This article aims to study efficient/trace optimal designs for crossover trials with multiple responses recorded from each subject in the time periods. A multivariate fixed effects model is proposed with direct and carryover effects corresponding to the multiple responses. The corresponding error dispersion matrix is chosen to be either of the proportional or the generalized Markov covariance type, permitting the existence of direct and cross-correlations within and between the multiple responses. The corresponding information matrices for direct effects under the two types of dispersions are used to determine efficient designs. The efficiency of orthogonal array designs of Type $I$ and strength $2$ is investigated for a wide choice of covariance functions, namely, Mat($0.5$), Mat($1.5$) and Mat($\infty$). To motivate these multivariate crossover designs, a gene expression dataset in a $3 \times 3$ framework is utilized.

翻译：本文旨在研究针对每个受试者在不同时间段记录多个响应的交叉试验的高效/迹最优设计。提出了一种多元固定效应模型，该模型包含与多个响应相对应的直接效应和残留效应。相应的误差协方差矩阵选择为比例型或广义马尔可夫协方差型，允许在多个响应内部及之间存在直接相关性和交叉相关性。利用两种协方差类型下直接效应对应的信息矩阵来确定高效设计。针对广泛的协方差函数选择，即Mat($0.5$)、Mat($1.5$)和Mat($\infty$)，研究了I型强度为$2$的正交阵列设计的效率。为说明这些多元交叉设计的应用，本文采用了一个$3 \times 3$框架下的基因表达数据集。