  This article aims to study efficient/trace optimal designs for crossover
trials, with multiple response recorded from each subject in each time period.
A multivariate fixed effect model is proposed with direct and carryover effects
corresponding to the multiple responses and the error dispersion matrix
allowing for correlations to exist between and within responses. Two
correlation structures, namely the proportional and the generalized markov
covariances are studied. The corresponding information matrices for direct
effects under the two covariances are used to determine efficient designs.
Efficiency of orthogonal array designs of Type $I$ and strength $2$ is
investigated for the two covariance forms. To motivate the multivariate
crossover designs, a gene expression data in a $3 \times 3$ framework is
utilized.

Methodology