Mixed-level Factorial Designs Research Articles

An algorithm for generating good mixed level factorial designs

An algorithm for the creation of mixed level arrays with generalized minimum aberration (GMA) is proposed. GMA mixed level arrays are particularly useful for experiments involving qualitative factors: for these, the number of factor levels is often a consequence of subject matter requirements, while a priori assumptions on a statistical model are not made, apart from assuming lower order effects to be more important than higher order effects. The proposed algorithm creates GMA arrays using mixed integer optimization with conic quadratic constraints. Fully achieving GMA is feasible for small problems; for larger problems, the optimization task is reduced to considering the confounding of low-order effects only. Lower bounds for the lowest-order confounding are provided (given the number of experimental runs). Where one of these bounds is actually attainable, the algorithm is often fast in providing an array which attains it. Examples illustrate the scope and usefulness of the algorithm, which is implemented in an R package, using one of two commercial optimizers.

Frequency tables for the coding-invariant quality assessment of factorial designs

ABSTRACTQuality assessment of factorial designs, particularly mixed-level factorial designs, is a nontrivial task. Existing methods for orthogonal arrays include generalized minimum aberration, a modification thereof that was proposed by Wu and Zhang for mixed two- and four-level arrays, and minimum projection aberration. For supersaturated designs, E(s2) or χ2-based criteria are widely used. Based on recent insights by Grömping and Xu regarding the interpretation of the projected aR values used in minimum projection aberration, this article proposes three new types of frequency tables for assessing the quality of level-balanced factorial designs. These are coding invariant, which is particularly important for designs with qualitative factors. The proposed tables are used in the same way as those used in minimum projection aberration and behave more favorably when used for mixed-level arrays. Furthermore, they are much more manageable than the above-mentioned approach by Wu and Zhang. The article justifies the proposed tables based on their statistical information content, makes recommendations for their use, and compares them with each other and with existing criteria. As a byproduct, it is shown that generalized minimum aberration refines the established expected χ2 criterion for level-balanced supersaturated designs.

On the equivalence of definitions for regular fractions of mixed-level factorial designs

The notion of regularity for fractional factorial designs was originally defined only for two-level factorial designs. Recently, rather different definitions for regular fractions of mixed-level factorial designs have been proposed by Collombier [1996. Plans d’Expérience Factoriels. Springer, Berlin], Wu and Hamada [2000. Experiments. Wiley, New York] and Pistone and Rogantin [2008. Indicator function and complex coding for mixed fractional factorial designs. J. Statist. Plann. Inference 138, 787–802]. In this paper we prove that, surprisingly, these definitions are equivalent. The proof of equivalence relies heavily on the character theory of finite Abelian groups. The group-theoretic framework provides a unified approach to deal with mixed-level factorial designs and treat symmetric factorial designs as a special case. We show how within this framework each regular fraction is uniquely characterized by a defining relation as for two-level factorial designs. The framework also allows us to extend the result that every regular fraction is an orthogonal array of a strength that is related to its resolution, as stated in Dey and Mukerjee [1999. Fractional Factorial Plans. Wiley, New York] to mixed-level factorial designs.

Construction of Efficient Mixed-Level Fractional Factorial Designs

Mixed-level factorial experimental designs involve factors with different numbers of levels. Full factorial designs require runs at all possible combinations of the factor levels. As the number of factors and/or factor levels increases, the total number of experiments increases dramatically. As a result, interest has focused on developing orthogonal or near-orthogonal mixed-level fractional factorial designs. Currently existing mixed-level designs are all balanced. However, relaxing the requirement of balance may result in a reduced number of experimental runs in practice. The objective of this paper is to develop mixed-level fractional factorial designs with economical run sizes that are as nearly balanced and orthogonal as possible. A new criterion is developed to assess the degree of near-balance for comparing and constructing designs. A modified J2-optimality criterion is used for evaluating design near orthogonality. These criteria are combined and used to assess different design alternatives. Three algorithms are then compared and used to build designs with desirable combinations of near balance and near orthogonality.

An approach for studying aliasing relations of mixed fractional factorials based on product arrays

Mixed-level fractional factorial designs are commonly used in industries but its aliasing relations have not been studied in full rigor. These designs take the form of a product array. Aliasing patterns of mixed level factorial designs, in the form of product arrays, are discussed here.

Minimum aberration blocking of regular mixed factorial designs

It is known by Zhang and Park (J. Statist. Plann. Inference 91 (2000) 107) that there are no minimum aberration (MA) designs with respect to both treatments and blocks for blocked regular mixed-level factorial designs. So it should be compromised between the block wordlength pattern and treatment wordlength pattern. Two methods are considered in this article. The first is MA blocking scheme of an MA design. The other is to combine the components of the two wordlength pattern vectors into one combined wordlength pattern according to the modified hierarchical assumptions and an appropriate ordering of the numbers of alias or confounding relations. The relationship between the two types of optimal blocked designs is investigated. A complete catalogue of optimal blocked regular mixed factorial designs of the above two types with 16 or 32 runs is given.

Theory of minimum aberration blocked regular mixed factorial designs

Chen and Cheng (Ann. Statist. 27 (1999) 1948) got the relationships between a blocked symmetrical factorial design and its residual design. In this paper, we extend the study to the case of blocked mixed-level factorial designs. By introducing a concept of blocked consulting design and using MacWilliams identities and Krawtchouk polynomials in coding theory, we obtain combinatorial identities that govern the relationships between the partitioned wordlength patterns of a blocked regular mixed factorial design and that of its blocked consulting design. Based on these identities, we furthermore establish some general rules for identifying minimum aberration blocked mixed factorial designs in terms of their blocked consulting designs. As applications, some blocked 4 1 2 n and 4 2 2 n designs with 16 and 32 runs are tabulated.

Trend-Free Run Orders of Mixed-Level Fractional Factorial Designs

Coster and Cheng presented a generalized foldover scheme for the construction of systematic run orders of fractional factorial designs, with all factors having the same prime power number of levels, for which all the main effects components of the factors are orthogonal to a polynomial trend present in every block of the design. In this paper, we present modifications to the foldover method that allow polynomial trend-free run orders to be constructed in the following more general settings: Designs for which the number of levels of each factor is not a prime power; mixed-level factorial designs with factors at different numbers of levels; cases in which some or all two- and higher-factor interactions, not just the main effects, are required to be orthogonal to the polynomial trend.