# **openCR** 2.0 - open population capture--recapture' In openCR: Open Population Capture-Recapture


mycache <- FALSE
Sys.setenv(RCPP_PARALLEL_BACKEND = "tinythread")  ## to dodge CRAN ASAN issue


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The R package openCR fits both non-spatial and spatial capture--recapture models to data from open animal populations, where there is turnover during sampling. The interface generally resembles that of secr (Efford 2020) upon which openCR depends for some functions. This document explains the purpose and general features of openCR. Help pages should be consulted for more detail on particular functions. Worked examples using published datasets are given in another vignette openCR-examples.pdf, and simulation-based examples are in a further vignette openCR-simulations.pdf[^footnote1]. The spatial model was described by Efford and Schofield (2020).

This is still something of a work in progress, so be careful to check results 'make sense' and be aware of limitations.

[^footnote1]: These supplementary vignettes are not included with the package. It is intended to distribute them on the website https://www.otago.ac.nz/density/. Otherwise contact the author.

# Outline

## Model types

openCR fits nonspatial open-population models of the Cormack-Jolly-Seber (CJS) and Jolly-Seber-Schwarz-Arnason (JSSA[^footnote2] or 'POPAN') types. JSSA models are offered in both full and conditional likelihood forms, each with several parameterizations of recruitment, and incorporating Pollock's robust design. Conditional-likelihood JSSA models are also called Pradel--Link--Barker (PLB) models. Pradel analyses are also provided.

[^footnote2]: As far as I know, this abbreviation was first used by Pledger et al. (2010). Recognising the contributions of Crosbie and Manly, Schofield and Barker (2009) and Cowen et al. (2010) referred to it the Crosbie-Manly-Arnason-Schwarz (CMAS) model. Link and Barker (2010) used 'Crosbie-Manly-Schwarz-Arnason' (CMSA) for the same model. CMSA has since been used by various authors, including Schofield and Barker (2016). JSSA is used in openCR because this highlights its evolution from the widely known Jolly-Seber model. POPAN refers to the software of Schwarz and Arnason (1996), recycled as the name of a data type in MARK.

Spatial versions of the CJS and JSSA model types are also provided[^footnote2a]. The spatial models allow for 'multi', 'proximity' or 'count' detectors as defined in secr. Several functions are implemented for the decline in hazard of detection with distance. Movement between primary sessions may be modelled (cf Ergon and Gardner 2014; Glennie et al. 2019), but particular care is needed, especially with respect to kernel truncation.

[^footnote2a]: The utility of the spatial CJS model type (CJSsecr) is in doubt because the distribution of detected animals is not uniform at first detection, but rather biased towards the vicinity of the detectors.

## Data

Data are assumed to be from a robust design. Secondary sampling sessions are nested within primary sessions and all turnover (births, deaths, immigration or emigration) is between primary sessions (Pollock 1982). There may be a single secondary session per primary session (this limits identifiability of some parameters).

## Model specification and fitting

Models are specified using formula notation as in secr. Possible predictors include both pre-defined variables for learned responses, trend over time, etc., and user-provided covariates. Models are fitted by numerically maximizing the log likelihood. The likelihood is formed as a product over capture histories (Pledger at al. 2010) rather than from summary statistics. The fitted model is an object of class 'openCR' for which generic methods are implemented (print, predict, AIC, plot etc.).

Variation in a parameter between primary sessions is modelled as e.g., model = phi ~ session[^footnote3]. Within-session variation in detection parameters may also be modelled (see field vole example in openCR-examples.pdf).

[^footnote3]: This is equivalent of ~t in Lebreton et al. (1992) or ~time in RMark, and openCR recognises ~ t as a synonym of ~ session.

## Parameterization

A selection of parameterizations is offered for recruitment in JSSA models. Models can also be parameterized in terms of the time-specific population size (non-spatial models) or density (spatial models), avoiding the super-population parameter.

Super-population size (or density in the case of secr models) may be computed as a derived parameter from 'CL' models with the function derived(), which also computes time-specific population sizes and densities.

## Features and limitations

openCR has definite limitations that may or may not be addressed in future versions. Important differences between secr and openCR are noted here. Online help is not guaranteed: users should attempt to solve their own problems, or seek help from other users via phidot or secrgroup.

# Dipper example

We start with a simple nonspatial example. Lebreton et al. (1992) demonstrated Cormack-Jolly-Seber methods with a dataset on European Dipper (Cinclus cinclus) collected by Marzolin (1988). The object dipperCH distributed with openCR provides these data in the secr 'capthist' format. See the Examples section of its help page ?dipperCH for code to input the data from other sources.

library(openCR)                   # also loads secr
options(digits = 4, width = 90)   # for more readable output


Dippers were captured annually over 1981--1987.

m.array(dipperCH, never.recap = T)   # compare Lebreton et al. 1992 Table 10


We can fit a Cormack-Jolly-Seber model directly with openCR.fit and display the estimates:

dipper.phi.t <- openCR.fit(dipperCH, type = 'CJS', model = phi~t)
predict(dipper.phi.t)
plot(dipper.phi.t, par = 'phi', ylim = c(0,1), pch = 16, col = 'red')


From this example you can see some of the virtues of openCR

• accessible data summaries
• compact model specification
• direct plotting and tabulation of results.

See openCR-examples.pdf for more extensive analyses of this dataset.

# A brief survey of open population capture--recapture models

There is a large literature on open-population capture--recapture modelling. Almost all modern models derive from the Cormack-Jolly-Seber (CJS) or Jolly-Seber (JS) models (Seber 1982), with refinements by Crosbie and Manly (1985), Schwarz and Arnason (1996), Pradel (1996) and others. The MARK software (White and Burnham 1999) implemented many of these developments and remains the standard. This section describes differences among models as they relate to openCR.

## CJS vs JS

The split between the CJS and JS model lineages is fundamental. CJS models do not model the first capture of each animal; they condition on that capture and model subsequent recapture probabilities $p$ and apparent survival $\phi$. CJS estimates of apparent survival are robust and useful (Lebreton et al. 1992), but CJS models stop short of estimating abundance, recruitment or population trend.

JS models model the first capture of each animal, and lead either directly or indirectly to estimates of abundance and recruitment. The modern development of JS methods rests heavily on Schwarz and Arnason (1996), so openCR follows Pledger et al. (2010) in using the label 'JSSA'. JSSA models were the basis of the POPAN software, which led to the POPAN data type in MARK. JSSA models are the main focus of openCR.

