Minimum aberration blocking schemes for two- and three-level fractional factorial designs

The concept of minimum aberration has been extended to choose blocked fractional factorial designs (FFDs). The minimum aberration criterion ranks blocked FFDs according to their treatment and block wordlength patterns, which are often obtained by counting words in the treatment defining contrast subgroups and alias sets. When the number of factors is large, there are a huge number of words to be counted, causing some difficulties in computation. Based on coding theory, the concept of minimum moment aberration, proposed by Xu [Statist. Sinica, 13 (2003) 691–708] for unblocked FFDs, is extended to blocked FFDs. A method is then proposed for constructing minimum aberration blocked FFDs without using defining contrast subgroups and alias sets. Minimum aberration blocked FFDs for all 32 runs, 64 runs up to 32 factors, and all 81 runs are given with respect to three combined wordlength patterns.     

Construction of minimum aberration blocked two-level regular factorial designs

ABSTRACT

In this article, we consider the construction of minimum aberration 2^{n ? k}: 2^p designs with respect to some existing combined wordlength patterns, where a 2^{n ? k}: 2^p design is a blocked two-level design with n treatment factors, 2^p blocks, and N = 2^q runs with q = n ? k. Two methods are proposed for two situations: n ? 2^{q ? p} ? 1 and n > N/2. These methods enable us to obtain some new minimum aberration 2^{n ? k}: 2^p designs from existing minimum aberration unblocked and blocked designs. Examples are included to illustrate the theory.     

Minimum aberration blocking schemes for 128-run designs

Several criteria have been proposed for ranking blocked fractional factorial designs. For large fractional factorial designs, the most appropriate minimum aberration criterion was one proposed by Cheng and Wu (2002). We justify this assertion and propose a novel construction method to overcome the computational challenge encountered in large fractional factorial designs. Tables of minimum aberration blocked designs are presented for N=128 runs and n=8–64 factors.     

Minimum aberration and model robustness for two-level fractional factorial designs

The performance of minimum aberration two-level fractional factorial designs is studied under two criteria of model robustness. Simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived. It is also shown that a minimum aberration design of resolution III or higher maximizes the number of two-factor interactions which are not aliases of main effects and, subject to that condition, minimizes the sum of squares of the sizes of alias sets of two-factor interactions. This roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform. It follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here. Examples are given to show that minimum aberration designs are indeed highly efficient.     

Bayesian optimal blocking of factorial designs

The presence of block effects makes the optimal selection of fractional factorial designs a difficult task. The existing frequentist methods try to combine treatment and block wordlength patterns and apply minimum aberration criterion to find the optimal design. However, ambiguities exist in combining the two wordlength patterns and therefore, the optimality of such designs can be challenged. Here we propose a Bayesian approach to overcome this problem. The main technique is to postulate a model and a prior distribution to satisfy the common assumptions in blocking and then, to develop an optimal design criterion for the efficient estimation of treatment effects. We apply our method to develop regular, nonregular, and mixed-level blocked designs. Several examples are presented to illustrate the advantages of the proposed method.     

Robust two-level regular fractional factorial designs

Abstract

In this paper, we introduce the concept of model quality for two-level regular fractional factorial designs. Under the effect hierarchy principle, this paper raises the definition of model quality and introduces robust model-number pattern (RP) to choose the optimal robust design. Some theoretical results on this optimality and comparisons with GMC and MEC criterion are given.     

Theory of optimal blocking of nonregular factorial designs

The authors introduce the notion of split generalized wordlength pattern (GWP), i.e., treatment GWP and block GWP, for a blocked nonregular factorial design. They generalize the minimum aberration criterion to suit this type of design. Connections between factorial design theory and coding theory allow them to obtain combinatorial identities that govern the relationship between the split GWP of a blocked factorial design and that of its blocked consulting design. These identities work for regular and nonregular designs. Furthermore, the authors establish general rules for identifying generalized minimum aberration (GMA) blocked designs through their blocked consulting designs. Finally they tabulate and compare some GMA blocked designs from Hall's orthogonal array OA(16,2¹5,2) of type III.     

Minimum aberration majorization in non-isomorphic saturated designs

In this paper we propose a new criterion, minimum aberration majorization, for comparing non-isomorphic saturated designs. This criterion is based on the generalized word-length pattern proposed by Ma and Fang (Metrika 53 (2001) 85) and Xu and Wu (Ann. Statist. 29 (2001) 1066) and majorization theory. The criterion has been successfully applied to check non-isomorphism and rank order saturated designs. Examples are given through five non-isomorphic L₁₆(2¹⁵) designs and two L₂₇(3¹³) designs.     

Uniformity in factorial designs with mixed levels

The uniformity can be utilized as a measure for comparing factorial designs. Fang and Mukerjee (Biometrika 87 (2000) 193–198) and Fang et al. (in: K.T. Fang, F.J. Hickernell, H. Niederreiter (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer, Berlin, 2002) found links among uniformity in terms of some non-uniformity measures, orthogonality and aberration for regular symmetric factorials. In this paper we extend their results to asymmetric factorials by considering a so-called wrap-around L₂-discrepancy to evaluate the uniformity of factorials. Furthermore, a lower bound of wrap-around L₂-discrepancy is obtained for asymmetric factorials and two new ways of construction of factorial designs with mixed levels are proposed.     

Minimum number of runs for two-level factorial search designs

A lower bound is given for the number of experimental runs required in search designs for two-level factorial models.     

