Bayesian Analysis of Multivariate Matched Proportions with Sparse Response

Multivariate matched proportions (MMP) data appears in a variety of contexts including post-market surveillance of adverse events in pharmaceuticals, disease classification, and agreement between care providers. It consists of multiple sets of paired binary measurements taken on the same subject. While recent work proposes non-Bayesian methods to address the complexities of MMP data, the issue of sparse response, where no or very few "yes" responses are recorded for one or more sets, is unaddressed. The presence of sparse response sets results in underestimates of variance, loss of coverage, and lowered power in existing methods. Bayesian methods have not previously been considered for MMP data but provide a useful framework when sparse responses are present. In particular, the Bayesian probit model provides an elegant solution to the problem of variance underestimation. We examine three approaches built on that model: a naive analysis with flat priors, a penalized analysis using half-Cauchy priors on the mean model variances, and a multivariate analysis with a Bayesian functional principal component analysis (FPCA) to model the latent covariance. We show that the multivariate analysis performs well on MMP data with sparse responses and outperforms existing non-Bayesian methods. In a re-analysis of data from a study of the system of care (SOC) framework for children with mental and behavioral disorders, we are able to provide a more complete picture of the relationships in the data. Our analysis provides additional insights into the functioning on the SOC that a previous univariate analysis missed.