199-30: Testing the Hypothesis of a Kroneckar Product Covariance Matrix in Multivariate Repeated Measures Data

We consider the problem of testing of a Kronecker product structured variance covariance matrix against the unstructured variance covariance matrix in the context of multivariate repeated measures data. A likelihood ratio test statistic is developed for this purpose. However, the test statistic for this test cannot be expressed in a closed form. An algorithm, is suggested for the maximum likelihood estimation of the variance covariance matrices to test the proposed hypothesis. The algorithm, developed by using IML procedure of SAS R ©, successfully converged in all the data sets on which it was tried. The proposed hypothesis can also be in principle, tested in two stages by the use of MIXED procedure of SAS. However, it was our experience that PROC MIXED often fails to converge. Three real data examples illustrating the algorithm and calculations are also presented. INTRODUCTION Data that contain multiple measurements over time on the same response variable for each subject or experimental unit are very common, in many fields such as biomedical, pharmaceutical, industrial engineering, business etc. This type of data are commonly called (univariate) repeated measures data. Multivariate repeated measures data or doubly multivariate data are those where multiple measurements are made over time on more than one response variable on each subject or unit. Suppose there are q response variables and on each response variable, observations are taken over p time points. We represent information on a typical subject by y, a pq × 1 dimensional column vector obtained by stacking all q responses at the first time point, then stacking all q responses at the second time point below it and so on. Assume that y follows a multivariate normal distribution with mean μ and with a pq × pq positive definite variance covariance matrix Ω. Without any assumption on covariance structures, Ω has pq(pq + 1) 2 number of unknown parameters. Estimation of Ω is not possible when the number of subjects, say, n is less than or equal to pq. This is a common scenario, especially when p is large. In order to circumvent this problem one may assume some covariance structure on Ω and a convenient common structure that can be assumed is the Kronecker product structure. Several authors, e.g, Boik (1991), Galecki (1994), Naik and Rao (2001), and Chaganty and Naik (2002) have used this Kronecker product structure variance covariance matrix in their analyses. Roy (2002), Roy and Khattree (2003, 2005) have recently used this structure in the classification problems. To be specific, we assume Ω to be of the form Ω = V ⊗Σ where V is a p×p symmetric positive definite and Σ is q×q also symmetric positive definite. The matrix V represents the correlation between repeated measures on a given subject and for a given response variable. Likewise, Σ represents the covariance between all response variables on a given subject and for a given time point. The above covariance structure makes an implicit assumption that for all the response variables, the correlation structure between repeated measures remains the same and that for covariance between all the response variables does not depend on time and remains constant for all time points. The matrix of correlation of repeated measures V , for a given response variables, can have any structure or can even be unstructured. Two covariance structures for V are very common. These are, the 1 Statistics and Data Analysis SUGI 30