## Parameterization of recruitment in JSSA models

The JSSA model appears in several different forms whose unity is obscured by differing parameterizations of recruitment. The classic POPAN formulation uses entry probabilities: the members of a notional superpopulation enter the population with time-specific probability $\beta_j$ (PENT in MARK), an idea from Crosbie and Manly (1985). Other parameterizations are

• number of new entrants at each time $j$
• per capita fecundity (new entrants at time $j$ scaled by 1/number in population at $j-1$)
• seniority (reverse-time survival Pradel 1996, Nichols 2016)
• population growth rate $\lambda$
• (relative) number in population at each time $j$

Estimates of recruitment or implied recruitment from any one of these six parameterizations can be used to infer the others[^footnote4]. The choice of parameterization rests on which is more natural for the problem in hand (and allows the desired constraints to be applied) and on practicalities (some are more likely to give numerical problems than others).

[^footnote4]: except for some mostly trivial differences relating to removals

Schwarz (2001) is illuminating (see also chapter on Jolly-Seber models by Schwarz and Arnason in the MARK book, Cooch and White 2019). Pradel (1996), Williams, Nichols and Conroy (2002: p.518 et seq.), Pledger et al. (2003, 2010) and Link and Barker (2005) also comment on and compare JS parameterizations. See also the MARK help page on 'Recruitment Parameters in Jolly-Seber models' ('Recruitment Parameters' in the help index).

## Conditional (PLB) vs full likelihood JSSA

For each JSSA recruitment parameterization there is a choice between models that include the total number of detected individuals ($u_\cdot$ or $n$ in different notations), and models that condition on this number. Conditional-likelihood models do not directly estimate abundance; abundance is estimated as a derived parameter (Schwarz and Arnason 1996). Full-likelihood models include abundance as a parameter. The choice of formulation has virtually no effect on the parameter estimates[^footnote5]. The conditional likelihood form is somewhat faster and easier to fit (Schwarz and Arnason 1996), and it focuses on parameters that are estimated robustly (apparent survival, seniority, population growth rate).

The conditional models discussed by Pradel (1996), Link and Barker (2005), Schofield and Barker (2016) and others lack a distinguishing label to indicate their collective similarity. The label Pradel--Link--Barker PLB was suggested by Efford and Schofield (2020).

[^footnote5]: this may not be true for spatial models with spatially varying density, but these models are not considered in openCR.

## Sufficient statistics vs capture histories

Historically the CJS and JS likelihoods have been expressed in terms of 'sufficient statistics' that are time-specific counts of animals in different categories, such as the number caught, the number marked etc. This approach is used in the openCR function JS.direct and with the Pradel model type in openCR.fit. The likelihood may also be computed as a product over terms, one for each observed capture history[^footnote6]. Modelling of individual capture histories, is slower, but it is extremely flexible, allowing direct inclusion of censoring, learned responses, individual covariates, secondary sessions and other extensions. This is the approach used in MARK and openCR.fit.

[^footnote6]: strictly, the product over observed histories is only one component of the likelihood

## Robust design

Most published formulations of CJS and JSSA models admit only one secondary session per primary session. Data collected according to a robust design with multiple secondary sessions must be collapsed to a single sample per primary session. However, it is simple to adapt the capture-history models for multiple secondary occasions, and this makes better use of the data. MARK offers many specific robust design models. A robust design is assumed in openCR; data with a single secondary session per primary session are merely a special case.

## Spatial vs nonspatial

Models may be spatially explicit or not. Nonspatial models ignore the spatial distribution of animals. Spatial models use the spatially explicit capture--recapture paradigm of Efford (2004), Borchers and Efford (2008) and Royle et al. (2014). Open population spatial models using MCMC were published by Gardner et al. (2010), Chandler and Clark (2014), Ergon and Gardner (2014), Whittington and Sawaya (2015) and others. Glennie et al. (2019) proposed a frequentist hidden Markov formulation. The spatial models in openCR are described by Efford and Schofield (2020) and provide very similar estimates to those of Glennie et al. (2019).

There are three major motivations for open spatial models

• allowance for varying extent of sampling area
• separation of emigration and mortality

openCR fits spatial analogues of CJS and JSSA models by maximizing the likelihood. The abundance parameter is density $D$ (animals per hectare) rather than population size $N$.

Recruitment in spatial models may be modelled using parameterizations to those described above for non-spatial models, replacing 'number' by 'density'. The locations at which animals recruit are not modelled.

## Home-range shifts between primary sessions

By definition, the interval between primary sessions is long enough for turnover due to births and deaths. It is also possible that resident animals shift their home ranges (i.e. disperse). Spatial models may either ignore such movement (Gardner et al. 2010, Chandler and Clark 2014, Whittington and Sawaya 2015) or attempt to model it (Ergon and Gardner 2014). There are good arguments for modelling movement:

• Movement that is ignored inflates estimates of the within-session scale of detection $\sigma$, with flow-on effects on demographic parameters.
• If the distribution of dispersal distances can be inferred from the detection histories of residents then it is possible in principle to separate actual mortality from losses due emigration (Ergon and Gardner 2014). However, the robustness and data requirements of movement models have yet to be fully understood.

# Data structure and input

Data should be provided to openCR.fit as secr 'capthist' objects. The occasions of a single-session dataset are treated as open-population temporal samples. For spatial analyses, the capthist object should use a point detector type ('multi', 'proximity' or 'count').

openCR mostly uses the terminology of primary and secondary sessions (Pollock 1982) rather than 'session' and 'occasions' as in secr. Where 'session' appears without qualifier it refers to a primary session composed of one or more secondary sessions.

The optional intervals attribute of the capthist object defines the structure. If intervals are not specified then they default to 1.0 and each occasion is treated as a primary session. If intervals are specified then some may be zero; occasions separated by 'zero' intervals are treated as secondary sessions within the same primary session, as in MARK.

# load code from Appendix 1

# run code from Appendix 1


Fig. 1. Structure of data for open-population analysis in openCR. Primary sessions initially correspond to the sessions (components) of a multi-session secr capthist object; each primary session may have one or more secondary sessions as numbered (top). For model fitting in openCR.fit the multi-session capthist is 'joined' to form a single-session capthist with an 'intervals' attribute; non-zero intervals indicate breaks between primary sessions (bottom). The join step is automatic when a multi-session capthist is provided to openCR.fit if stratified = FALSE. (See Appendix 1 for code to make this figure).

To construct your own capthist objects --

1. Consult secr-datainput.pdf, or
2. Convert a dataframe in RMark input format using secr::unRMarkInput, or
3. Read a MARK .inp input file with read.inp.

