the Arnason-Schwarz provides an adequate fit to the data is clearly rejected ($\chi^2_{197}=982.59$, $P < 0.01$). In a second step, we further explore each component of the overall test:
test3Gsr(ch,n,verbose=FALSE) # transience
test3Gsm(ch,n,verbose=FALSE)
test3Gwbwa(ch,n,verbose=FALSE) # memory
testMitec(ch,n,verbose=FALSE) # short-term trap-dependence
testMltec(ch,n,verbose=FALSE) # long-term trap-dependence
It appears that all components are significant but the test for a long-term trap-dependence effect. By setting the verbose argument to TRUE (by default argument), one could closely examine the individual contingency
tables and better understand the reasons for the departure to the null hypotheses. For example, let us redo the test for transience Test 3.GSR:
test3Gsr(ch,n,verbose=TRUE)
By inspecting the data.frame containing the details of the test, we see that there is no transients in site 2.
Future directions
R2ucare allows evaluating the quality of fit of standard capture-recapture models for open populations.
Future developments will focus on implementing goodness-of-fit tests for models combining different
sources of data (McCrea et al. 2014) and residual-based diagnostics (Choquet et al. 2013, Warton et al. 2017).
Availability
The current stable version of the package requires R r getRversion() and is distributed under the GPL license.
It can be installed from CRAN (https://cran.r-project.org/web/packages/R2ucare/) and loaded into a R session as follows:
install.packages("R2ucare",dependencies=TRUE)
library("R2ucare")
The repository on GitHub https://github.com/oliviergimenez/R2ucare hosts the development version of the package, it can be installed as follows:
if(!require(devtools)) install.packages("devtools")
library("devtools")
install_github("oliviergimenez/R2ucare")
We also maintain a forum at https://groups.google.com/forum/#!forum/esurge_ucare to which questions can be asked.
Acknowledgments
\small{Replication files (paper and code) are available on the first author's Github account (https://github.com/oliviergimenez). This work was supported by a grant from the French National Research Agency, reference ANR-16-CE02-0007. We warmly thank E. Marboutin and J. Hestbeck for sharing the wolf and geese datasets, respectively}.
Authors' contributions
OG, JDL and RP conceived the ideas and designed methodology; OG, JDL, RC and RP wrote the code; OG and RP led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication.
References
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Choquet, R., Reboulet, A.-M., Lebreton, J.-D., Gimenez, O., and R. Pradel. (2005). U-CARE 2.2 user's manual. CEFE, UMR 5175, Montpellier.
Choquet, R., Lebreton, J.-D., Gimenez, O., Reboulet, A.-M., and R. Pradel. (2009). U-CARE: Utilities for performing goodness of fit tests and manipulating CApture-REcapture data. \textit{Ecography} \textbf{32}: 1071-1074.
Choquet, R., Rouan, L., Pradel, R. (2009). Program E–SURGE: A software application for fitting multievent models. Modeling Demographic Processes in Marked Populations. D. L. Thomson, E. G. Cooch and M. J. Conroy. Berlin, Germany, Springer. \textbf{3}: 845-865.
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Fletcher, D., Lebreton, J.-D., Marescot, L., Schaub, M., Gimenez, O., Slooten, E. and S. Dawson (2012). Bias in estimation of adult survival and asymptotic population growth rate caused by undetected capture heterogeneity. \textit{Methods in Ecology and Evolution} \textbf{3}: 206-216.
Gimenez O., Choquet R. and J.-D. Lebreton (2003). Parameter redundancy in multistate capture-recapture models. \textit{Biometrical Journal} \textbf{45}: 704-722.
Gimenez, O., Cam, E., and J.-M. Gaillard (2017). Individual heterogeneity and capture-recapture models: what, why and how? \textit{Oïkos}. In press. doi: 10.1111/oik.04532.
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Laake, J. L., Johnson, D. S. and Conn, P. (2013). marked: An R package for maximum-likelihood and MCMC analysis of capture-recapture data. \textit{Methods in Ecology and Evolution} \textbf{4}: 885-890.
