Methods for Combining Probability and Nonprobability Samples Under Unknown Overlaps

TL;DR

This paper introduces a Bayesian hierarchical method to combine probability and nonprobability samples, estimating propensity scores directly from observed data, and compares it with existing pseudo likelihood approaches through simulations.

Contribution

It proposes a novel likelihood-based approach for combining samples, allowing direct specification of the likelihood for observed data, improving over pseudo likelihood methods.

Findings

Bayesian hierarchical model effectively estimates propensity scores.

The new method outperforms pseudo likelihood approaches in simulations.

Direct likelihood specification enhances sample integration accuracy.

Abstract

Nonprobability (convenience) samples are increasingly sought to reduce the estimation variance for one or more population variables of interest that are estimated using a randomized survey (reference) sample by increasing the effective sample size. Estimation of a population quantity derived from a convenience sample will typically result in bias since the distribution of variables of interest in the convenience sample is different from the population distribution. A recent set of approaches estimates inclusion probabilities for convenience sample units by specifying reference sample-weighted pseudo likelihoods. This paper introduces a novel approach that derives the propensity score for the observed sample as a function of inclusion probabilities for the reference and convenience samples as our main result. Our approach allows specification of a likelihood directly for the observed sample as opposed to the approximate or pseudo likelihood. We construct a Bayesian hierarchical formulation that simultaneously estimates sample propensity scores and the convenience sample inclusion probabilities. We use a Monte Carlo simulation study to compare our likelihood based results with the pseudo likelihood based approaches considered in the literature.