| Abstract: | For longitudinal data, the modeling of a correlation matrix ?R can be a difficult statistical task due to both the positive definite and the unit diagonal constraints. Because the number of parameters increases quadratically in the dimension, it is often useful to consider a sparse parameterization. We introduce a pair of prior distributions on the set of correlation matrices for longitudinal data through the partial autocorrelations (PACs), which vary independently over (?1,1). The first prior shrinks each of the PACs toward zero with increasingly aggressive shrinkage in lag. The second prior (a selection prior) is a mixture of a zero point mass and a continuous component for each PAC, allowing for a sparse representation. The structure implied under our priors is readily interpretable for time-ordered responses because each zero PAC implies a conditional independence relationship in the distribution of the data. Selection priors on the PACs provide a computationally attractive alternative to selection on the elements of ?R or ?R? 1 for ordered data. These priors allow for data-dependent shrinkage/selection under an intuitive parameterization in an unconstrained setting. The proposed priors are compared to standard methods through a simulation study and illustrated using a multivariate probit data example. Supplemental materials for this article (appendix, data, and R code) are available online. |
