Embedded multilevel regression and poststratification: Model-based inference with incomplete auxiliary information.

Health disparity research often evaluates health outcomes across demographic subgroups. Multilevel regression and poststratification (MRP) is a popular approach for small subgroup estimation as it can stabilize estimates by fitting multilevel models and adjust for selection bias by poststratifying on auxiliary variables, which are population characteristics predictive of the analytic outcome. However, the granularity and quality of the estimates produced by MRP are limited by the availability of the auxiliary variables' joint distribution; data analysts often only have access to the marginal distributions. To overcome this limitation, we embed the estimation of population cell counts needed for poststratification into the MRP workflow: embedded MRP (EMRP). Under EMRP, we generate synthetic populations of the auxiliary variables before implementing MRP. All sources of estimation uncertainty are propagated with a fully Bayesian framework. Through simulation studies, we compare different methods of generating the synthetic populations and demonstrate EMRP's improvements over alternatives on the bias-variance tradeoff to yield valid subpopulation inferences of interest. We apply EMRP to the Longitudinal Survey of Wellbeing and estimate food insecurity prevalence among vulnerable groups in New York City. We find that all EMRP estimators can correct for the bias in classical MRP while maintaining lower standard errors and narrower confidence intervals than directly imputing with the weighted finite population Bayesian bootstrap (WFPBB) and design-based estimates. Performances from the EMRP estimators do not differ substantially from each other, though we would generally recommend using the WFPBB-MRP for its consistently high coverage rates.