Constructing D-Efficient Mixed-Level Foldover Designs Using Hadamard Matrices

Abstract

This article introduces a new class of Hadamard matrix-based mixed-level foldover designs (MLFODs) and an algorithm which facilitates the construction of MLFODs. Our new MLFODs were constructed by converting some 2-level columns of a Hadamard matrix to 3-level ones. Like the 2-level foldover designs (FODs), each new MLFOD was constructed by a half fraction and its foldover. Our Hadamard-matrix based MLFODs are compared with the conference matrix-based FODs of Jones and Nachtsheim (2014) in terms of the Defficiencies and the maximum of the absolute values of the correlation coefficients among the columns of the model matrix. Like the latter, our designs are also definitive in the sense that the estimates of all main effects are unbiased with respect to any active second-order effects. In addition, they require fewer runs and can be used to study the presence of the second-order effects more efficiently. Examples illustrating the use of our new MLFODs are given.

Designs for Screening Experiments with Quantitative Factors

“…Another method of constructing this type of design was discussed in [12]. When there are more 2-level factors than 3-level ones, the readers are encouraged to use the Hadamard matrix-based designs discussed in [13]. Note that when there is a need to block a design into two blocks, one of the 2-level factors can be used as a blocking factor.…”

Response Surface Designs Robust against Nuisance Factors

“…Designs constructed by this approach can supplement the existing catalogue of designs in the literature. Although 3-level designs are used in this paper to illustrate our blocking approach, ours can also be used with 2-level designs (the factorial and fractional factorial designs) or mixed-level designs [13][14][15] or mixture designs [16].…”

Orthogonal Minimally Aliased Response Surface Designs for Three-Level Quantitative Factors and Two-Level Categorical Factors

“…A commonality of the work of Jones and Nachtsheim (2013) and that of Nachtsheim et al (2017) is that the majority of the factors in their mixed-level designs are quantitative and have three levels. Nguyen et al (2020) also proposed a method to generate mixed-level designs for experiments involving three-level quantitative and two-level categorical factors simultaneously. More specifically, they presented a heuristic foldover algorithm based on two-level orthogonal matrices that converts some of the two-level columns into three-level ones.…”

“…More specifically, they presented a heuristic foldover algorithm based on two-level orthogonal matrices that converts some of the two-level columns into three-level ones. The goal of Nguyen et al (2020) was to obtain designs with the same kind of correlation structure as DSD-A designs, while minimizing the aliasing between the quadratic effects. They named their newly constructed designs mixed-level foldover designs (MLFO designs).…”