Spatially Explicit Capture–Recapture Models

Description

A family of capture–recapture models (e.g. SECR) may include submodels that allow for contingent variation in core parameters, including the effects of covariates. The language of generalised linear models is a convenient framework for describing submodels (e.g. Huggins 1989, Lebreton et al. 1992). Each parameter is treated as a linear combination of effects on its transformed ('link') scale. This is useful for combining effects because, given a suitable link function, any combination maps to a feasible value of the parameter. The logit scale has this property for probabilities in (0,1), and the natural log scale works for positive parameters i.e. (0, +Inf). A 'sin' link (strictly asin(2x-1)) can also be used for probabilities in [0,1], and including the boundary values is sometimes useful (see MARK help). Only these functions and the null 'identity' link have so far been implemented in secr.fit.

Submodels for spatially explicit capture–recapture in secr are defined symbolically using the R formula notation. A separate linear predictor is used for each core parameter. Core parameters are 'real' parameters in the terminology of MARK, and secr uses that term to reduce confusion. Four real parameters are modelled in secr 1.3: D (density), g0, sigma and z. Only the last three real parameters, the ones jointly defining detection probability as a function of location, can be estimated directly when the model is fitted by maximizing the conditional likelihood. D is then a derived parameter. 'z' is a shape parameter used only for a 'hazard-rate' detection function (Hayes and Buckland 1983).

Each real parameter has a linear predictor of the form

y = X * beta,

where y is vector of parameter values on the link scale, X is a design matrix of predictor values, beta is a vector of coefficients, and '*' stands for matrix multiplication. The elements of beta are estimated when we fit the model; in MARK these are called 'beta parameters' to distinguish them from the 'real' parameter values in y. X has one column for each element of beta. To repeat: there is an X and a beta for each real parameter; elsewhere in the documentation we use 'beta' to refer to a concatenation of all the parameter-specific beta's. We now describe design matrices in more detail.

[Some variations on the basic SECR model do not fit easily into this framework. An example is the choice of detection function (halfnormal vs hazard-rate). These are treated as higher-level choices.]

Design matrices

The design matrix contains a column of '1's (for the constant or intercept term) and additional columns as needed to describe the effects in the submodel. Depending on the model, these may be continuous predictors (e.g. air temperature to predict occasion-to-occasion variation in g0), indicator variables (e.g. 1 if animal i was caught before occasion s, 0 otherwise), or coded factor levels.

Within secr.fit, a design matrix is constructed automatically from the input data (capthist) and the model formula (e.g. model$g0) in a 2-stage process. First, a data frame is built containing 'design data' with one column for each variable in the formula. Second, the R function model.matrix() is used to construct the design matrix. This process is hidden from the user. The design matrix will have at least one more column than the design data, and more if the formula includes interactions or factors with more than two levels. For a good description of the general approach see the documentation for RMark (Laake and Rexstad 2008). The key point is that the necessary design data can be either extracted from the inputs (capthist and mask) or generated automatically (e.g. indicator of previous capture, mentioned in the previous paragraph).

Real parameters fall into two groups: density (D) and detection (g0, sigma and z). Density and detection parameters are subject to different effects, so they use different design matrices and are described separately here:

secr detection models, secr density models

Note

The structure of secr precludes certain types of model. Unlike density, detection parameters (g0, sigma etc.) cannot be modelled as varying in space per se, whether continuously or discretely (e.g. as a function of habitat class). However, such variation may be modelled between detectors or between sessions. As an example, consider a measure of vegetation cover in a 50-m circle centred on each detector. This may be used as a detector covariate in models for g0 or sigma. A 'detector-centred' view of habitat effects is almost as sensible as an 'animal-centred' view; the one reservation is that the spatial scale (radius of the circle) is arbitrary rather than being driven by sigma as you might like. Perhaps this could be fixed in future versions by computing the trap covariate 'on the fly' from covariates in the habitat mask, given the current magnitude of sigma.

Author(s)

Murray Efford murray.efford@otago.ac.nz

References

Laake, J. and Rexstad E. (2008) Appendix C. RMark - an alternative approach to building linear models in MARK. In: Cooch, E. and White, G. (eds) Program MARK: A Gentle Introduction. 6th edition. Available online at www.phidot.org.