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Péron, G., Crochet, P.A.C., Choquet, R., Pradel, R., Lebreton, J.-D. and O. Gimenez. (2010). Capture-recapture models with heterogeneity to study survival senescence in the wild. \textit{Oïkos} \textbf{119}: 524-532.
Péron, G., J.-M. Gaillard, C. Barbraud, C. Bonenfant, A. Charmantier, R. Choquet, T. Coulson, V. Grosbois, A. Loison, G. Marzolin, N. Owen-Smith, D. Pardo, F. Plard, R. Pradel, C. Toïgo, O. Gimenez (2016). Evidence of reduced individual heterogeneity in adult survival of long-lived species. \textit{Evolution} \textbf{70}: 2909-2914.
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Introduction
Capture–recapture (CR) models have become a central tool in population ecology for estimating demographic parameters under imperfect detection of individuals (Lebreton et al. 1992; 2009). These methods rely on the longitudinal monitoring of individuals that are marked (or identifiable) and then captured or sighted alive over time.
Single-state CR models, and the Cormack-Jolly-Seber (CJS) model in particular (Lebreton et al. 1992), have been used to assess the effect of climate change (e.g. Guéry et al. 2017) or study senescence (e.g. Péron et al. 2016). The extension of single-state models to situations where individuals are detected in several geographical sites or equivalently states (e.g. breeding/non-breeding or sane/ill) are called multi-state CR models (Lebreton et al. 2009). Multistate CR models, and the Arnason-Schwarz model in particular (Lebreton et al. 2009), are appealing for addressing various biological questions such as metapopulation dynamics (e.g. Spendelow et al. 2016) or life-history trade–offs (e.g. Supp et al. 2015).
A necessary step for correct inference about demographic parameters is to assess the fit of single- and multi-state models to CR data, regardless of whether a Bayesian or a frequentist framework is adopted.
Two families of methods exist to perform goodness-of-fit (GOF) tests for CR models. First, an omnibus test of the null hypothesis that a given model fits the data adequately can be conducted using resampling methods and the deviance as a metric (White 2002). However when the null hypothesis is rejected, this omnibus approach does not inform about an alternative model that could be fitted. Second, specialized tests have been built to address biologically meaningful causes of departure from the null hypothesis. A global test for single- and multi-state CR models is decomposed into several interpretable components based on contingency tables, for example the presence of transients (Pradel et al., 1997; Pradel et al. 2003) or that for trap–dependence (Pradel, 1993; Pradel et al. 2003). These GOF tests are implemented in the Windows application U-CARE (Choquet et al. 2009).
Here, we introduce the R (R Development Core Team 2014) package R2ucare to perform GOF tests for single- and multi-state CR models. R2ucare also includes various functions to help manipulate CR data. As a package in the CRAN database, R2ucare provides full advantage of R’s many features (e.g. simulations, model fitting), while being multi-platform. We go through the theory first, then illustrate the use of R2ucare with wolves in France for single-state models and geese in the U.S. for multi-state models.
Theory
Once a model has been specified, GOF testing is the procedure that controls model assumptions. GOF testing and model fitting are two complementary procedures that share and compete for the information contained in the data. More liberal models require more information to be fitted (there are more parameters to estimate) but also fewer assumptions need to be verified. For instance, the time-dependent CJS model is merely content with the numbers of individuals captured at each occasion and the numbers never seen again from those released at each occasion when it comes to estimating its parameters. These summary statistics leave much of the details of the capture histories available to test its assumptions.
There are several ways in which this remaining information may be exploited to test the assumptions. The implementation retained in R2ucare builds on the optimal approach originally devised by Pollock et al. (1985) and later modified by Pradel (1993). It is based on contingency tables and aims at testing with chi-squared tests (and Fisher's exact tests when needed) for transients and trap-dependence. These aspects are examined specifically in two independent component tests called respectively Test 3.SR and Test 2.CT. The component tests directed at transients and trap-dependence actually address features of the data that are consequences of respectively the presence of transients and trap-dependence, so that these features may also be caused by other, completely different phenomena. They do verify respectively that:
-
Newly encountered individuals have the same chance to be later reobserved as recaptured (previously encountered) individuals; this is the null hypothesis of
Test 3.SR. -
Missed individuals have the same chance to be recaptured at the next occasion as currently captured individuals; this is the null hypothesis of
Test 2.CT.
