  Health disparity research often evaluates health outcomes across demographic
subgroups. Multilevel regression and poststratification (MRP) is a popular
approach for small subgroup estimation due to its ability to stabilize
estimates by fitting multilevel models and to 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 and demonstrate EMRP's improvements over
alternatives on the bias-variance tradeoff to yield valid subpopulation
inferences of interest. As an illustration, 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 WFPBB and design-based
estimates. Performances from the EMRP estimators do not differ substantially
from each other, though we would generally recommend the WFPBB-MRP for its
consistently high coverage rates.
