A Multiple‐Record Systems Estimation Method that Takes Observed and Unobserved Heterogeneity into Account

Abstract

Summary. We present a model to estimate the size of an unknown population from a number of lists that applies when the assumptions of (a) homogeneity of capture probabilities of individuals and (b) marginal independence of lists are violated. This situation typically occurs in epidemiological studies, where the heterogeneity of individuals is severe and researchers cannot control the independence between sources of ascertainment. We discuss the situation when categorical covariates are available and the interest is not only in the total undercount, but also in the undercount within each stratum resulting from the crossclassification of the covariates. We also present several techniques for determining confidence intervals of the undercount within each stratum using the profile log likelihood, thereby extending the work of Cormack (1992, Biometrics 48, 567-576).

A Lower Bound Model for Multiple Record Systems Estimation with Heterogeneous Catchability

“…Following Stanghelli and van der Heijden (2004) we use the EM algorithm to fit this model. The unobserved complete data vector has entries y x,u , the frequency of capture history x in class u, u = 0,1.…”

“…The M step fits a Poisson regression to the dependent vector y while the E step estimates the dependent vector using y x,u = n x ׈y x,u /(y x,0 +ˆy x,1 ), andˆy x,u stands for the predicted value for (x,u) obtained at the previous M step. The conditional estimator for N is n+exp(γ)+exp(γ +γ U ) and the variance is calculated as in Stanghelli and van der Heijden (2004).…”

“…Many analyses consist of a stepwise search of important log-linear list interactions, see Chao, Tsay, Lin, Shau, and Chao (2001), Cormack, Chang, and Smith (2000) and Aaron, Chang, Markovic, and LaPorte (2003). More recently, Forcina (2001, 2006), Stanghelli and van der Heijden (2004) use a few latent classes to account for the heterogeneity; they use discrete distributions for the heterogeneous catchability.…”

“…This work considers a mixture model similar to that presented in Bartolucci and Forcina (2001) and Stanghelli and van der Heijden (2004), with list dependencies and heterogeneous capture probabilities. First a lower bound for the population size for a known set of list dependencies that generalizes the bound of Chao (1998) is presented.…”

“…Models obtained with simple catchability distributions such as the logit normal of Coull and Agresti (1999) and the latent class model of Stanghelli and van der Heijden (2004), are considered. Situations where the maximum likelihood estimator of N for the latent class model does not exist are identified.…”