Post-Stratification: A Modeler's Perspective

Abstract

Post-stratification is a common technique in survey analysis for incorporating population distributions of variables into survey estimates. The basic technique divides the sample into post-strata, and computes a post-stratification weight w ih = rP h /r h for each sample case in post-stratum h, where r h is the number of survey respondents in post-stratum h, P h is the population proportion from a census, and r is the respondent sample size. Survey estimates, such as functions of means and totals, then weight cases by w h . Variants and extensions of the method include truncation of the weights to avoid excessive variability and raking to a set of two or more univariate marginal distributions. Literature on post-stratification is limited and has mainly taken the randomization (or design-based) perspective, where inference is based on the sampling distribution with population values held fixed. This article develops Bayesian model-based theory for the method. A basic normal post-stratification model is introduced which yields the post-stratified mean as the posterior mean, and a posterior variance that incorporates adjustments for estimating variances. Modifications are then proposed for small sample inference, based on (a) changing the Jeffreys prior for the post-stratum parameters to borrow strength across post-strata, and (b) ignoring partial information about the post-strata. In particular, practical rules for collapsing post-strata to reduce posterior variance are developed and compared with frequentist approaches. Methods for two post-stratifying variables are also considered. Raking sample counts and respondent counts is shown to provide approximate Bayesian inferences when the margins of the two post-stratifiers are available but their joint distribution is not. When the joint distribution is available, raking effectively ignores the information it contains, and hence can be compared with other techniques that ignore information such as collapsing. For inference about means, it is suggested that raking is most appropriate when post-stratum means have an additive or near-additive structure, whereas collapsing is indicated when interactions are present.