# Multivariate models for the analysis of crossover designs

Crossover designs are widely used experimental plans in which each subject is given a sequence of treatments. Thus, the response of each subject is observed under each treatment in a sequence. The means of the within subject treatment differences are then compared so that the differences among subjects are eliminated as a component of variability in the evaluation of treatment effects. In the 2 x 2 x 2 crossover design, there are two treatment sequences, two treatments and two periods of treatment. In the first group, treatment A is administered in the first period and then treatment B in the second period and the treatment sequence is referred to as AB. Similarly, the second group is administered treatments in the sequence BA. Experimental subjects are randomly assigned to one of the sequence groups. Data collected in a crossover design are inherently multivariate because, even in the simplest case, more than one response is observed for each subject. However, most of the research conducted on methods of analysis for these designs has been on univariate response in each treatment period. This research is focused on the development of statistical methods for multivariate response in each treatment period One special type of multivariate structure in crossover designs occurs when a single variate is observed repeatedly in each treatment period. If the covariance matrix for this structure has a circular pattern, traditional univariate methods can be applied to provide statistical inferences for the experiment. If the covariance pattern is multivariate circular, then the traditional multivariate methods estimate more parameters than required for a valid analysis. This dissertation develops two new tests for evaluating the plausibility of the assumption that a covariance matrix is multivariate circular. Multivariate models are developed for analyzing multivariate repeated measurements in the 2 x 2 x 2 crossover design. In particular, a multivariate mixed model is developed to take advantage of the gains in power achieved when the assumption of multivariate circularity of the residual covariance matrix is justified. Methods of statistical inference are advanced in terms of tests of the general linear hypothesis and simultaneous confidence intervals for the multivariate effects