An Alternative Expression of the Fractional Factorial Designs for Two-level and Three-level Factors

Abstract

Two-level and three-level fractional factorial designs are widely and effectively applied in many practical fields for finding active factors. We introduce an operation for obtaining interaction effects in the design matrices of the experiments in this paper. The relationship between two factors and their interaction can be found by applying group theory with the operation, which is called the Hadamard product for two-level factors, and the generalized Hadamard product for three-level factors, based on the columns of the design matrix. The group structure can be applied to solve many related problems including the fold-over technique. The operation, i.e. the Hadamard product and the expression of the design matrices for three-level factorial experiment, are established by introducing the complex cubic root of 1 and some other required generalization.