Although these components are often called 'test of transience' and 'test of trap-dependence', when it comes to interpretation, one should keep in mind that transience and trap-dependence are just two specific reasons why the tests respectively called 3.SR and 2.CT might be significant.
Interestingly, these two components provide formal tests for comparing the CJS model with more general models, namely a model with an interaction between age (2 classes) and time in the survival probability for Test 3.SR (Pradel et al. 1997) and a model allowing for a different recapture probability of individuals just released for Test 2.CT (Pradel 1993).
Beyond these two oriented components, the remaining information is distributed and structured into two additional components: Test 3.Sm and Test 2.CL. Those examine long-term features of the data:
-
Among those individuals seen again, when they were seen does not differ among previously and newly marked individuals; this is the null hypothesis of
Test 3.Sm. -
There is no difference in the timing of reencounters between the individuals encountered and not encountered at occasion $i$, conditional on presence at both occasions $i$ and $i+2$; this is the null hypothesis of
Test 2.CL.
Data are generally sparse for these components and scattered over many occasions. Despite the implementation of some automatic pooling (see Choquet et al. 2005 for more details about the pooling rules), they are rarely significant alone.
Although many situations can lead to similar test results, we propose here a decision tree (Figure \ref{fig:dtcjs}) that should lead to reasonable solutions in most cases.
The theory for the GOF test of the multistate Arnason-Schwarz model was developed along similar lines as for the CJS model (Pradel et al. 2003). This test has yet more components and some components have a more complex structure (hence our non attempt to build a decision tree as for the CJS model), but for all that concerns us, the reasoning remains very similar. The test implemented in R2ucare is actually a test of the Jolly-Move model, a slightly more general model than the Arnason-Schwarz model in that it allows detection parameters to depend on the previous state occupied. This is biologically irrelevant in most common situations (Pradel et al. 2003), so that we may reason as if we were examining the Arnason-Schwarz model. Components here have been designed to detect transients, trap-dependence, and the memory of past states. This last point means that the component examines whether transitions to a new state depend on previous states beyond the current one. The corresponding components are respectively Test 3.GSR, Test M.ITEC, and Test WBWA. Like for the CJS case, they actually examine features of the data, namely that:
-
Newly encountered individuals have the same chance to be later reobserved as recaptured (i.e. previously encountered) individuals; this is the null hypothesis of
Test 3.GSRwhich is the exact equivalent of3.SR. -
Individuals currently in the same state, whether captured or missed, have the same chance to be recaptured in each state at the next occasion; this is the null hypothesis of
Test M.ITEC. -
Individuals currently captured in the same state have the same chance to be next reobserved in the different states independently of their observed state at the most recent capture; this is the null hypothesis of
Test WBWA.
These interpretable components are complemented by two composite components with no clearly identified interpretation, Test 3.GSm and Test M.LTEC. We do not attempt to give a description of these; let it suffice to say that Test 3.GSm is concerned with comparing newly and previously encountered, while Test M.LTEC contrasts missed and encountered individuals. Fortunately, these components play a secondary role as they are usually not significant alone.
For more details about the theory of GOF testing for CR models, we strongly encourage users to read Pradel et al. (2005) and Cooch and White (2006).
The R2ucare package
The R2ucare package contains R functions to perform GOF tests for CR models as well as various functions to manipulate CR data (see Table \ref{tab:functions} and the vignette of the package named vignette_R2ucare). It ensures reproducibility which was not possible with the U-CARE (Choquet et al. 2009) Windows standalone application. Besides, it can be used in combination with other R packages for fitting CR data like RMark (Laake 2013) or marked (Laake et al. 2013) or to carry out simulations to assess statistical power (e.g. Bromaghin et al. 2013; Fletcher et al. 2012).
