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.

翻译：本文旨在研究交叉试验中的高效/迹最优设计，其中每个受试者在每个时间段记录多个响应。提出一个多变量固定效应模型，包含相应于多响应的直接效应和残留效应，以及允许响应之间和响应内部存在相关性的误差分散矩阵。研究了两种相关结构，即比例协方差和广义马尔可夫协方差。基于这两种协方差下的直接效应信息矩阵，以确定高效设计。针对两种协方差形式，探讨了强度为2的I型正交阵列设计的效率。为激发多变量交叉设计，采用了$3 \times 3$框架下的基因表达数据。