  Multilevel regression and poststratification (MRP) is a popular method for
addressing selection bias in subgroup estimation, with broad applications
across fields from social sciences to public health. In this paper, we examine
the inferential validity of MRP in finite populations, exploring the impact of
poststratification and model specification. The success of MRP relies heavily
on the availability of auxiliary information that is strongly related to the
outcome. To enhance the fitting performance of the outcome model, we recommend
modeling the inclusion probabilities conditionally on auxiliary variables and
incorporating flexible functions of estimated inclusion probabilities as
predictors in the mean structure. We present a statistical data integration
framework that offers robust inferences for probability and nonprobability
surveys, addressing various challenges in practical applications. Our
simulation studies indicate the statistical validity of MRP, which involves a
tradeoff between bias and variance, with greater benefits for subgroup
estimates with small sample sizes, compared to alternative methods. We have
applied our methods to the Adolescent Brain Cognitive Development (ABCD) Study,
which collected information on children across 21 geographic locations in the
U.S. to provide national representation, but is subject to selection bias as a
nonprobability sample. We focus on the cognition measure of diverse groups of
children in the ABCD study and show that the use of auxiliary variables affects
the findings on cognitive performance.
