Use of the Ratio Plot in Capture-Recapture Estimation

Statistical graphics are a fundamental, yet often overlooked, set of components in the repertoire of data analytic tools. Graphs are quick and efficient, yet simple instruments for preliminary exploration of a dataset to understand its structure and to provide insight into influential aspects of inference such as departures from assumptions and latent patterns. In this article, we present and assess a graphical device for choosing a method for estimating population size in capture recapture studies of closed populations. The basic concept is derived from a homogeneous Poisson distribution where the ratios of neighboring Poisson probabilities multiplied by the value of the larger neighbor count are constant. This property extends to the zero-truncated Poisson distribution, which is of fundamental importance in capture recapture studies. In practice, however, this distributional property is often violated. The graphical device developed here, the ratio plot, can be used for assessing specific departures from a Poisson distribution. For example, simple contaminations of an otherwise homogeneous Poisson model can be easily detected and a robust estimator for the population size can be suggested. Several robust estimators are developed and a simulation study is provided to give some guidance on which one should be used in practice. More systematic departures can also easily be detected using the ratio plot. In this article, the focus is on Gamma-mixtures of the Poisson distribution that leads to a linear pattern (called structured heterogeneity) in the ratio plot. More generally, the article shows that the ratio plot is monotone for arbitrary mixtures of power series densities. This article has online supplementary materials.