Function | Description
:--------------- | :------------------------------------------------------------------------
marray | build a m-array for single-site/state capture-recapture data
multimarray | build a m-array for multi-site/state capture-recapture data
group_data | pool together individuals with the same encounter capture-recapture history
ungroup_data | split encounter capture-recapture histories into individual ones
read_inp | read MARK formated files
read_headed | read E-SURGE formated files
test3sr | implement Test 3.SR for single-site/state models (presence of transients)
test3sm | implement Test 3.Sm for single-site/state models
test2ct | implement Test 2.CT for single-site/state models (presence of trap-dependence)
test2cl | implement Test 2.CL for single-site/state models
test3Gsr | implement Test 3.GSR for multi-site/state models (presence of transients)
test3Gsm | implement Test 3.GSm for multi-site/state models
test3Gwbwa | implement Test WBWA for multi-site/state models (presence of memory)
testMitec | implement Test M.ITEC for multi-site/state models (presence of trap-dependence)
testMltec | implement Test M.LTEC for multi-site/state models
Table: The main functions of R2ucare and their description. See main text for more details.\label{tab:functions}
Goodness-of-fit tests for single-site/state models
We illustrate the use of R2ucare to assess the GOF of the CJS model to a dataset on wolves (Canis lupus) in France (e.g., Fletcher et al. 2012). Briefly, the data consist of capture histories for 160 individuals, partitioned into 35 3-month intervals (from spring 1995 to autumn 2003).
We first load the R2ucare package:
library(R2ucare)
Then we read in the wolf data that is provided with the package. To do so, R2ucare contains two functions that accomodate the most frequent CR formats: read_inp deals with the MARK format (Cooch and White 2006) while read_headed deals with the E-SURGE format (Choquet et al. 2009). The wolf dataset has the MARK format, therefore:
wolf = system.file("extdata", "wolf.inp", package = "R2ucare") wolf = read_inp(wolf)
We then get the matrix and number of CR encounter histories:
ch = wolf$encounter_histories n = wolf$sample_size
Following the procedure described in Figure \ref{fig:dtcjs}, we first assess the overall fit of the CJS model by using the function overall_CJS:
overall_CJS(ch,n)
Clearly, the CJS model does not fit the data well ($\chi^2_{115}=180.73$, $P < 0.01$). We then test for an effect of trap-dependence:
test2ct(ch,n,verbose = FALSE)
Test 2.CT is significant ($\chi^2_{31}=64.45$, $P < 0.01$). We also provide the signed square root (sign_test) of the Pearson chi–square statistic as a directional test of the null hypothesis (Pradel et al. 2005), which is negative when there is an excess of individuals encountered at a given occasion among the individuals encountered at the previous occasion.
Note that, by default, the GOF functions in R2ucare returns all the contingency tables that compose the test under scrutiny, which might not be of immediate use and rather cumbersome on screen, hence the use of verbose=FALSE in the call to the test2ct function above. Now we ask whether there is a transient effect:
test3sr(ch,n,verbose = FALSE)
Test 3.SR is also significant ($\chi^2_{29}=65.41$, $P < 0.01$). We also provide the signed square root (sign_test) of the Pearson chi–square statistic (Pradel et al. 2005), which is positive when there is an excess of never seen again among the newly marked.
Navigating through the decision tree in Figure \ref{fig:dtcjs} suggests we should perform the two remaining tests:
test3sm(ch,n,verbose = FALSE) test2cl(ch,n,verbose = FALSE)
Neither Test 3.Sm ($\chi^2_{25}=22.98$, $P = 0.58$) nor Test 2.CL ($\chi^2_{30}=27.89$, $P = 0.58$) is significant, therefore we recommend fitting a CJS model incorporating both a transience effect and a trap-dependence effect and start the analysis from there. In passing, it is possible to calculate a GOF test for this new model by removing the two components Test 3.SR and Test 2.CT to the overall GOF test (Pradel et al. 2005):
# substract the components 3SR and 2CT to the CJS test statistic stat_new = overall_CJS(ch,n)$chi2 - (test3sr(ch, n)$test3sr[[1]] + test2ct(ch, n)$test2ct[[1]]) # calculate degree of freedom associated with the new test statistic df_new = overall_CJS(ch,n)$degree_of_freedom - (test3sr(ch, n)$test3sr[[2]] + test2ct(ch, n)$test2ct[[2]]) # compute p-value 1 - pchisq(stat_new, df_new)
This new model incorporating transient and trap-dependence effects fits the wolf data well ($\chi^2_{55}=50.87$, $P = 0.63$).