Examples of data input code also appear on the help pages for data objects FebpossumCH, fieldvoleCH, microtusCH and dipperCH.

A multi-session capthist object will be converted automatically to a single-session object using function secr::join unless stratified = TRUE (see below). An appropriate intervals attribute is constructed, using the intervals attribute of the multi-session object for the intervals between primary sessions (1.0 if not specified), and setting other intervals to zero.

Table 1. Input formats for \textbf{openCR} 2.0

| Input | stratified | Interpretation | |:-----------------|:----------|:----------------------------------------------------| | single-session capthist | not used | secondary sessions split into primary sessions by 'intervals' | | multi-session capthist| FALSE | single stratum (each 'session' is one primary session) | | multi-session capthist| TRUE | multi-stratum (each 'session' is one stratum) |

## Stratification

From openCR 2.0 onwards any model may be stratified. For stratified models (stratified = TRUE in openCR.fit) each session of a multi-session capthist object is interpreted as an independent stratum that contributes one component of the log likelihood. Each stratum (session) has its own detectors and capture data. This assumes that primary sessions within each stratum have previously been joined manually in a nominally 'single-session' capthist. The function stratify helps you construct stratified capthist objects from collections of single-session objects.

# run code from Appendix 1


Fig. 2. Structure of data for stratified open-population analysis in openCR. Each stratum is a pre-joined (single-session) component of a multi-session capthist object. The internal structure and detector may differ between strata.

Stratified models may use 'stratum' as a factor-valued predictor. Groups of strata may be contrasted using stratum-level covariates as described later.

# Model types

The various models available in openCR are named to encode the distinctions made in the 'Brief survey'. Names are formed by concatenating four components:

1. 'CJS' vs 'JSSA'
2. Spatial ('secr') vs non-spatial (default, blank)
3. JSSA recruitment parameterization ('f','l','b','g','BN','BD','N','D' - see following)
4. JSSA likelihood conditional ('CL') vs full (default, blank)

Thus 'JSSAsecrfCL' is a spatial JSSA model parameterized in terms of per capita recruitment $f$ and fitted by maximizing the conditional likelihood (a spatial version of Link and Barker (2005), minus parameter covariation). Any movement model is specified separately with the 'movementmodel' argument of openCR.fit.

Models of the form 'JSSA...CL' are variations on the Pradel--Link--Barker models. openCR 2.0.0 recognises labels of the form 'PLB...' as an alias for each of these models. Thus 'PLBf' is synonymous with 'JSSAfCL', and 'PLBsecrl' is synonymous with 'JSSAsecrlCL'.

Parameters vary with the type of model, as listed below. Each of these primary parameters ('real' parameters in MARK) may also be modelled as a linear combination of predictors on a suitable link scale, allowing the inclusion of covariates and constraints. The coefficients of the parameter-specific linear combinations are called 'beta' parameters in MARK; the likelihood is maximized with respect to the concatenated list of beta parameters.

## Non-spatial openCR models

### Parameters and model types

Table 2. Parameter definitions and default link functions (nonspatial models)

| Parameter | Symbol | Link | Description | |:--------|:-------|:-------|:----------------------------------------------------| | p | $p$ | logit | capture probability (recapture probability for CJS) | | phi | $\phi$ | logit | apparent survival | | b | $b$ | mlogit | entry probability cf PENT in MARK | | f | $f$ | log | per capita recruitment rate | | gamma | $\gamma$ | logit | seniority (Pradel 1996) | | lambda | $\lambda$ | log | population growth rate (finite rate of increase)| | superN | $N$ | log | superpopulation size| | BN | $B_N$ | log | number of entrants| | N | $N_j$ | log | time-specific population size|

* parameters marked with an asterisk are scaled by the interval between primary sessions.

Table 3. Parameters of nonspatial openCR models

| Type | Alias | p | phi | b | f | gamma | lambda | superN | BN | N | |:--------|----|---|-----|---|---|---|---|---|---|---|
| CJS | |+ | + | | | | | | | | | JSSAbCL | PLBb | + | + | + | | | | | | | | JSSAfCL | PLBf | + | + | | + | | | | | | | JSSAgCL | PLBg | + | + | | | + | | | | | | JSSAlCL | PLBl | + | + | | | | + | | | | | JSSAb | | + | + | + | | | | + | | | | JSSAf | | + | + | | + | | | + | | | | JSSAg | | + | + | | | + | | + | | | | JSSAl | | + | + | | | | + | + | | | | JSSAB | | + | + | | | | | | + | | | JSSAN | | + | + | | | | | | | + |

Models with type ending in CL are of the Pradel--Link--Barker type, with aliases as shown.

### Non-spatial models using sufficient statistics

openCR mostly fits models by modelling capture histories one-by-one. An alternative faster method is to evaluate the likelihood expressed in terms of sufficient statistics. Sufficient statistics vary among models, but they are typically counts such as provided by the function JS.counts. The 'sufficient statistics' approach is not compatible with individual covariates. The non-spatial model types 'Pradel' and 'Pradelg' are implemented in openCR using sufficient statistics (Pradel 1996) and therefore fall outside the main framework (Table 3). They correspond to 'JSSAlCL' and 'JSSAgCL' respectively, and estimate the same parameters as those models. Estimates should coincide except when there are losses on capture. 'Pradel' is parameterized in terms of population growth rate (lambda) and 'Pradelg' is parameterized in terms of seniority (gamma).

Additionally, the function JS.direct computes classic Jolly--Seber estimates using the sufficient statistics.

## Spatial openCR models

Table 4. Parameter definitions and default link functions (spatial models)

| Parameter | Symbol | Link | Description | |:--------|:-------|:-----------|:--------------------------------------------------| | lambda0 | $\lambda_0$ | log | detection function intercept | | sigma | $\sigma$ | log | detection function scale (m) | | z | $z$ | log | detection function shape parameter (HHR, HAN, HCG, HVP) | | phi | $\phi$ | logit | apparent survival | | b | $b$ | mlogit | entry probability (beta) | | f | $f$ | log | per capita recruitment rate| | gamma | $\gamma$ | logit | seniority (Pradel 1996) | | lambda | $\lambda$ | log | population growth rate (finite rate of increase)| | superD | $D$ | log | superpopulation density| | BD | $B_D$ | log | entrants per hectare | | D | $D_j$ | log | time-specific population density|

* parameters marked with an asterisk are scaled by the interval between primary sessions.

