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 $\rm{I}$ orthogonal array of strength $2$ is proved to be universally optimal over the class of binary designs, for the direct treatment effects.

翻译：本文研究了多元交叉设计的普适最优性问题。多元响应交叉设计的提出源于一个 $3 \times 3$ 交叉试验设置，该设置旨在研究口服药物 $3$ 种剂量对黏膜炎症相关基因表达的影响。受试者被分配到三种处理序列中，并在 $3$ 个时间周期内从每位受试者采集 $5$ 种不同基因表达的反应测量值。为了在交叉试验设置中对多个（或 $g$ 个，其中 $g>1$）响应进行建模，本文考虑了一个包含直接处理效应和残留处理效应的多元固定效应模型。假设模型存在非零的响应内相关性，而响应间相关性则设为零。本文推导了对应于直接处理效应的信息矩阵，并研究了一些相关结果。研究表明，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 $g>1$、$t$ 种处理、$p$ 个周期且 $p=t \geq 3$ 的情形，本文证明了由强度为 $2$ 的 $\rm{I}$ 型正交表所表示的设计，在二元设计类中对于直接处理效应是普适最优的。