Modeling hierarchical variance with Kronecker structure, with application to quality measures in Medicare Advantage
Studying covariance matrices in hierarchical models can reveal meaningful relationships among variables, but these become difficult to interpret as the number of variables grows. Conventional factor analysis reduces the dimension by mapping onto a set of one-dimensional factors, but does not accommodate variables with a cross-classified layout. For such applications, we develop hierarchical models with Kronecker-product (separable) covariance structure at the second level. This is motivated by survey measures of health care quality in Medicare Advantage contracts, from the Consumer Assessments of Healthcare Providers and Systems (CAHPS) survey. Annual surveys measure contract quality on aspects such as customer service, physician communication, and access to care. We model variation across contracts of 11 quality measures over 5 years. Our variables are cross-classified by measure and year, so we specify a covariance matrix that is the Kronecker product of a measure factor and a year factor. This structure allows separate analyses for the two dimensions. We apply one-way factor analysis to the across-measure Kronecker factor and time series analysis to the across-year Kronecker factor. An extended model incorporating multiple Kronecker-product terms, analogous to principal components, further improves fit. Discrepancies between Kronecker and unstructured covariance estimates provide further insight into patterns of variation.