Table 5. Parameters of spatial openCR models

| Type | Alias | lambda0 | sigma | z | phi | b | f | gamma | lambda | superD | BD | D | |:------------|----|---|---|---|---|---|---|---|---|---|---|---|
| CJSsecr | | + | + | + | + | | | | | | | | | JSSAsecrbCL | PLBsecrb | + | + | + | + | + | | | | | | | | JSSAsecrfCL | PLBsecrf | + | + | + | + | | + | | | | | | | JSSAsecrgCL | PLBsecrg | + | + | + | + | | | + | | | | | | JSSAsecrlCL | PLBsecrl | + | + | + | + | | | | + | | | | | JSSAsecrb | | + | + | + | + | + | | | | + | | | | JSSAsecrf | | + | + | + | + | | + | | | + | | | | JSSAsecrg | | + | + | + | + | | | + | | + | | | | JSSAsecrl | | + | + | + | + | | | | + | + | | | | JSSAsecrB | | + | + | + | + | | | | | | + | | | JSSAsecrD | | + | + | + | + | | | | | | | + | | secrCL | | + | + | + | | | | | | | | | | secrD | | + | + | + | | | | | | + | | |

Spatial models with type ending in CL have features in common with the Pradel--Link--Barker models, hence the aliases as shown.

# Model formulae

Formulae define a linear model for each 'real' parameter (p, phi, sigma etc.) on the link scale (logit, log etc.). Alternative link functions not shown in Tables 2 and 4 are 'loglog' and 'sin', both as defined in MARK.

The default linear combination for each parameter is a constant, null model (~1, parameter constant over time, unaffected by individual differences etc.). To include other effects build formulae using either predefined (built-in) predictors listed here, or the names of covariates.

## Built-in predictors

Table 6. Built-in predictors ('sessions' refers to primary sessions)

| Predictor | Parameters | Description | |:-----|--------------------| -------------------------------------------|
| stratum | all | Factor, one level per stratum (stratified = TRUE) | | session | all except 'superN', 'superD' | Factor, one level per primary session | | t | all except 'superN', 'superD' | synonym of 'session' | | Session | all except 'superN', 'superD' | Continuous time | | b | p, phi, lambda0, sigma | learned response (persists across sessions) | | B | p, lambda0, sigma | transient (Markovian) response across sessions| | bk | p, phi, lambda0, sigma | detector-specific learned response (persists across sessions) | | bsession | p, lambda0, sigma | learned response within sessions | | Bsession | p, lambda0, sigma | transient (Markovian) response within sessions| | bksession | p, lambda0, sigma | detector-specific learned response within sessions | | Bksession | p, lambda0, sigma | detector-specific transient (Markovian) response within sessions| | h2 | all except abundance | 2-class finite mixture | | h3 | all except abundance | 3-class finite mixture | | age | all except abundance | age factor | | Age | all except abundance | linear effect on age | | Age2 | all except abundance | linear effect on age$^2$ |

Differences among the various learned responses may be understood by examining their effect on the parameter index array (PIA). This table illustrates the PIA slice corresponding to an individual with the non-spatial detection history shown (4 primary sessions, each of 4 secondary sessions). The values '1' and '2' refer to different parameter combinations, most commonly to levels of lambda0.

| Detection history : | 0100 0000 0000 0100 | | |:--------------| --------------------------|------------| | ~bsession | 1122 1111 1111 1122 | persistent within primary session | | ~Bsession | 1121 1111 1111 1121 | transient within primary session | | ~b | 1122 2222 2222 2222 | persistent | | ~B | 1122 2222 1111 1122 | transient across primary sessions |

IMPORTANT NOTE: Learned response predictors ('b', 'bsession' etc.) were re-defined in openCR 1.3.0. Models fitted with earlier versions should be re-fitted.

## User-provided covariates

The rules for covariates largely follow secr (secr-overview.pdf). Covariates may be at the level of stratum, primary session, secondary session (detection parameters only), individual (CL models only), or detector (spatial models only). Further complexity may be modelled by providing custom design data cutting across these categories (see below).

Individual and detector covariates are named columns in the 'covariates' attributes of the respective capthist and traps object. Covariate names should differ from the built-in predictors (Table 6).

Stratum covariates are provided to openCR.fit in the argument 'stratumcov'. That should be a dataframe with one row per stratum; the name of any column may be used in a model formula.

Primary session covariates are provided to openCR.fit in the argument 'sessioncov', rather than associated with a data object. If 'sessioncov' is a vector (length equal to number of primary sessions) rather than a dataframe then it may be referenced as 'scov' in model formulae. For stratified data, 'sessioncov' may be a list with one component per stratum (the lazy option of providing a single vector or dataframe works only if all strata have the same sessions).

Covariates for detection parameters in secondary sessions are provided in the 'timecov' argument. If 'timecov' is a vector (length equal to total number of secondary sessions) rather than a dataframe then it may be referenced as 'tcov' in model formulae. For stratified data, 'timecov' may be a list with one component per stratum (the lazy option of providing a single vector or dataframe works only if all strata have the same primary and secondary sessions).

# More on modelling

## Closed populations

The types 'secrD' and 'secrCL' cause openCR.fit to treat the data as if from a closed population (no mortality, no recruitment, no movement); the intervals attribute is ignored.

msk <- make.mask(traps(captdata), buffer = 100, type = 'trapbuffer')

secr <- secr.fit(captdata, detectfn = 'HHN', mask = msk, trace = FALSE)
openCR <- openCR.fit(captdata, detectfn = 'HHN', mask = msk, type = 'secrD')

