On the use of linear models in the estimation of the size of a population using capture–recapture data

Abstract

In the analysis of capture–recapture data it is often assumed that the population is closed and any variability in the expected number of captures between occasions is due to changes in the capture probabilities. However, it is also possible that the catchable population changes in a systematic way between occasions. Here a martingale estimating equation approach is used to extend closed population estimators to models with systematic changes in the population size that may be explained by covariates. The procedure is applied to capture–recapture data collected on a population of mountain pygmy possums (Burramys Parvus) with capture occasions before and after a major bushfire, to examine the effect of the fire on the population size.

Introduction

Suppose that the size of the catchable population on each occasion may depend on covariates. For example, the catchable population in a given area may be larger at night than during the day, the number of animals resident in an area may be larger in the breeding season compared to the nonbreeding season or as in our example there may be a detrimental environmental event. Whilst open population models are available, as the changes in the population size may be explained by covariates it is preferable to include these covariates in the model for the population size. For simplicity we consider model ${\mathbf{M}}_t$ (Darroch, 1958, Otis et al., 1978, Chao and Huggins , 2005), which allows capture probabilities that vary with capture occasion but are homogeneous within the capture occasion. Here simple martingale estimating equations are used to develop closed-form estimators for the parameters in linear models for the population size.

Recall that under the model ${\mathbf{M}}_t$ the size of a closed population may be estimated from capture–recapture data on discrete capture occasions $0⩽t_1<t_2,\dots ,<t_J⩽\tau$ by exploiting the hypergeometric distribution to construct martingale estimating equations (Yip, 1993). It is not necessary to model or estimate the capture probabilities. Recall that if N is the size of a closed population, $n_j$ the number of individuals captured on occasion j, $m_j$ the number of captures of previously marked individuals on occasion j, $u_j$ the number of captures of previously unmarked individuals and $M_j={\sum }_{k=1}^{j-1}{u}_k$ the number of marked individuals in the population just before occasion j then given N, $n_j$ and $M_j$, $m_j$ has a hypergeometric distribution so that for each j

$$E(m_j|N,n_j,M_{j-1})=n_j{M}_j/N$$

which gives rise to the estimating equations ${\sum }_{j=1}^J{m}_j({\mathit{Nm}}_j-n_j{M}_j)=0$, along the lines of Yip (1993). Then a closed-form estimator for N is $\tilde{N}={\sum }_{j=1}^J{m}_j{n}_j{M}_j/{\sum }_{j=1}^J{m}_j^2$. Our aim here is to extend these estimators to the situation where the population size $N_j$ on occasion j are not constant but vary as a function of some explanatory variables. For computational simplicity we consider closed-form estimators and do not consider the optimal estimating equations of Yip (1993) as simulations suggest there is little efficiency to be gained by this.