An aberration criterion for conditional models

The growing demand and application of factorial designs especially when we center on factor screening, require a more in-depth understanding of such designs. In view of this need, we take a look at conditional models elaborated from the concept put out by Mukerjee et al. [2017]. In their work, Mukerjee and his colleagues proposed conditioned models with one pair of conditional and conditioned factors. This paper seeks to build on this by introducing conditioned models with two pairs.

In this paper, we refer to the factorial effects of one factor defined conditionally on each fixed level of another factor as conditional models. This has been seen in sliding level experiments in engineering as discussed by Wu and Hamada [2009]. These types of experiments are sometimes used in a traditional model then converted to a conditional one for specific purposes. Generally, conditional models have been preferably used because of their efficiency and the strategy for finding minimum aberration designs.

1 Introduction

The body of work in this paper revolves around two-level factorials with two pairs of conditional and conditioned factors. These factors are characterized such that the main effect and interaction effects involving one factor are defined conditionally on each fixed level of the other factor. Here, we refer to the first factor as the conditional factor and the second factor as the conditioned factor. The other factors are traditional factors.

This paper seeks to extend the parametrization, effect hierarchy order, aberration and complementary set theory in Mukerjee et al. [2017] to this setting. Catalogues of 16-run and 32-run minimum aberration designs under a condition model are also provided.

2 Parametrization and Effect hierarchy

Consider a two-tone factorial design with at least four factors each at levels 0 and 1. These factors consist of two pairs of conditional and conditioned factors. The factorial effect for every non-null combination of these factors under a conditional model with conditional and conditioned factors can be defined.

Reparametrizing the coefficients of these interactions gives us a chance to present these interactions in a whole new light. We can define unconditional and conditional factorial effects of various orders. We proceed to formulate the effect hierarchy of these interactions using a Gaussian random function with mean zero.

Conclusion

This paper focused on the value of conditional models. It introduces a new system of understanding factorial effect under the conditional model framework using two pairs of conditional and conditioned factors. It builds from Mukerjee et al. [2017] parametrization and complementary set theories while providing efficient computational procedures to engineer minimum aberration designs for any number of factors. In bringing these ideas together, we have a better appreciation of factorial interactions and the value of conditional models.