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A polynomial indicator function of designs is first introduced by Fontana, Pistone and Rogantin (2000) for two-level designs. They give the structure of the indicator function of two-level designs, especially from the viewpoints of the orthogonality of the designs. Based on these structure, they use the indicator functions to classify all the orthogonal fractional factorial designs with given sizes using computational algebraic software. In this paper, generalizing the results on two-level designs, the structure of the indicator functions for multi-level designs is derived. We give a system of algebraic equations for the coefficients of indicator functions of fractional factorial designs with given orthogonality. We also give another representation of the indicator function, a contrast representation, which reflects the size and the orthogonality of the corresponding design directly. The contrast representation is determined by a contrast matrix, and does not depend on the level-coding, which is one of the advantages of it. We use these results to classify orthogonal $2^3times 3$ designs with strength $2$ and orthogonal $2^4times 3$ designs with strength $3$ by computational algebraic software.
The minimum aberration criterion has been frequently used in the selection of fractional factorial designs with nominal factors. For designs with quantitative factors, however, level permutation of factors could alter their geometrical structures and
We consider conditional exact tests of factor effects in designed experiments for discrete response variables. Similarly to the analysis of contingency tables, Markov chain Monte Carlo methods can be used for performing exact tests, especially when l
It is known that a Markov basis of the binary graph model of a graph $G$ corresponds to a set of binomial generators of cut ideals $I_{widehat{G}}$ of the suspension $widehat{G}$ of $G$. In this paper, we give another application of cut ideals to sta
A new class of two-level non-regular fractional factorial designs is defined. We call this class an {it affinely full-dimensional factorial design}, meaning that design points in the design of this class are not contained in any affine hyperplane in
In this paper we study optimality aspects of a certain type of designs in a multi-way heterogeneity setting. These are ``duals of plans orthogonal through the block factor (POTB). Here by the dual of a main effect plan (say $rho$) we mean a design in