Bayesian Mixed Effects Model Estimation under Informative Sampling
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When random effects are correlated with the response variable of interest, the usual approach of employing 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 modelling formulations and sampling designs. This work culminates recent developmental efforts in combining traditional survey estimation approaches with the Bayesian modeling paradigm and provides a bridge across the two rich but disparate sub-fields.
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