We discuss the statistical analysis of data from two clinical trials using crossover designs. In both studies, response was observed repeatedly over time in each treatment period. The first study involves repeated measurements of a single response variable whereas the second involves bivariate response. Methods are described for fitting polynomial growth curves to achieve data reduction in a two-stage approach to the analysis of crossover designs. Thus, a multivariate parametric analysis frequently can be conducted even when the sample sizes are somewhat small as is the case in many crossover designs. Hypotheses that are usually of interest in crossover designs can be tested in the second stage of the analysis. Methods for testing the multivariate general linear hypothesis as a basis for statistical inference in such problems are discussed.