A covariate adjustment for zero-truncated approaches to estimating the size of hidden and elusive populations

TL;DR

This paper enhances zero-truncated population size estimation by extending the Zelterman estimator with covariate adjustment, demonstrating improved robustness in cases with unobserved heterogeneity through case studies.

Contribution

It introduces a covariate-adjusted Zelterman estimator based on a maximum likelihood interpretation, extending its applicability to include observed heterogeneity.

Findings

The covariate-adjusted estimator performs well in real case studies.

It is more robust than existing methods under unobserved heterogeneity.

The approach is demonstrated with drug user and immigrant population data.

Abstract

In this paper we consider the estimation of population size from one-source capture--recapture data, that is, a list in which individuals can potentially be found repeatedly and where the question is how many individuals are missed by the list. As a typical example, we provide data from a drug user study in Bangkok from 2001 where the list consists of drug users who repeatedly contact treatment institutions. Drug users with 1, 2, 3$,...$ contacts occur, but drug users with zero contacts are not present, requiring the size of this group to be estimated. Statistically, these data can be considered as stemming from a zero-truncated count distribution. We revisit an estimator for the population size suggested by Zelterman that is known to be robust under potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a locally truncated Poisson likelihood which is equivalent to a binomial likelihood. This result allows the extension of the Zelterman estimator by means of logistic regression to include observed heterogeneity in the form of covariates. We also review an estimator proposed by Chao and explain why we are not able to obtain similar results for this estimator. The Zelterman estimator is applied in two case studies, the first a drug user study from Bangkok, the second an illegal immigrant study in the Netherlands. Our results suggest the new estimator should be used, in particular, if substantial unobserved heterogeneity is present.