Optimal run order for order-of-addition experiments

Order-of-addition experiments have emerged as a cornerstone in modern experimental design, yet the critical role of run-order has been entirely neglected in the literature. This oversight is surprising, given that the run order can significantly influence the cost, efficiency, and validity of the experiment. Certain run orders are inherently more economical and effective, while suboptimal orders may introduce unnecessary complexities or compromise results. To ensure experimental integrity, an optimal run order must minimize factor level changes, especially when transitions between levels are costly–and safeguard against the detrimental effects of time trends, which can skew factor effect estimates and undermine conclusions. Despite its importance, the relationship between run order and order-of-addition experiments remains poorly understood. Our research addresses this critical gap by deriving the theoretical lower bound on the total number of factor level changes for ordered order-of-addition designs. We introduce novel construction methods that achieve this minimum, ensuring both cost efficiency and practicality. Additionally, we develop a rigorous framework to eliminate the bias introduced by linear time trends, proposing robust methods to construct time trend-free ordered designs. These methods lead to the establishment of optimal ordered designs that excel under both criteria. The proposed designed orders go beyond theoretical elegance–they are robust to lurking variables in the external environment, making them invaluable for real-world applications. This study not only shines the spotlight on an overlooked dimension of order-of-addition experiments but also delivers solutions that redefine how these experiments should be conducted.

添加顺序实验（order-of-addition experiments）已成为现代实验设计领域的重要基石，但现有学术文献却完全忽视了运行顺序（run-order）的关键作用。这一疏漏颇令人意外，毕竟运行顺序可对实验的成本、效率与有效性产生显著影响。部分运行顺序天生具备更高的经济性与有效性，而次优顺序则可能引入不必要的复杂度，甚至损害实验结果的可靠性。为保障实验严谨性，最优运行顺序需尽可能减少因子水平变换（factor level changes）的次数，尤其当不同水平间的转换成本高昂时，同时还需规避时间趋势（time trends）带来的不利影响——这类趋势会扭曲因子效应估计值，动摇实验结论。尽管运行顺序的重要性不言而喻，但目前学界对运行顺序与添加顺序实验之间的内在关联仍知之甚少。本研究针对这一关键空白，推导了有序添加顺序设计中因子水平变换总次数的理论下界。我们提出了可实现该下界的新型构造方法，兼顾成本效率与实际应用可行性。此外，我们构建了一套严谨的分析框架以消除线性时间趋势引入的偏差，并提出了可构造无时间趋势有序设计的稳健方法。上述方法最终确立了同时满足两项评价准则的最优有序设计。所提出的设计顺序不仅具备理论美感，还可抵御外部环境中潜在变量的干扰，因此在实际应用中极具价值。本研究不仅凸显了添加顺序实验中一个长期被忽视的研究维度，更提供了可重新定义此类实验实施方式的系统性解决方案。