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Register a new userUniform fractional factorial designs
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Added by
Yu Tang
Publication date
2012
fields
Mathematical Statistics
and research's language is
English
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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 statistical properties. In this paper uniformity is used to further distinguish fractional factorial designs, besides the minimum aberration criterion. We show that minimum aberration designs have low discrepancies on average. An efficient method for constructing uniform minimum aberration designs is proposed and optimal designs with 27 and 81 runs are obtained for practical use. These designs have good uniformity and are effective for studying quantitative factors.
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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.
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 large-sample approximations of the null distributions are poor and the enumeration of the conditional sample space is infeasible. To construct a connected Markov chain over the appropriate sample space, a common approach is to compute a Markov basis. Theoretically, a Markov basis can be characterized as a generator of a well-specified toric ideal in a polynomial ring and is computed by computational algebraic softwares. However, the computation of a Markov basis sometimes becomes infeasible even for problems of moderate sizes. In this paper, we obtain the closed form expression of minimal Markov bases for the main effect models of $2^{p-1}$ fractional factorial designs of resolution $p$.
In this paper, we develop a general approach to proving global and local uniform limit theorems for the Horvitz-Thompson empirical process arising from complex sampling designs. Global theorems such as Glivenko-Cantelli and Donsker theorems, and local theorems such as local asymptotic modulus and related ratio-type limit theorems are proved for both the Horvitz-Thompson empirical process, and its calibrated version. Limit theorems of other variants and their condition
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 statistics. We show that a set of binomial generators of cut ideals is a Markov basis of some regular two-level fractional factorial design. As application, we give a Markov basis of degree 2 for designs defined by at most two relations.
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 a multi-way heterogeneity setting obtained from $rho$ by interchanging the roles of the block factors and the treatment factors. Specifically, we take up two series of universally optimal POTBs for symmetrical experiments constructed in Morgan and Uddin (1996). We show that the duals of these plans, as multi-way designs, satisfy M-optimality. Next, we construct another series of multiway designs and proved their M-optimality, thereby generalising the result of Bagchi and Shah (1989). It may be noted that M-optimality includes all commonly used optimality criteria like A-, D- and E-optimality.
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