# massage the predict.openCR results to the same format as predict.secr
pred_openCR <- plyr::rbind.fill(predict(openCR))
pred_openCR <- pred_openCR[c(2,1,3), !(names(pred_openCR) %in% c('stratum','session'))]
rownames(pred_openCR) <- secr$realnames # compare estimates predict(secr)[,-1] pred_openCR  # compare timings in seconds c(secr = secr$proctime, openCR = openCR$proctime)  The maximised log likelihoods differ because openCR does not include the multinomial constant. secr has function logmultinom that lets us add it back: # compare maximised log likelihoods c(secr.logLik = logLik(secr), openCR.logLik = logLik(openCR) + logmultinom(captdata))  ## Finite mixtures Two- and three-class finite mixtures (h2, h3) allow for individual heterogeneity in detection and turnover parameters (Pledger et al. 2003, 2010). Using one of these predictors in a formula causes a further real parameter 'pmix' to be added. pmix is the proportion in latent mixture class 2 for h2, and the proportions in classes 2 and 3 for h3 (the proportion in class 1 is obtained by subtracting from 1). The implementation in openCR assumes that class membership applies across all parameters. The posterior probabilities of class membership for all detected individuals are returned as the 'posterior' component of the fitted model. Finite mixture likelihoods are prone to multimodality. Misleading estimates result when the numerical maximization settles on a local maximum (see also [secr-finitemixtures.pdf]. ## Age If age is modelled as a factor then it is useful to group older animals in a maximum age class ('maximumage'). 'minimumage', 'maximumage' and 'initialage' are optional components of the 'details' argument of openCR.fit. 'initialage' can name an individual covariate to avoid the assumption that all animals are the minimum age at first detection. The agecov argument of openCR.fit may be used to specify a recoding of numerical age, analogous to the sessioncov argument of secr.fit. Thus numerical ages may be bracketed into 'young', 'middle' and 'old'. The length of agecov should match the number of numerical ages (maximumage - minimumage + 1). For a quadratic relationship with age, specify an additive model with both Age and Age2 terms (e.g., model = phi ~ Age + Age2). ## Sampling intervals We have seen the role of the intervals attribute in defining primary and secondary sessions. Between-session intervals need to be specified only if they vary, or if you would like rates (phi, gamma, lambda, f) to be reported in time units other than the (implicitly constant) sampling interval. Scaling from the standardised parameter$\theta_j$to the interval-specific value$\theta^\prime_j$uses$\theta^\prime_j = \theta_j^{T_j}$where$\theta_j$is one of$\phi_j$or$\lambda_j$, and$T_j$is the duration of interval$j$. Scaling$\gamma$follows the same pattern except that the relevant duration for$\gamma_j$is$T_{j-1}$. Scaling per capita recruitment$f_j$is more tricky. We use$f^\prime_j = (\phi_j + f_j)^{T_j} - \phi_j^{T_j}$. ## Custom design data Occasionally there is a need for covariates that do not relate specifically to individuals, sessions or detectors, and are not included as canned predictors. For this you must construct your own dataframe of design data and pass it as the 'dframe' argument of openCR.fit. Design data are used as input to the model.matrix function (the 'data' argument); model.matrix generates the design matrix for each real parameter. Design data are usually constructed internally in openCR.fit from named covariates and other predictors that appear in model formulae; if 'dframe' is provided then the internally constructed design data are added as extra columns, overwriting any custom columns of the same name. The same design dataframe is used for all parameters. Constructing 'dframe' is fiddly. The dataframe should have one row for each combination of unique capture history, secondary session, detector and latent class (mixture). For nonspatial models without finite mixtures this collapses to one row for each capture history and secondary session. The order of rows follows that of the elements in an array with dimensions ($n$,$S$,$K$,$X$) for$n$unique capture histories,$S$secondary sessions,$K$detectors and$X$latent classes[^footnote8]. The secr function insertdim can help to expand data into the correct row order. [^footnote8]: This rectangular (or cuboidal) configuration includes cells that are redundant and unused for a particular model type (e.g., cells corresponding to sessions at or before first capture in CJS models). However, the full complement of rows is required in dframe. A warning: by default openCR.fit replaces the input capthist with a more compact version using only unique capture histories (the number of each is kept in the individual covariate 'freq'; see the function squeeze). Design data are in terms of the 'squeezed' capture histories. In this example we define a function to construct custom design data for a learned response. makedf.b <- function (ch, spatial = FALSE, nmix = 1, naive = FALSE) { R <- 1 # assume single stratum ch <- squeeze(ch) # Construct matrix of logical values TRUE iff caught before detected <- apply(abs(ch),1:2,sum)>0 detected <- t(apply(detected, 1, cumsum)>0) if (naive) b <- rep(FALSE, prod(dim(ch)[1:2])) else b <- t(apply(detected, 1, function(x) {x[which.max(x)] <- FALSE; x})) # For a simple non-spatial case: data.frame(customb = as.vector(b)) # More generally: n <- nrow(ch) S <- ncol(ch) K <- if (spatial) dim(ch)[3] else 1 data.frame(customb = insertdim(b, c(2,3,1), c(R,n,S,K,nmix))) }  Now compare the result with the canned predictor 'b' for a persistent learned response. ovenj <- join(ovenCH) fitb <- openCR.fit(ovenj, model = p ~ b) fitbc <- openCR.fit(ovenj, model = p ~ customb, dframe = makedf.b(ovenj)) AIC(fitb, fitbc)  Our custom model gives exactly the same result as the canned predictor 'b' when type = 'CJS' because the precise secondary session of first capture is irrelevant for CJS models (recaptures are modelled only for subsequent primary sessions unless details$CJSp1 == TRUE).

Discrepancies can arise with non-CJS models because these account for animals never detected. The corresponding likelihood component uses a distinct design matrix for a 'naive' animal. To customize non-CJS models a separate dframe should be provided that applies to naive animals:

fitb2 <- openCR.fit(ovenj, model = p ~ b, type = 'JSSAfCL', start = fitb)
fitbc2 <- openCR.fit(ovenj, model = p ~ customb,  type = 'JSSAfCL',
dframe = makedf.b(ovenj), dframe0 = makedf.b(ovenj, naive = TRUE))
AIC(fitb2, fitbc2)


## Transience

makedf.resident <- function (ch, spatial = FALSE, nmix = 1) {
nstrata <- 1 # assume single stratum
ch <- squeeze(ch)
n <- nrow(ch)
S <- ncol(ch)
K <- if (spatial) dim(ch)[3] else 1
primary <- primarysessions(intervals(ch))
detected <- apply(abs(ch),1:2,sum)>0
nprimary <- apply(detected, 1, function(x) length(unique(primary[x])))
data.frame(resident = insertdim(nprimary>1, 1, c(nstrata, n, S, K, nmix)))
}


A simpler approach is to code an individual covariate that scores whether an individual was detected in more than one primary session.

addresidentcov <- function (ch) {
primary <- primarysessions(intervals(ch))
detected <- apply(abs(ch), 1:2, sum)>0
nprimary <- apply(detected, 1, function(x) length(unique(primary[x])))
covariates(ch) <- data.frame(residentcov =  nprimary>1)
ch
}


Results are identical:

ovenj <- join(ovenCH)
fitnull <- openCR.fit(ovenj, model = phi ~ 1)
fitcov  <- openCR.fit(ovenj, model = phi ~ residentcov)
fitdf   <- openCR.fit(ovenj, model = phi ~ resident, dframe = makedf.resident(ovenj))
fits <- openCRlist(fitnull, fitcov, fitdf)
AIC(fits)
pred <- predict(fits, newdata = data.frame(resident = TRUE, residentcov = TRUE))
do.call(rbind, lapply(pred, '[[', 'phi'))


