On the use of linear models in the estimation of the size of a population using capture–recapture data
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 (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 the size of a closed population may be estimated from capture–recapture data on discrete capture occasions 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, the number of individuals captured on occasion j, the number of captures of previously marked individuals on occasion j, the number of captures of previously unmarked individuals and the number of marked individuals in the population just before occasion j then given N, and , has a hypergeometric distribution so that for each jwhich gives rise to the estimating equations , along the lines of Yip (1993). Then a closed-form estimator for N is . Our aim here is to extend these estimators to the situation where the population size 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.
Section snippets
Population size
The martingale estimating equation approach to model is easily extended to allow the use of covariates to model changes in the population size. It is reasonable to suppose that in an open population the numbers of marked individuals, in the population just before occasion j also change over time. Denote this quantity by . Note that (1) does not depend on N being constant across occasions. Let be the population size on occasion j as a function of the row vector covariates
Example
We consider annual captures of the mountain pygmy possum (Burramys Parvus) at one site at Mt. Hotham in Victoria, Australia from 2000 to 2004. See Heinze et al. (2004) for some discussion of this species. In each year trapping was conducted in November and the traps were placed in the same grid. For the present exercise we consider annual captures, that is whether or not the animals were captured in that year. This gives five capture occasions. There was a major bushfire in the area in 2003
Discussion
It has been shown that linear models for the population size may be used to model a population size that systematically varies between occasions. The approach may be extended to more complex estimators and models for the capture probabilities. The analytic derivation of standard errors is possible but as the numbers of marked individuals at j are estimated by a procedure using information on past, present and future captures this is not straightforward and shall be considered elsewhere. A more
Acknowledgement
The author is grateful to Professor Anne Chao of the National Tsing Hua University, Hsin-Chu, Taiwan for helpful discussions on the model and method and her insights into what is actually being estimated in capture–recapture studies. The author is also grateful to Dean Heinze for providing the data on the pygmy possums.
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