Partial identification and dependence-robust confidence intervals for capture-recapture surveys

Abstract

Capture-recapture (CRC) surveys are widely used to estimate the size of a population whose members cannot be enumerated directly. When k capture samples are obtained, counts of unit captures in subsets of samples are represented naturally by a 2 k contingency table in which one element -the number of individuals appearing in none of the samples -remains unobserved. In the absence of additional assumptions, the population size is not point-identified. Assumptions about independence between samples are often used to achieve point-identification. However, real-world CRC surveys often use convenience samples in which independence cannot be guaranteed, and population size estimates under independence assumptions may lack empirical credibility. In this work, we apply the theory of partial identification to show that weak assumptions or qualitative knowledge about the nature of dependence between samples can be used to characterize a non-trivial set in which the true population size lies with high probability. We construct confidence sets for the population size under bounds on pairwise capture probabilities, and bounds on the highest order interaction term in a log-linear model using two methods: test inversion bootstrap confidence intervals, and profile likelihood confidence intervals. We apply these methods to recent survey data to estimate the number of people who inject drugs in Brussels, Belgium.

The Central Role of the Identifying Assumption in Population Size Estimation

“…The other option is to study LCMs through the framework of partial identification (Tamer, 2010;Gustafson, 2010), which was recently used in multiple-systems estimation by Sun et al (2020) for frequentist inference for partially-identified log-linear models. This would require both: 1) a better technical understanding of what parameters, or functions of parameters, of LCMs are not identified, and 2) placing substantively meaningful priors on the non-identified parameters (i.e.…”