Generalizing the two sources capture–recapture estimator for covariate information

Dual system estimation is a statistical tool used to estimate the hidden population where each of the two sources covers part of the population. In estimation, the widely used Lincoln–Petersen and Chapman estimators are used under the assumption of two-source independence. However, this situation is often unrealistic, and the traditional estimators might have a large bias, especially if the two sources experience a large dependence. To improve the methodology, covariate information is used for stratification. We formulate the strata-adjusted Chapman estimator for estimating the population size. Furthermore, the log-linear modeling approach is adopted to cope with covariate information. Two semi-parametric (imputed) bootstrap methods using the log-linear model with and without incorporating the model selection procedure are proposed to quantify the uncertainty surrounding the estimates of the unknown population size. Across all designs, methods that exploit covariate information—either via stratification or within the log-linear modeling framework—consistently reduce bias and mean squared error relative to unadjusted estimators that ignore heterogeneity. For interval estimation, the imputed bootstrap that incorporates model selection in each replicate achieves coverage close to the nominal level across heterogeneous and sparse regimes, typically outperforming both the fixed-model bootstrap and normal-theory intervals for the strata-adjusted Chapman. We illustrate these approaches in an application to estimate the hidden number of suicides in Cambodia and heroin users in Thailand.