6.1. Introduction

The multivariate and the univariate approaches to analyzing the repeated measures data presented in Chapter 5 represent the extremes of the assumptions made on covariance structures. The former has absolutely no requirements (except that the variance covariance matrix of repeated measures be positive definite) and hence requires one to estimate the maximum possible number of variance and covariance parameters. By contrast, the latter imposes stringent requirements (except when sphericity is required) when the entire covariance structure is governed only by two parameters (in the case of compound symmetry and a few more in the of Huynh-Feldt structure).

Nonetheless, each of these two approaches permits us to use the least squares and the (univariate or multivariate) analysis of variance based approach for data analysis. What is desirable is to assume a covariance structure on repeated measures, which is not as liberal as the multivariate approach but at the same time is not as restrictive and hence not as unrealistic as the univariate approach of Chapter 5. Unfortunately, as soon as any deviation from the two is allowed, the analysis of variance based approach is no longer valid. However, in such a situation one can use an alternative (but not necessarily equivalent) approach based on likelihoods. While the likelihood theory regarding estimation and testing of hypothesis is well established, it must be noted that most of the likelihood based statistical test procedures are asymptotic in nature and hence are only approximate for the finite sample cases.

Some of the useful covariance structures other than compound symmetry for the repeated measure data are the first order autoregressive, unstructured covariance, Toeplitz, and banded Toeplitz. The MIXED procedure provides all of these with several other covariance structures as options. Further, the univariate split plot analysis of repeated measures can be performed by using the MIXED procedure, which makes available several alternative covariance structures including those listed above and in addition to compound symmetry. This kind of analysis under the general mixed effects linear model is the major theme of this chapter. Specifically, this chapter explains how to formulate problems in repeated measures data analysis as problems in the general mixed effects linear model and to utilize the MIXED procedure to solve these problems.