To date, no GOF test exists for models with individual covariates (unless we discretize them and use groups), individual time-varying covariates (unless we treat them as states) or temporal covariates; therefore, these covariates should be removed from the dataset before using R2ucare. For groups, we recommend treating the groups separately (see e.g. the example in the help file for overall_CJS).
Goodness-of-fit tests for the Arnason-Schwarz model
We now wish to assess the GOF of the Arnason-Schwarz model to a dataset on Canada Geese (Branta canadensis) (Pradel et al. 2005). Briefly, the data consist of capture histories for 28,849 individuals marked and re–observed at wintering locations in the US between 1984 and 1986.
We first read in the geese data that are provided with the package:
geese = system.file("extdata", "geese.inp", package = "R2ucare") geese = read_inp(geese)
We then get the matrix and number of CR encounter histories:
ch = geese$encounter_histories n = geese$sample_size
Then we assess the quality of fit of the Arnason-Schwarz model to the geese CR data with the overall_JMV function. Beware that it takes a minute or so to run the test because an iterative optimization procedure is involved to perform Test M.ITEC and Test M.LTEC (Pradel et al. 2003) that is repeated several times to try and avoid local minima.
overall_JMV(ch,n)
The null hypothesis that the Arnason-Schwarz provides an adequate fit to the data is clearly rejected ($\chi^2_{197}=982.59$, $P < 0.01$). In a second step, we further explore each component of the overall test:
test3Gsr(ch,n,verbose=FALSE) # transience test3Gsm(ch,n,verbose=FALSE) test3Gwbwa(ch,n,verbose=FALSE) # memory testMitec(ch,n,verbose=FALSE) # short-term trap-dependence testMltec(ch,n,verbose=FALSE) # long-term trap-dependence
It appears that all components are significant but the test for a long-term trap-dependence effect. By setting the verbose argument to TRUE (by default argument), one could closely examine the individual contingency
tables and better understand the reasons for the departure to the null hypotheses. For example, let us redo the test for transience Test 3.GSR:
test3Gsr(ch,n,verbose=TRUE)
By inspecting the data.frame containing the details of the test, we see that there is no transients in site 2.
Future directions
R2ucare allows evaluating the quality of fit of standard capture-recapture models for open populations.
Future developments will focus on implementing goodness-of-fit tests for models combining different
sources of data (McCrea et al. 2014) and residual-based diagnostics (Choquet et al. 2013, Warton et al. 2017).
Availability
The current stable version of the package requires R r getRversion() and is distributed under the GPL license.
It can be installed from CRAN (https://cran.r-project.org/web/packages/R2ucare/) and loaded into a R session as follows:
install.packages("R2ucare",dependencies=TRUE) library("R2ucare")
The repository on GitHub https://github.com/oliviergimenez/R2ucare hosts the development version of the package, it can be installed as follows:
if(!require(devtools)) install.packages("devtools") library("devtools") install_github("oliviergimenez/R2ucare")
We also maintain a forum at https://groups.google.com/forum/#!forum/esurge_ucare to which questions can be asked.
Acknowledgments
\small{Replication files (paper and code) are available on the first author's Github account (https://github.com/oliviergimenez). This work was supported by a grant from the French National Research Agency, reference ANR-16-CE02-0007. We warmly thank E. Marboutin and J. Hestbeck for sharing the wolf and geese datasets, respectively}.
Authors' contributions
OG, JDL and RP conceived the ideas and designed methodology; OG, JDL, RC and RP wrote the code; OG and RP led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication.
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