Bayesian Estimation Under Informative Sampling with Unattenuated Dependence

TL;DR

This paper demonstrates that Bayesian pseudo-posterior inference remains consistent under complex, dependent sampling designs by relaxing traditional independence assumptions, with practical validation on national survey data.

Contribution

It introduces a relaxed condition for consistency of Bayesian pseudo-posteriors under residual sampling dependence, extending applicability to complex survey designs.

Findings

Pseudo-posterior remains consistent with unattenuated dependence.

Relaxed dependence conditions broaden survey analysis models.

Empirical validation on the National Survey on Drug Use and Health.

Abstract

An informative sampling design leads to unit inclusion probabilities that are correlated with the response variable of interest. However, multistage sampling designs may also induce higher order dependencies, which are typically ignored in the literature when establishing consistency of estimators for survey data under a condition requiring asymptotic independence among the unit inclusion probabilities. We refine and relax this condition of asymptotic independence or asymptotic factorization and demonstrate that consistency is still achieved in the presence of residual sampling dependence. A popular approach for conducting inference on a population based on a survey sample is the use of a pseudo-posterior, which uses sampling weights based on first order inclusion probabilities to exponentiate the likelihood. We show that the pseudo-posterior is consistent not only for survey designs which have asymptotic factorization, but also for designs with residual or unattenuated dependence. Using the complex sampling design of the National Survey on Drug Use and Health, we explore the impact of multistage designs and order based sampling. The use of the survey-weighted pseudo-posterior together with our relaxed requirements for the survey design establish a broad class of analysis models that can be applied to a wide variety of survey data sets.