Methodology for pooling subpopulation regressions when sample sizes are small and there is uncertainty about which subpopulations are similar

Inference for parameters associated with small geographical areas or domains of study requires considerable care because the subpopulation sample sizes are usually very small. Since sample survey data are usually clustered, hierarchical models are often appropriate. However, the customary hierarchical models may specify more exchangeability than is warranted. Thus, we propose an alternative model that is more flexible. We consider the case of a set of multiple linear regressions, one for each subpopulation. The objective is to make inference about one or more regression coefficients, β_i. We derive the posterior mean and variance of β_i, and obtain simplified versions of these moments by using reference-type prior distributions. We use a set of numerical examples to contrast our method with the more conventional hierarchical analysis, and to exhibit the large gains in precision that are possible.