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

Technical Summary: Efficient and Robust Block Designs for Order-of-Addition Experiments

Problem Statement
Order-of-Addition (OofA) experiments investigate how the sequence of component addition influences a response variable. While significant methodological progress has been made in constructing OofA designs under the Pairwise Ordering (PWO) and Component-Position (CP) models, a critical gap remains in handling heterogeneous experimental units. In practical settings—such as pharmaceutical batch processing or multi-environment biochemical assays—experiments often involve distinct groups of units (blocks) where variation between groups exceeds variation within groups. Existing literature lacks robust frameworks for constructing and evaluating block OofA designs. Without appropriate blocking, variation arising from batch-to-batch or environment-to-environment differences can obscure the true effects of the addition sequence. Furthermore, standard heuristic search algorithms used for fractional factorial designs (e.g., coordinate-exchange) are computationally inefficient for OofA designs because every run must be a valid permutation of components, creating a vast and constrained candidate space.

Methodology
The paper proposes a unified framework for constructing and selecting optimal block OofA designs based on position models and an extended indicator function approach.

1. Position Models and Indicator Functions:
   The authors adopt the position-based (PB) model introduced by Stokes and Xu (2022), which utilizes orthogonal polynomial contrasts to represent component positions. This model is more parsimonious than PWO or CP models. The study extends the indicator function framework (originally by Cheng and Ye, 2004, for factorial designs) to OofA designs.

   • For block designs, the indicator function is expanded to include a block factor $B$.
   • The function decomposes into pure-type words (representing aliasing between the mean and position effects) and mixed-type words (representing confounding between block effects and position effects).

2. Word Length Pattern (WLP) Criterion:
   A new criterion, the Word Length Pattern (WLP), is proposed to evaluate design quality.

   • The WLP is a vector $(w^P_1, w^B_1, w^P_2, w^B_2, \dots)$, where $w^P_l$ measures the severity of aliasing for $l$-th order position effects, and $w^B_l$ measures the severity of confounding between block effects and $l$-th order position effects.
   • Designs are compared using a minimum aberration criterion: a design is superior if it has a lexicographically smaller WLP. This prioritizes the reduction of confounding for lower-order effects and prioritizes reducing aliasing over confounding for effects of the same order.

3. Construction Algorithms:
   To overcome the computational intractability of exhaustive searches, the authors develop algorithms that utilize Orthogonal Latin Squares (LSs) and Component Orthogonal Arrays (COAs).

   • Candidate Generation: For $m$ components (where $m$ is a prime or prime power), the method generates a set of mutually orthogonal Latin squares over a Galois field. These are permuted to create a large set of candidate COAs and LSs.
   • Search Strategy: Instead of swapping individual runs, the algorithm constructs designs by stacking candidate COAs and LSs. The search proceeds by:
     • Assigning whole COAs or LSs to blocks.
     • Exchanging entire LSs between blocks.
     • Exchanging specific rows between blocks.
   • This "exchange of blocks/rows" strategy significantly reduces the search space compared to point-exchange algorithms, allowing for the identification of optimal designs with minimal aberration.

Key Contributions

• Theoretical Extension: The paper successfully adapts the indicator function framework to OofA designs, specifically extending it to handle block factors and defining a composite WLP that accounts for both aliasing (position effects) and confounding (block effects).
• Algorithmic Efficiency: The development of algorithms based on stacking Latin squares and COAs provides a computationally efficient alternative to exhaustive search or standard coordinate-exchange methods for constructing block OofA designs.
• Design Selection Criterion: The introduction of the WLP criterion offers a systematic, quantitative method for selecting robust block OofA designs that minimize the confounding of block effects with important position effects.

Results and Validation

• Case Studies: The paper illustrates the construction process using examples, such as a 5-component design with 3 blocks of size 12. The algorithm successfully identifies designs with specific WLPs by exchanging LSs and rows.
• Correlation Analysis: Correlation plots demonstrate that the proposed designs effectively manage confounding and aliasing between effects, confirming the theoretical properties of the WLP criterion.
• Simulation Studies: Simulations indicate that designs constructed using the proposed criterion and algorithms achieve statistical power and Type I error rates comparable to those of full block OofA designs. This suggests that the fractional block designs are efficient surrogates for full designs in terms of hypothesis testing performance.

Significance and Claims
The paper claims to provide a practical and efficient method for constructing block OofA designs, addressing a previously unexplored area in the literature. By leveraging position models and orthogonal Latin squares, the study offers a way to handle variability in heterogeneous experimental units without resorting to computationally prohibitive exhaustive searches. The authors assert that their approach ensures designs accurately reflect addition sequence effects while effectively managing associated variability. The work is presented as a foundation for future research and applications in complex experimental settings where blocking is necessary, such as in pharmaceutical manufacturing and materials science.