Abstract

We show how to use supplemental information from a small probability sample (ps) to do Bayesian predictive inference for finite population means of small areas using a relatively larger non-probability sample (nps). We focus on the most practical situation when there are common covariates; the nps has the study variable but no survey weights and the ps has survey weights but no study variable. We assume that the population model is correct and any functional relation between the study variable and the covariates is unspecified. Data preparation is necessary, and there are three steps, which are a double mass imputation, stratification of the population (not the samples), and creating a spatial structure to accommodate the covariates. Our main Bayesian analysis uses the conditional auto-regressive model, which helps to accommodate the covariates without incorporating them into the model, thereby avoiding a functional relation between the study variable and the covariates. However, the actual small areas are not part of the model, but we need to keep track of them, and the strata are modeled as the small areas. Our procedure allows a small area (not a stratum) to participate in several strata, and this helps to mitigate over-shrinkage, which is common in small area models. Using an illustrative example on body mass index data, our method appears to work well when compared to other methods.