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

Abstract

Orthogonal minimally aliased response surface or OMARS designs constitute a new family of three-level experimental designs for studying quantitative factors. Many experiments, however, also involve one or more two-level categorical factors. In this work, we derive necessary conditions for the existence of mixed-level OMARS designs and present three construction methods based on integer programming. Like the original three-level OMARS designs, the new mixed-level designs are orthogonal main-effect plans in which the main effects are also orthogonal to the second-order effects. These properties distinguish the new designs from definitive screening designs with additional two-level categorical factors and other mixed-level designs recently presented in the literature. To demonstrate the flexibility of our construction methods, we provide 587 mixed-level OMARS designs as supplementary material.

Blocking OMARS designs and definitive screening designs

“…Instead, it minimizes the confounding of both the interaction effects and the quadratic effects with the blocks. We also demonstrate that the new method for blocking also works for the mixed-level OMARS designs with three-level quantitative factors and two-level categorical factors from Núñez Ares et al (2023).…”

“…For one published case study, we were able to identify a blocking arrangement of a DSD that is superior to the one used. The flexibility of the MILP blocking procedure implies that the large catalog of OMARS designs constructed by Núñez Ares and Goos (2020) as well as the catalog of mixed-level OMARS designs of Núñez Ares et al (2023) can now also be used for blocked experiments and that blocked DSDs are no longer the only design options for small orthogonally blocked response surface experiments. A remarkable result of our work is that our MILP blocking approach yields blocking arrangements of DSDs that outperform those produced by JMP 16 and created using the blocking procedure of Jones and Nachtsheim (2016).…”

“…In short, the family of OMARS designs involves many useful experimental designs with appealing statistical properties. Recently, Núñez Ares et al (2023) showed how to construct mixed-level OMARS designs containing a combination of two-and three-level factors.…”

Analysis of data from orthogonal minimally aliased response surface designs

“…Many OMARS designs offer a better power for detecting quadratic effects than DSDs and allow all full second-order models in more than three factors to be fitted. Recently, Núñez Ares et al (2023) extended the initial set of OMARS designs with a set of mixed-level OMARS designs, involving two-level categorical and three-level quantitative factors and possessing the same orthogonality properties as the initial set of pure three-level OMARS designs. Due to the recent nature of the introduction of OMARS designs in the literature, except for the design used by Maestroni et al (2018), there are no published applications of OMARS designs other than DSDs yet.…”

“…Another advantage is that our method does not rely on columns dropped from an initial design with too many columns. It is therefore applicable to DSDs that are not derived from conference matrices (such as the DSDs enumerated by Schoen et al (2022)) and to OMARS designs which are built from scratch using the methodology presented in Núñez Ares and Goos (2020) and Núñez Ares et al (2023). Finally, our method does not only work for foldover designs, but for every design in which the main effects are orthogonal to the second-order effects.…”