We consider the optimal experimental design problem of allocating subjects to treatment or control when subjects participate in multiple, separate controlled experiments within a short time-frame and subject covariate information is available. Here, in addition to subject covariates, we consider the dependence among the responses coming from the subject's random effect across experiments. In this setting, the goal of the allocation is to provide precise estimates of treatment effects for each experiment. Deriving the precision matrix of the treatment effects and using D-optimality as our allocation criterion, we demonstrate the advantage of collaboratively designing and analyzing multiple experiments over traditional independent design and analysis, and propose two randomized algorithms to provide solutions to the D-optimality problem for collaborative design. The first algorithm decomposes the D-optimality problem into a sequence of subproblems, where each subproblem is a quadratic binary program that can be solved through a semi-definite relaxation based randomized algorithm with performance guarantees. The second algorithm involves solving a single semi-definite program, and randomly generating allocations for each experiment from the solution of this program. We showcase the performance of these algorithms through a simulation study, finding that our algorithms outperform covariate-agnostic methods when there are a large number of covariates.

翻译：我们研究了在受试者于短时间内参与多个独立受控实验且可获得受试者协变量信息时，将受试者分配至处理组或对照组的最优实验设计问题。在此，除受试者协变量外，我们还考虑了源于受试者在不同实验间随机效应的响应依赖性。在此设定下，分配的目标是为每个实验提供精确的处理效应估计。通过推导处理效应的精度矩阵并以D-最优性作为分配准则，我们论证了协作式设计与分析多个实验相较于传统独立设计与分析的优势，并提出两种随机化算法以解决协作式设计的D-最优性问题。第一种算法将D-最优性问题分解为一系列子问题，每个子问题均为可通过具有性能保证的半定松弛随机化算法求解的二次二元规划。第二种算法涉及求解单一半定规划，并基于该规划的解为每个实验随机生成分配方案。我们通过模拟研究展示了这些算法的性能，发现当协变量数量较多时，我们的算法优于协变量无关方法。