Hines et al. (2003) suggested extending the definition of residence to include animals captured at least $d$ days apart within a primary session; either of the approaches here may be modified accordingly. Here is the code for two individual covariates:

addresidentcov2 <- function (ch, d = 1) {
primary <- primarysessions(intervals(ch))
secondary <- secondarysessions(intervals(ch))
detected <- apply(abs(ch), 1:2, sum)>0
nprimary <- apply(detected, 1, function(x) length(unique(primary[x])))
dsecondary <- apply(detected, 1, function(x)
max(by(secondary[x], primary[x], function(y) diff(range(y)))))
covariates(ch) <- data.frame(residentcov1 = nprimary>1,
residentcov2 = nprimary>1 | dsecondary>=d)
ch
}


## Factor coding

Factor predictors take a number of discrete values (levels). These are usually represented by columns of 0's and 1's in the design matrix, where the number of columns (and coefficients) relates to the number of levels. The default in R is to use 'treatment contrasts'; one coefficient describes a reference class (level) and other coefficients represent the effect size (difference from the reference class on the link scale). By default the first level is used as the reference: for time effects (t, session) the first primary session is the reference level[^footnote9].

This may lead to trouble if the parameter is not identifiable in the reference class. One workaround is to specify a session covariate with differently ordered levels. Another is to switch from treatment contrasts to dummy variable coding in which each coefficient represents the magnitude of one real parameter on the link scale (useful in itself). Dummy variable coding is achieved by overriding the default contrasts and removing the intercept from the formula (-1). The following model fits yield the same estimates of 'real' parameters and the same log-likelihood, but with different 'beta' parameters:

fit0 <- openCR.fit(ovenCH, model = p~t)
contr.none <- function(n) contrasts(factor(1:n), contrasts = FALSE)
fitd <- openCR.fit(ovenCH, model = p ~ -1+t,
details = list(contrasts = list(t = contr.none)))
coef(fit0)
coef(fitd)


[^footnote9]: This does not apply for times when a parameter can never be estimated -- for example, openCR understands that seniority (gamma) is not estimated for the first session and uses the second session for the reference level.

## Mean of a parameter across levels of a factor

Suppose you wish to estimate the average of a parameter across levels of a factor such as time (session). Cooch and White (2019 Section 6.15) advocate modifying the design matrix so that one beta parameter (coefficient) relates directly to the mean. This is achieved very simply in openCR.fit[^footnote10] by setting the contrast function for the factor to contr.sum in the details argument[^footnote11]. With the resulting factor coding the first coefficient corresponds to the mean. Applying this to estimate the average time-specific survival rate for the dippers assuming constant recapture probability:

fit <- openCR.fit(dipperCH, model = phi~t, details = list(contrasts = list(t = contr.sum)))
invlogit(coef(fit)['phi',c('beta','lcl','ucl')])


The mean is backtransformed from the link scale. This results in some bias owing to the nonlinearity of link functions other than the identity function. Cooch and White take the position that the bias is often ignorable.

[^footnote10]: This also works in secr.fit. [^footnote11]: Helmert contrasts (contr.helmert) also yield the mean as the first coefficient, but the coding is more obscure.

# Movement models

Potential movement of home ranges between primary sessions (= dispersal) is a critical part of open-population models. The argument movementmodel of openCR.fit allows the possibilities in Table 7. Two of these do not model movement at all. The default 'static' is a null model in which each animal retains the same home range. The 'uncorrelated' option models the locations of an animal independently in each primary session; information is sacrificed and no particular movement model is implied.

The remaining options (normal, exponential, t2D, uniform, and user-supplied function) fit a dispersal kernel (Nathan et al. 2012) to represent movement between primary sessions. This usually requires at least one more parameter to represent the spatial scale of dispersal.

Table 7. Models for movement between primary sessions.

| Movement model | Parameter(s) | Description (aliases in parentheses) | |-------------|------------|--------------------------------------------| | static | (none) | Centres constant across primary sessions | | uncorrelated | (none) | Centres unconstrained | | normal | move.a | Bivariate normal kernel (Gaussian, BVN)| | exponential | move.a | Bivariate negative exponential kernel (Laplace, BVE) | | t2D | move.a, move.b | Bivariate $t$-distribution kernel (2Dt, BVT)| | annular | move.a | non-zero only at centre and edge cells | | uniform | (none) | Uniform within arbitrary kernel radius | | (user function) | move.a, move.b | User-supplied kernel function (ncores = 1 only)|

## Movement kernels {#kernels}

All movement kernels are radially symmetrical. Relative probability of movement is specified in terms of radial distance $r$ from the point of origin (Table 8).

The extent of the kernel is controlled by the argument 'kernelradius' that gives the radius in terms of mask cells. The default radius (10) results in a discretized kernel of 349 cells (square of 441 cells minus corners). Cell-specific values are normalised so that they sum to 1.0 across the kernel. Dispersal probability effectively falls to zero at the boundary of the kernel, so the kernel radius is a critical part of the model. The 'uniform' kernel has no parameters but depends critically on the user-specified kernel radius, as does the 'annular' kernel.

Table 8. Kernel probability density functions. Based in part on Nathan et al. (2012, Table 15.1) and Clark et al. (1999) with adjustment for parameterisation in openCR.

| Kernel | move.a | move.b | pdf | Expected movement | Proportion $r > R$ | |-----------|------|------|-----------------------|--------------------------|---------------------| | normal | $\alpha_g$ | | $\frac{1}{2 \pi \alpha_g^2} \exp\left( \frac{-r^2}{2\alpha_g^2} \right)$ | $\frac{\alpha_g \sqrt \pi}{\sqrt 2}$ | $\exp \left( - \frac{R^2}{2 \alpha_g^2}\right)$| | exponential | $\alpha_l$ | | $\frac{1}{2 \pi \alpha_l^2} \exp \left( \frac{-r}{\alpha_l} \right)$ | $2 \alpha_l$ | $\left(\frac{R}{\alpha_l} + 1\right) \exp\left(-\frac{R}{\alpha_l}\right)$ | | t2D | $\alpha_t$ | $\beta$ | $\frac{\beta}{\pi \alpha_t^2} \left( 1 + \frac{r^2}{\alpha_t^2} \right)^{-(\beta+1)}$ | $\alpha_t \frac{\sqrt \pi}{2} \frac{\Gamma \left( \beta-0.5 \right)}{\Gamma (\beta)}$, $\beta > 0.5$| $\left(\frac{\alpha_t^2}{\alpha_t^2 + R^2} \right)^\beta$| | annular | $p_0$ | | $p_0 \delta$, $r = 0$ | $(1-p_0) c$|---| | | | | $\frac{1-p_0}{\pi (c_2^2 - c_1^2)}$, $c_1 \le r < c_2$ ||| | | | | 0 otherwise ||| * Continuous, untruncated, kernel. Expected values for the discretized and truncated kernel will be less (see summary.kernel).

