Log-Linear Models for Capture-Recapture

Abstract

SUMMARY Log-linear models are developed for capture-recapture experiments, and their advantages and disadvantages discussed. Ways in which they can be extended, sometimes with only partial success, to open populations, subpopulations, trap dependence, and long chains of recapture periods are presented. The use of residual patterns, and analysis of subsets of data, to identify behavioural patterns and acceptable models is emphasised and illustrated with two examples. 1. Capture-Recapture The biological problem with which we are concerned is that of estimating the parameters of a free-living population of unknown size, from a sequence of s samples, in each of which any individual either is or is not observed. The models in this paper require that each animal be marked, naturally or artificially, in such a way that we can observe the numbers of animals which have any particular pattern of being or not being observed through the sequence of samples. There are t = (2s - 1) different observable patterns, which we shall refer to as capture histories, although there is no requirement for physical capture, and denote each pattern by a sequence of l's and 2's representing, respectively, seen or not seen in a particular sample. A general pattern of capture history will be denoted a, with ra, the number of individuals observed to have this pattern. Thus, for example, with s = 3 samples and a = 121, r,2, is the number of animals seen in the first and third samples but not in the second. The pattern with all 2's is by definition unobservable. In general, the population will be subject to demographic change by birth or immigration and by death or emigration. The simplest assumption for the response of an animal to the observer, one which underlies the classical Petersen, Schnabel, or Jolly-Seber estimators, is that every animal alive in the population at a sampling time has the same probability of being observed. This probability may vary from one sample to another or be constant over all, or a number of, samples. If this assumption fails it may do so for either or both of two reasons. The first is that of trap dependence, the probability of an individual being observed in a sample depending on its past history of capture. The second is that of heterogeneity between individuals who, because of differences in their behaviour or in where they live, inherently have different probabilities of being observed in any sample. The basic idea of heterogeneity refers to properties of individuals which remain constant over time, but individual animals may also have time-varying behaviour, the aspect most commonly considered being temporary departure of an individual from the population. Various excellent texts describe theoretical and practical aspects of capture-recapture, notably the comprehensive one by Seber (1982), the introduction by Begon (1979), the