APPLICATION OF GRÖBNER BASES TO THE ANALYSIS OF CERTAIN TWO OR THREE LEVEL FACTORIAL DESIGNS

H. Evangelaras (Greece), I. Kotsireas (Canada) and C. Koukouvinos (Canada)

Abstract

Screening designs are useful for situations where a large number of factors (q) is examined but only few (k) of these are expected to be important. Hadamard matrices have traditionally been used for this purpose. Since these designs are only main effects plans and since the number of runs are greater than the number of active factors (main effects), there are plenty degrees of freedom unused for identifying and estimating interactions of factors. Computational Algebraic Geometry can be used to solve identifiability problems in design of experiments in Statistics. The key idea is to view the design as a solution set of a system of non-linear polynomial equations. Then, the theory of Gröbner bases allows one to identify the whole set of estimable effects (main or interactions) of the factors of the design. Modern Computer Algebra Systems such as Maple, Magma, Mathematica and Computational Algebra packages such as CoCoA and Singular contain efficient implementations of the algorithms needed to carry out the computations to solve these problems. In this paper, we discuss the application of Gröbner bases theory in certain two and three level factorial designs.

Keywords and phrases: Gröbner bases, Hadamard matrices, inequivalent projections, ideals, leading terms, divisibility condition, estimable effects.