  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.
