A family of capture--recapture models (e.g. SECR) may include submodels that constrain variation in core parameters and include the effects of covariates. The language of generalised linear models is convenient 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).
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 commonly modelled in secr: 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 when the detection function requires three parameters. Other real parameters are used for acoustic models (beta0, beta1; http://www.otago.ac.nz/density/pdfs/secr-sound.pdf) and for the mixture proportion (pmix) in finite mixture models (http://www.otago.ac.nz/density/pdfs/secr-sound.pdf).
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 the vector got by concatenating 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
types of effect, so they use different design matrices and are described
separately here secr detection models and here
secr density models.
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 http://www.phidot.org.