Combining Cluster Sampling and Link-Tracing Sampling to Estimate Totals and Means of Hidden Populations in Presence of Heterogeneous Probabilities of Links

We propose Horvitz-Thompson-like and Hájek-like estimators of the total and mean of a response variable associated with the elements of a hard-to-reach population, such as drug users and sex workers. A portion of the population is assumed to be covered by a frame of venues where the members of the population tend to gather. An initial cluster sample of elements is selected from the frame, where the clusters are the venues, and the elements in the sample are asked to name their contacts who belong to the population. The sample size is increased by including in the sample the named elements who are not in the initial sample. The proposed estimators do not use design-based inclusion probabilities, but model-based inclusion probabilities which are derived from a Rasch model and are estimated by maximum likelihood estimators. The inclusion probabilities are assumed to be heterogeneous, that is, they depend on the sampled people. Variance estimates are obtained by bootstrap and are used to construct confidence intervals. The performance of the proposed estimators and confidence intervals is evaluated by two numerical studies, one of them based on real data, and the results show that their performance is acceptable.