A Bayesian test of independence in a two-way contingency table using surrogate sampling

We consider a Bayesian approach to the study of independence in a two-way contingency table which has been obtained from a two-stage cluster sampling design. If a procedure based on single-stage simple random sampling (rather than the appropriate cluster sampling) is used to test for independence, the p-value may be too small, resulting in a conclusion that the null hypothesis is false when it is, in fact, true. For many large complex surveys the Rao–Scott corrections to the standard chi-squared (or likelihood ratio) statistic provide appropriate inference. For smaller surveys, though, the Rao–Scott corrections may not be accurate, partly because the chi-squared test is inaccurate. In this paper, we use a hierarchical Bayesian model to convert the observed cluster samples to simple random samples. This provides surrogate samples which can be used to derive the distribution of the Bayes factor. We demonstrate the utility of our procedure using an example and also provide a simulation study which establishes our methodology as a viable alternative to the Rao–Scott approximations for relatively small two-stage cluster samples. We also show the additional insight gained by displaying the distribution of the Bayes factor rather than simply relying on a summary of the distribution.