  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.
