In this article, universally optimal multivariate crossover designs are studied. The multiple response crossover design is motivated by a $3 \times 3$ crossover setup, where the effect of $3$ doses of an oral drug are studied on gene expressions related to mucosal inflammation. Subjects are assigned to three treatment sequences and response measurements on 5 different gene expressions are taken from each subject in each of the $3$ time periods. To model multiple or $g$ responses, where $g>1$, in a crossover setup, a multivariate fixed effect model with both direct and carryover treatment effects is considered. It is assumed that there are non zero within response correlations, while between response correlations are taken to be zero. The information matrix corresponding to the direct effects is obtained and some results are studied. The information matrix in the multivariate case is shown to differ from the univariate case, particularly in the completely symmetric property. For the $g>1$ case, with $t$ treatments and $p$ periods, for $p=t \geq 3$, the design represented by a Type $I$ orthogonal array of strength $2$ is proved to be universally optimal over the class of binary designs, for the direct treatment effects.

翻译：本文研究了通用最优多元交叉设计。多响应交叉设计的动机源于一个$3 \times 3$交叉设置，其中研究了口服药物的$3$种剂量对黏膜炎症相关基因表达的影响。受试者被分配到三个治疗序列中，并在每个$3$个时间段内从每位受试者采集5种不同基因表达的响应测量值。为建模交叉设置中的多个或$g$个响应（其中$g>1$），考虑了一个包含直接效应和残留治疗效应的多元固定效应模型。假设响应内部存在非零相关性，而响应间相关性设为零。推导出直接效应的信息矩阵并研究了一些结果。结果表明，多元情况下的信息矩阵与单变量情况不同，特别是在完全对称性方面。对于$g>1$情况，当$t$种治疗和$p$个时间段，且$p=t \geq 3$时，由强度为$2$的I型正交阵列表示的设计被证明在二元设计类中对于直接治疗效应是通用最优的。