Pseudo Bayesian Mixed Models under Informative Sampling

TL;DR

This paper introduces a Bayesian method for mixed models under informative sampling that provides unbiased inference for hyperparameters, outperforming traditional frequentist approaches especially when sample design variables are correlated with random effects.

Contribution

It develops a weight-exponentiated Bayesian approach for unbiased hyperparameter estimation in mixed models under informative sampling, bridging survey estimation and Bayesian modeling.

Findings

Bayesian method achieves unbiased hyperparameter estimation.

Simulation and real data show improved performance over frequentist methods.

Approach is robust across different sampling designs.

Abstract

When random effects are correlated with sample design variables, the usual approach of employing individual survey weights (constructed to be inversely proportional to the unit survey inclusion probabilities) to form a pseudo-likelihood no longer produces asymptotically unbiased inference. We construct a weight-exponentiated formulation for the random effects distribution that achieves unbiased inference for generating hyperparameters of the random effects. We contrast our approach with frequentist methods that rely on numerical integration to reveal that only the Bayesian method achieves both unbiased estimation with respect to the sampling design distribution and consistency with respect to the population generating distribution. Our simulations and real data example for a survey of business establishments demonstrate the utility of our approach across different modeling formulations and sampling designs. This work serves as a capstone for recent developmental efforts that combine traditional survey estimation approaches with the Bayesian modeling paradigm and provides a bridge across the two rich but disparate sub-fields.