\vspace{12pt}

The 't2D' kernel is the same as '2Dt' of Clark et al. (1999) and Nathan et al. (2012), renamed to avoid initial digit. The parameter $\alpha_t$ (move.a) corresponds to $a$ in Nathan et al. (2012) and $\sqrt u$ in Clark et al. (1999); the parameter $\beta$ (move.b) corresponds to $b-1$ in Nathan et al. (2012) and $p$ in Clark et al. (1999). Defining move.b as $\beta \equiv b-1$ is handy because the default link for move.b (log) then ensures $b>1$. The degrees of freedom of the corresponding $t$-distribution are given by $\nu = 2\beta$.

The t2D' kernel approaches bivariate normal as $\beta \to \infty$ and Cauchy as $\beta \to 0$ (e.g., Clark et al. 1999). Clark et al. (1999 p. 1485) found it hard to fit this kernel to seed dispersal data.

The annular' kernel introduced in openCR 2.0.0 places proportion $p_0$ of the mass at zero (no movement) and the rest in an annulus bounded by the radii $c_1$ and $c_2$. The discrete implementation in openCR places all probability mass either at the central point (zero movement) or in the edge cells; $c_2$ is defined by the kernel radius and $c_2 - c_1$ by the kernel spacing. Cells with zero mass are dropped from the kernel.

# Extras

## Sampling variance warning {#warning}

Full models (not CL or Pradel) include superpopulation size $N$ as a variable. The default in openCR for both non-spatial and spatial models is to treat $N$ as a Poisson variable, from which it follows that the number of individuals detected at least once ($n$) is also Poisson. This is also the default in secr. However, estimates from POPAN models in MARK treat $N$ as fixed and $n$ as binomial. The assumption of fixed $N$ leads to narrower confidence intervals and estimates of detection and turnover parameters that differ slightly from conditional likelihood models (see e.g. Schofield and Barker 2016). To obtain JSSA estimates from openCR that match those from MARK it is necessary to set distribution = "binomial".

## Example datasets

Several examples of analyses with openCR are given in the associated vignette openCR-examples.pdf. These use data already formatted as secr capthist objects in R; the objects are provided in one or other package. All are available immediately openCR is loaded with library. Each has its own help page.

Table 9. Data objects in openCR. 'RD' indicates robust design with multiple secondary sessions. See openCR-examples.pdf for references.

| Data object | Spatial | RD | Species etc. | Source | |:--------------|:-------|:----|:--------------------------------------|:----------------------------------| | microtusCH etc. | No | Yes | Meadow vole Microtus pennsylvanicus USA | Nichols et al. (1984), Williams et al. (2002) | | FebpossumCH | No | Yes | Brushtail possum Trichosurus vulpecula New Zealand | M. Efford unpubl. | | dipperCH | No | No | European dipper Cinclus cinclus France | Lebreton et al. (1992), MARK | | gonodontisCH | No | No | Moth Gonodontis bidentata England | Bishop et al. (1978), Crosbie (1979) | | fieldvoleCH | Yes | Yes | Field vole Microtus agrestis* Norway | Ergon and Lambin (2013) |

Table 10. Multi-session data objects in secr.

| Data object | Spatial | RD | Species etc. | Source | |:--------------|:-------|:----|:--------------------------------------|:----------------------------------| | OVpossumCH | Yes | Yes | Brushtail possum Trichosurus vulpecula New Zealand | M. Efford unpubl. | | ovenCHp | Yes | Yes | Ovenbird Seiurus aurocapilla USA | D. Dawson and M. Efford unpubl. |

## Testing assumptions

This is generally an undeveloped field for spatially explicit capture--recapture models. Demonstrating that assumptions were not satisfied may also be of no consequence: we would usually ignore such a finding if the estimator is reasonably robust.

For Cormack-Jolly-Seber (nonspatial) models there is an established suite of tests following Burnham et al. (1987). The tests have been implemented in the U-CARE software of Choquet et al. (2009), recently translated into R by Gimenez et al (2018) as package R2ucare. Program RELEASE (Burnham et al. 1987) also implements the core CJS tests and is available through MARK.

The openCR function ucare.cjs is a wrapper for relevant functions in R2ucare, which should be installed. We briefly demonstrate it here for the dipper data of Marzolin (1988).

if (requireNamespace("R2ucare"))
ucare.cjs(dipperCH, verbose = FALSE, by = 'sex')


This invocation of ucare.cjs calls the R2ucare functions test3sr, test3sm, test2ct, test2cl and overall_CJS for each sex and provides a condensed report. For interpretation see the original papers, the R2ucare vignette, and Chapter 5 of the MARK book (Cooch and White 2019). Lebreton et al. (1992: 86) indicate only Test 3SR is meaningful for these data (see also openCR-examples.pdf).

## Limitations of openCR {#limitations}

openCR does not do

1. Continuous random effects (consider finite mixtures as an alternative)
2. Parameter counting to adjust AIC
3. Overdispersion adjustment (chat, QAIC) or goodness-of-fit tests, except for ucare.cjs (above).
4. MCMC
5. Bootstrap confidence intervals
6. Temporary emigration parameterizations of non-spatial robust-design models
7. Age-specific survival curves (Weibull etc.)
8. Mark-resight
9. SE for derived parameters and estimates with mlogit link (to be fixed)

Parameter counting and overdispersion adjustment are probably the most critical omissions. See Cooch and White (2019) for detailed coverage in the context of MARK.

## Differences from secr {#differences}

Defaults for some arguments differ between openCR.fit and secr.fit. For openCR.fit --

1. trace = FALSE
2. By default the reported log likelihood and AIC do not include the multinomial constant (details$multinom = FALSE) 3. The default criterion for AIC.openCR is 'AIC', not 'AICc' as in secr. distribution has been elevated to a full argument rather than merely a component of details. This argument describes the distribution of the number of individuals detected (default distribution = "poisson") (see here). When details$LLonly = TRUE, openCR.fit returns a vector with the log likelihood in position 1, followed by the named starting values of the coefficients (beta parameters) (secr.fit returns only the log likelihood).