Interaction balance for symmetrical factorial designs with generalized minimum aberration

There are two different systems of contrast parameterization when analyzing the interaction effects among the factors with more than two levels, i.e., linear-quadratic system and orthogonal components system. Based on the former system and an ANOVA model, Xu and Wu (2001) introduced the generalized wordlength pattern for general factorial designs. This paper shows that the generalized wordlength pattern exactly measures the balance pattern of interaction columns of a symmetrical design ground on the orthogonal components system, and thus an alternative angle to look at the generalized minimum aberration criterion is given. This work is partially supported by NNSF of China grant No. 10231030.     

Minimum cost linear trend-free 12-run fractional factorial designs

Time trend resistant fractional factorial experiments have often been based on regular fractionated designs where several algorithms exist for sequencing their runs in minimum number of factor-level changes (i.e. minimum cost) such that main effects and/or two-factor interactions are orthogonal to and free from aliasing with the time trend, which may be present in the sequentially generated responses. On the other hand, only one algorithm exists for sequencing runs of the more economical non-regular fractional factorial experiments, namely Angelopoulos et al. [1 P. Angelopoulos, H. Evangelaras, and C. Koukouvinos, Run orders for efficient two-level experimental plans with minimum factor level changes robust to time trends, J. Statist. Plann. Inference 139 (2009), pp. 3718–3724. doi: 10.1016/j.jspi.2009.05.002[Crossref], [Web of Science ®] , [Google Scholar]]. This research studies sequential factorial experimentation under non-regular fractionated designs and constructs a catalog of 8 minimum cost linear trend-free 12-run designs (of resolution III) in 4 up to 11 two-level factors by applying the interactions-main effects assignment technique of Cheng and Jacroux [3 C.S. Cheng and M. Jacroux, The construction of trend-free run orders of two-level factorial designs, J. Amer. Statist. Assoc. 83 (1988), pp. 1152–1158. doi: 10.1080/01621459.1988.10478713[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] on the standard 12-run Plackett–Burman design, where factor-level changes between runs are minimal and where main effects are orthogonal to the linear time trend. These eight 12-run designs are non-orthogonal but are more economical than the linear trend-free designs of Angelopoulos et al. [1 P. Angelopoulos, H. Evangelaras, and C. Koukouvinos, Run orders for efficient two-level experimental plans with minimum factor level changes robust to time trends, J. Statist. Plann. Inference 139 (2009), pp. 3718–3724. doi: 10.1016/j.jspi.2009.05.002[Crossref], [Web of Science ®] , [Google Scholar]], where they can accommodate larger number of two-level factors in smaller number of experimental runs. These non-regular designs are also more economical than many regular trend-free designs. The following will be provided for each proposed systematic design:

• (1) The run order in minimum number of factor-level changes.

• (2) The total number of factor-level changes between the 12 runs (i.e. the cost).

• (3) The closed-form least-squares contrast estimates for all main effects as well as their closed-form variance–covariance structure.

In addition, combined designs of each of these 8 designs that can be generated by either complete or partial foldover allow for the estimation of two-factor interactions involving one of the factors (i.e. the most influential).     

Graph based isomorph-free generation of two-level regular fractional factorial designs

We provide a new necessary and sufficient check for testing the isomorphism of two 2-level regular fractional factorial designs. The approach is based on modeling fractional factorial designs as bipartite graphs. We employ an efficient canonical graph labeling approach to compare two designs for isomorphism. We then improve upon the existing non-isomorphic fractional factorial design generation algorithm by reducing the number of candidate designs from which isomorphs need to be removed. Not only does our method generate non-isomorphic designs much faster, it is also able to generate designs with run sizes of 2048 and 4096 runs, which were not generated by the existing methods.     

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.     

Indicator function and complex coding for mixed fractional factorial designs

In a general fractional factorial design, the n levels of a factor are coded by the nth roots of the unity. This device allows a full generalization to mixed-level designs of the theory of the polynomial indicator function which has already been introduced for two-level designs in a joint paper with Fontana. The properties of orthogonal arrays and regular fractions are discussed.     

Characterization of minimum aberration mixed factorials in terms of consulting designs

In this paper, by introducing a concept of consulting design and based on the connection between factorial design theory and coding theory, we obtain combinatorial identities that relate the wordlength pattern of a regular mixed factorial design to that of its consulting design. According to these identities, we further-more establish the general and unified rules for identifying minimum aberration mixed factorial designs through their consulting designs. It is an improvement and generalization of the results in Mukerjee and Wu (2001). This paper is supported by NNSF of China grant No. 10171051 and RFDP grant No. 1999005512.     

Creating catalogues of two-level nonregular fractional factorial designs based on the criteria of generalized aberration

We consider the problem of constructing good two-level nonregular fractional factorial designs. The criteria of minimum G and G₂ aberration are used to rank designs. A general design structure is utilized to provide a solution to this practical, yet challenging, problem. With the help of this design structure, we develop an efficient algorithm for obtaining a collection of good designs based on the aforementioned two criteria. Finally, we present some results for designs of 32 and 40 runs obtained from applying this algorithmic approach.     

A simple method for constructing confounded designs for mixed factorial experiments

This paper presents the sinplesr procedure that uses wodular aryithmetic for constructing confounded designs for mixed factorial experiments. The present procedure and the classical one for confounding in symmetrical factorial experiments are both at the same mathema.tical level. The present procedure is written for

practitioners and is lllustrared with several examples.