27 - Full Factorial Designs and Distracted Fractions

Summary

Abstract

Design of Experiments is an important branch of Statistics. One of its key problems is to find minimal Fractions of a Full Factorial Design, which identify a Complete Polynomial Model. This paper shows how to use Computer Algebra and Commutative Algebra techniques and results to produce good classes of solutions to the problem. It is known that most of them can be obtained by means of Gröbner bases, hence they generally depend on the term-order chosen; here we show how to use the Distracted Fractions to yield solutions independent of the term-order.

Introduction

Design of Experiments (DoE) is a branch of Statistics, which has a long tradition in the use of algebraic methods (see for example Box et al. 1978 and Collombier 1996). In general all these methods were developed in the case of binary experiments, with coding levels either {0,1} or {-1,1} and some generalizations to the non-binary case were also developed (see Collombier 1996).

More recently some connections were discovered between classical problems in Statistics and the methods of Computer Algebra. For instance in their recent work Pistone and Wynn (1996) address the problem of identifying polynomial models in general designs. In particular they point out the connection between DoE and Gröbner bases.