Analyzing multivariate data in crossover designs using permutation tests

Studies using crossover designs typically involve observations on a large number of response variables made on each of a relatively small number of subjects. Moreover, investigators often observe the responses longitudinally over time. As the number of variates approaches the number of subjects traditional multivariate statistics based on the concept of statistical distance often are not very powerful, and when that number exceeds the total number of subjects in the study, these tests are not defined. In these situations, statisticians frequently analyze each variate separately and adjust for the multiple testing using a technique suitable for correlated data. In the case of a single variate measured repeatedly, we often make the assumption of a patterned covariance matrix and then conduct a univariate mixed-model analysis. We discuss an alternative approach using a variety of data structures in 2 x 2 crossover designs with (1) univariate response in each treatment period, (2) multivariate response in each treatment period, and (3) longitudinal repeated measures on a single variate in each treatment period.