Latent multinomial models for extended batch-mark data

Batch marking is common and useful for many capture-recapture studies where individual marks cannot be applied due to various constraints such as timing, cost, or marking difficulty. When batch marks are used, observed data are not individual capture histories but a set of counts including the numbers of individuals first marked, marked individuals that are recaptured, and individuals captured but released without being marked (applicable to some studies) on each capture occasion. Fitting traditional capture-recapture models to such data requires one to identify all possible sets of capture-recapture histories that may lead to the observed data, which is computationally infeasible even for a small number of capture occasions. In this paper, we propose a latent multinomial model to deal with such data, where the observed vector of counts is a non-invertible linear transformation of a latent vector that follows a multinomial distribution depending on model parameters. The latent multinomial model can be fitted efficiently through a saddlepoint approximation based maximum likelihood approach. The model framework is very flexible and can be applied to data collected with different study designs. Simulation studies indicate that reliable estimation results are obtained for all parameters of the proposed model. We apply the model to analysis of golden mantella data collected using batch marks in Central Madagascar.