ANOVA Models and Polynomial Models on Designs

Abstract

Suppose we have unreplicated responses on fractional factorial designsof multilevel factors. In this setting, the response functions arerepresented by polynomial functions, and the identifiability of the parameter can be characterized by the Gröbner basis theory. This argument is one of the classical topics on the computational algebraic statistics introduced by Pistone and Wynn (1996). In Pistone and Wynn (1996), the concept of the confounding of factors, which is an important concept in the theory of the design of experiments, is defined as the ideal membership problem to the set of polynomials vanishing on the design points (i.e., the design ideal). In this framework, some of the concepts such as confoundingand resolution, which are usually considered in the settings of regular fractional factorial designs of two-level or three-level factors, can be generalized to general fractional factorial designs.In this paper, based on the computational algebraic methods, the relationship between the ANOVA models and the polynomial models on the designs is considered. The ANOVA models are usually considered in the setting of balanced multi-way layout, and are equivalent to the polynomial models in the two-level cases. On the other hand, the correspondence is not completely clear in the general settings. In this paper, we define the ANOVA model in the setting of unbalanced multi-way layout. We also propose some ideas to characterize the ANOVA models as the polynomial models, and how to apply the computational algebraic methods to the ANOVA models.