In secr the argument CL is used in secr.fit to switch between full- and conditional-likelihood models. In openCR conditional-likelihood models are given a separate type with the suffix CL (or see PLB alias).

The predictor 't' is used in secr models to indicate a factor with one level for each secondary session. In openCR it is a synonym for 'session', i.e. a factor with one level for each primary session. This is consistent with the use of 't' in Lebreton et al. (1992) and makes for more compact model specification. In the unlikely event that you want to code a model with one level for each secondary session in openCR, use the 'timecov' argument.

Parts of openCR are coded in C++, via the R package Rcpp, whereas secr uses C. The Rcpp interface requires less copying of data, and enables the use of multiple threads via RcppParallel. openCR also duplicates some C++ functions in native R code, which is useful for debugging. Select the R version by setting details = list(R = TRUE) in openCR.fit. This currently works for most models except those with detector type 'multi'.

Strata (openCR >=2.0) are analogous to sessions in secr in that they are treated as independent with no re-detections of animals between strata. The total log-likelihood in openCR is the sum of stratum log likelihoods, just as the total is the sum of session loglikelihoods in secr.

These features of secr are not available in openCR

1. Hybrid mixture models (hcov in secr)
2. Groups (use strata, or CL and individual covariates, or see marked)
3. Regression splines from mgcv
4. Model averaging
5. Density surfaces and other spatial density models
6. Post-hoc probability density of activity centres (fxi in secr)
7. Non-point detectors (polygon, polygonX etc. in secr)
8. 'collate' function (make.table may do the job)
9. Variable effort for nonspatial models (cf Efford, Borchers and Mowat 2013) (The 'usage' attribute of traps objects is applied in spatial openCR models).
10. Negative binomial counts (binomN<0)

## Relationship to other software

The non-spatial capability of openCR largely duplicates MARK and RMark. Several of the nonspatial model types have exact matches in MARK (Table 11).

Table 11. Relationship of non-spatial openCR models to MARK model types

| openCR type | MARK model | Reference | |:--------|------------|-------------------| | CJS | CJS | Seber (1982) | | JSSAb | POPAN | Schwarz and Arnason (1996) | | JSSAfCL | LinkBarker | Link and Barker (2005) | | Pradel | Pradlambda | Pradel (1996) | | Pradelg | Pradsen | Pradel (1996) |

The R package marked (Laake, Johnson and Conn 2013) also overlaps substantially with the non-spatial features of openCR. Its interface echoes RMark just as openCR echoes secr. marked has some fancy features for individual covariates and random effects, and promises fast processing of large datasets. marked 1.1.13 includes full-likelihood JSSA (POPAN) models parameterized in terms of entry probabilities (type JSSAb)[^footnote15], but not the other JSSA options in Table 3.

[^footnote15]: dipper example in openCR-examples.pdf.

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\pagebreak

# Appendix 1. Code for figures.

Code used to generate schematic diagrams of data structure.

onemulti <- function(st = c(0,6,11,15), le = c(5,4,3,5), yb = 7, col=col1, outer = TRUE) {
col <- rep(col, le)
xl <- unlist(mapply(":",st,le+st-1))
yb <- rep(yb,length(xl))
xr <- xl + width
yt <- yb + height
rect(xl,yb,xr,yt,col=col)
text(xl+width/2, yb+height/2, unlist(mapply(":", 1, le)))

xl <- st - margin
yb <- rep(yb[1], length(xl)) - margin
xr <- st+le-1+width+margin
yt <- yb+height+2*margin
rect(xl,yb,xr,yt)
text(st+le/2, rep(yb[1]+2*margin,length(st))+height+0.5, paste('session',1:length(st)))
if (outer) {
rect(st[1]-3*margin, yb[1]-2*margin, tail(st+le-1,1)+width+3*margin,
yb[1]+height+8*margin)
}
}

onejoined <- function(offset = 1.5, le = c(5,4,3,5), yb = 2.2, col=col1, intervals = TRUE,
intlabel = 'intervals', leftlabel = '', outer = TRUE) {
col <- rep(col, le)
xl <- 0:(sum(le)-1)+offset
yb <- rep(yb,length(xl))
xr <- xl + width
yt <- yb + height
rect(xl,yb,xr,yt,col=col)
text(xl+width/2, yb+height/2, c(1:length(xl)))
if (intervals) {
xi <- offset + (1:(length(xl)-1)) - (1-width)/2
xip <- cumsum(le)[-length(le)]   # intermediate between primary sessions
intervals <- rep(0,length(xi))
intervals[xip] <- 1
text(xi, yb [-1]-0.8, intervals)
text(-0.2, yb[1]-0.8, intlabel)
segments(xi[xip], rep(yb[1]-0.4,length(xip)), xi[xip], rep(yb[1]+0.4,
length(xip))+height)
}
text (0.4, yb[1]+height/2, leftlabel, adj = c(1,0.5))
if (outer) {
rect(offset-2*margin, yb[1]-2*margin, sum(le)-1+offset+width+2*margin,
yb[1]+height+2*margin)
}
}

# Fig. 1 Single-stratum data
par(cex=1, xpd = TRUE, mfrow = c(1,1), mar=c(1,4,1,4))
width <- 0.85
height <- 1.1
margin <- 0.15
col1 <- c('salmon','pink','brown', 'red')
col2 <- c('green','lightgreen','darkgreen', 'lightblue')
MASS::eqscplot(0,0,xlim=c(0,20), ylim=c(0,8), type='n', axes=F,xlab='',ylab='')
onemulti(col = col1)
text(9, 5.2, 'join()', cex=1.1)
arrows (10.7,6.2,10.7,4.2)
onejoined(leftlabel='')

# Fig. 2 Multi-stratum data
par(cex = 0.9, xpd = TRUE, mfrow = c(1,1), mar = c(1,4,1,4))
MASS::eqscplot(0,0,xlim=c(-3,20), ylim=c(-2,8), type='n', axes=FALSE, xlab = '',ylab='')
onejoined(leftlabel='stratum 1', yb = 6.5, intlabel='')
onejoined(leftlabel='stratum 2', yb = 3, intlabel='')
onejoined(leftlabel='stratum 3', yb = -0.5, le = c(4,3,4,4), intlabel='', col = col2)
rect(-3, -2, 19.3, 8.7)
`

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openCR documentation built on May 16, 2021, 1:06 a.m.