Efficient and Robust Block Designs for Order-of-Addition Experiments

Designs for Order-of-Addition (OofA) experiments have received growing attention due to their impact on responses based on the sequence of component addition. In certain cases, these experiments involve heterogeneous groups of units, which necessitates the use of blocking to manage variation effects. Despite this, the exploration of block OofA designs remains limited in the literature. As experiments become increasingly complex, addressing this gap is essential to ensure that the designs accurately reflect the effects of the addition sequence and effectively handle the associated variability. Motivated by this, this paper seeks to address the gap by expanding the indicator function framework for block OofA designs. We propose the use of the word length pattern as a criterion for selecting robust block OofA designs. To improve search efficiency and reduce computational demands, we develop algorithms that employ orthogonal Latin squares for design construction and selection, minimizing the need for exhaustive searches. Our analysis, supported by correlation plots, reveals that the algorithms effectively manage confounding and aliasing between effects. Additionally, simulation studies indicate that designs based on our proposed criterion and algorithms achieve power and type I error rates comparable to those of full block OofA designs. This approach offers a practical and efficient method for constructing block OofA designs and may provide valuable insights for future research and applications.

💡 Research Summary

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The paper addresses a notable gap in the design of Order‑of‑Addition (OofA) experiments: the lack of systematic methods for incorporating heterogeneous blocking structures. While OofA designs have been extensively studied under the pairwise‑ordering (PWO) and component‑position (CP) models, most existing work assumes a homogeneous set of experimental units. In practice, however, experiments often involve multiple blocks (e.g., batches, instruments, environmental conditions) that introduce additional sources of variability. Ignoring these blocks can lead to confounded estimates of the sequence effects, reducing the reliability of conclusions.

To tackle this problem, the authors first adopt the position‑based model introduced by Stokes and Xu (2022). In this framework each component (Z_j) is treated as a “position factor” taking values (1,\dots,m), and a run corresponds to a permutation of the positions. Orthogonal polynomial contrasts (p_1(z), p_2(z),\dots) are defined over the position levels, allowing linear, quadratic, and higher‑order effects to be expressed compactly. Because the full model contains non‑estimable constraints (e.g., (\sum p_1(z)=0)), the authors work with reduced models that omit redundant terms, thereby ensuring identifiability while retaining the essential structure of the OofA effects.

The second major contribution is the adaptation of Cheng and Ye’s (2004) indicator‑function methodology to OofA designs. For any design (D) the authors define a polynomial term \