Classical multilevel and Bayesian approaches to population size estimationusing multiple lists

One of the major objections to the standard multiple-recapture approach to population estimation is the assumption of homogeneity of individual 'captu re' probabilities. Modelling individual capture heterogeneity is complicate d by the fact that it shows up as a restricted form of interaction among li sts in the contingency table cross-classifying list memberships for all ind ividuals. Traditional log-linear modelling approaches to capture-recapture problems are well suited to modelling interactions among lists but ignore t he special dependence structure that individual heterogeneity induces. A ra ndom-effects approach, based on the Rasch model from educational testing an d introduced in this context by Darroch and co-workers and Agresti, provide s one way to introduce the dependence resulting from heterogeneity into the log-linear model; however, previous efforts to combine the Rasch-like hete rogeneity terms additively with the usual log-linear interaction terms sugg est that a more flexible approach is required. In this paper we consider bo th classical multilevel approaches and fully Bayesian hierarchical approach es to modelling individual heterogeneity and list interactions. Our framewo rk encompasses both the traditional log-linear approach and Various element s from the full Rasch model. We compare these approaches on two examples, t he first arising from an epidemiological study of a population of diabetics in Italy, and the second a study intended to assess the 'size' of the Worl d Wide Web. We also explore extensions allowing for interactions between th e Rasch and log-linear portions of the models in both the classical and the Bayesian contexts.