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Analysis of periodically administered health care surveys may reveal key patterns of variation in quality measures over time. Observations of multiple measures in multiple years yield an array-structured random variable. Its variance-covariance matrix may be constrained to be a Kronecker product of an across-measure matrix and across-year matrix. Imposing constraints such as these can aid interpretation of the patterns in correlations across dimensions (such as measure and year).
In this work, we take a Bayesian approach to estimation and inference of the covariance matrix implied by a multi-level model when the covariance is Kronecker-structured or a sum of Kronecker-structured matrices. We use a separation strategy, decomposing the covariance matrix into standard deviations and correlations. We consider shrinking the covariance matrix toward a Kronecker structure using informative priors as well as strictly enforcing the structure as a constraint. We use Markov chain Monte Carlo to make draws from the posterior distribution and explore graphical methods for displaying this posterior uncertainty about variance-covariance matrices. We illustrate with data from the CAHPS survey.
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