Despite the increasing prevalence of vector observations, computation of optimal experimental design for multi-response models has received limited attention. To address this problem within the framework of approximate designs, we introduce mREX, an algorithm that generalizes the randomized exchange algorithm REX (J Am Stat Assoc 115:529, 2020), originally specialized for single-response models. The mREX algorithm incorporates several improvements: a novel method for computing efficient sparse initial designs, an extension to all differentiable Kiefer's optimality criteria, and an efficient method for performing optimal exchanges of weights. For the most commonly used D-optimality criterion, we propose a technique for optimal weight exchanges based on the characteristic matrix polynomial. The mREX algorithm is applicable to linear, nonlinear, and generalized linear models, and scales well to large problems. It typically converges to optimal designs faster than available alternative methods, although it does not require advanced mathematical programming solvers. We demonstrate the application of mREX to bivariate dose-response Emax models for clinical trials, both without and with the inclusion of covariates.

翻译：尽管向量观测日益普遍，但多响应模型的最优实验设计计算尚未得到充分关注。为解决近似设计框架下的这一问题，我们提出了mREX算法，该算法推广了最初专用于单响应模型的随机交换算法REX（J Am Stat Assoc 115:529, 2020）。mREX算法包含多项改进：一种计算高效稀疏初始设计的新方法、对可微Kiefer最优准则的扩展，以及执行权重最优交换的高效方法。针对最常用的D最优准则，我们提出了一种基于特征矩阵多项式的权重最优交换技术。mREX算法适用于线性、非线性和广义线性模型，并能良好扩展至大规模问题。该算法通常比现有替代方法更快收敛至最优设计，且无需依赖高级数学规划求解器。我们通过临床试验中的双变量剂量响应Emax模型（包含与不包含协变量两种情况）展示了mREX算法的应用。