Finite Mixtures in Capture-Recapture Models

Abstract

Capture-recapture studies of closed populations may be analysed to give estimates of the population size, needed for evaluating control operations of pest species, for monitoring the size of endangered species, and for setting quotas of harvested species. The estimates may be biased if the model does not take account of possible influences on the probability of capture - for example, if the animals are heterogeneous, having intrinsically different probabilities of capture, analysis by a model which assumes homogeneity of capture will underestimate the true population size. Pollock (Ph.D. thesis, Cornell University, 1974) and Otis et al. (1978, Wildlife Monographs 62, 1-135) proposed models which allowed for three sources of variation in the capture probability, time (variation from one sample to the next), behaviour (the trap response of an animal, becoming trap-shy or trap-happy), and heterogeneity among the animals. The computer package CAPTURE provides a model selection proecedure and population estimates for these models. Some of the model fitting is from ad hoc models, and some from maximum likelihood. The model selection procedure itself, based on a discriminant analysis, has been found to be somewhat unreliable and lacking in power (Menkens and Anderson, 1988, Ecology 69, 1952-1959). In this thesis, closed population capture-recapture models are proposed in which heterogeneity of capture probabilities between animals is modelled as a random effect, using finite mixture distributions. The animals are partitioned into a fixed number of groups, within each of which they have homogeneous probability of capture. The models are fitted by maximum likelihood. This makes feasible all the models of Pollock (1974) and Otis et al. (1978) which involve heterogeneity: models Mh, Mbh, Mth and Mtbh. In a further development the time effect, usually assumed fixed, is instead modelled as a random effect, by partitioning the samples using finite mixtures. Models Mt and Mtb may be fitted with this random time effect, and the previously unavailable interactive version of Mtb is now feasible. A cross partition of animals and samples allows new fitting of Mth and Mtbh, with both heterogeneity and time as random effects. These new models have fewer parameters than the earlier ones with fixed time effects, and seem likely to prove more realistic and useful. We now have a general, unified maximum likelihood framework for fitting all eight models of Otis et al. (1978). Either Monte Carlo or modified likelihood ratio tests are available for model comparisons. The unified theory also covers removal models. In addition, covariates (if available) for animals and/or samples may be combined with this framework and the models compared via likelihood ratio tests. For most data sets, a simple dichotomy of animals and/or samples is enough to substantially correct for bias in the estimation of population size, although there is the option of fitting more than two groups if the data warrant it. Appraisal of the performance of the population estimate is included. In the region of the parameter space typical of small mammal studies, the root mean square error and the median absolute deviation show that the new estimators are comparable with those available in the CAPTURE mark-recapture package. Evaluation of the new maximum likelihood model selection procedure is needed, and a comparison should be made with the one in CAPTURE. This has not yet been done, as some of the higher models are slow to fit, making the simulations needed for power calculations too slow. The model fitting is being rewritten for speedier execution, which should make the power appraisals feasible.