(M,S)-optimality in selecting factorial designs

Use of the (M,S) criterion to select and classify factorial designs is proposed and studied. The criterion is easy to deal with computationally and it is independent of the choice of treatment contrasts. It can be applied to two-level designs as well as multi-level symmetrical and asymmetrical designs. An important connection between the (M,S) and minimum aberration criteria is derived for regular fractional factorial designs. Relations between the (M,S) criterion and generalized minimum aberration criteria on nonregular designs are also discussed. The (M,S) criterion is then applied to study the projective properties of some nonregular designs.     

Optimal block sequences for blocked fractional factorial split-plot designs

It is known that for blocked 2^n-k$2^{n-k}$ designs a judicious sequencing of blocks may allow one to obtain early and insightful results regarding influential parameters in the experiment. Such findings may justify the early termination of the experiment thereby producing cost and time savings. This paper introduces an approach for selecting the optimal sequence of blocks for regular two-level blocked fractional factorial split-plot screening experiments. An optimality criterion is developed so as to give priority to the early estimation of low-order factorial effects. This criterion is then applied to the minimum aberration blocked fractional factorial split-plot designs tabled in McLeod and Brewster [2004. The design of blocked fractional factorial split-plot experiments. Technometrics 46, 135–146]. We provide a catalog of optimal block sequences for 16 and 32-run minimum aberration blocked fractional factorial split-plot designs run in either 4 or 8 blocks.     

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.     

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.     

Invariance of generalized wordlength patterns

The generalized wordlength pattern (GWLP) introduced by Xu and Wu [2001. Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29, 1066–1077] for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang [2004. Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60, 279–285] defined the J$J$-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the J$J$-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the J$J$-characteristics are not. We briefly discuss some implications of these results.     

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.     

Optimal blocking and foldover plans for nonregular two-level designs

This paper discusses the issue of choosing optimal designs when both blocking and foldover techniques are simultaneously employed to nonregular two-level fractional factorial designs. By using the indicator function, the treatment and block generalized wordlength patterns of the combined blocked design under a general foldover plan are defined. Some general properties of combined block designs are also obtained. Our results extend the findings of Ai et al. (2010) from regular designs to nonregular designs. Based on these theoretical results, a catalog of optimal blocking and foldover plans in terms of the generalized aberration criterion for nonregular initial design with 12, 16 and 20 runs is tabulated, respectively.     

A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs

The minimum aberration criterion has been advocated for ranking foldovers of 2^k−p$2^{k-p}$ fractional factorial designs (Li and Lin, 2003); however, a minimum aberration design may not maximize the number of clear low-order effects. We propose using foldover plans that sequentially maximize the number of clear low-order effects in the combined (initial plus foldover) design and investigate the extent to which these foldover plans differ from those that are optimal under the minimum aberration criterion. A small catalog is provided to summarize the results.     

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.     

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.     

Minimum Aberration Two-Level Fractional Factorial Split-Plot Designs with Hard to Change Factors

In this article we will consider industrial experiments in which some experimental factors have hard to change levels and others have levels which are easy to change. In such situations, fractional factorial split plot designs are often used where the hard to change factors are included as a subset of the whole plot factors and the easy to change factors make up the subplot factors. Here we consider the problem of finding two-level split plot designs which have minimum aberration among those designs which also minimize the number of level changes for the hard to change factors.     

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.     

Optimal fractional factorial designs and their construction

Optimal symmetrical fractional factorial designs with n$n$ runs and m$m$ factors of s$s$ levels each are constructed. We consider only designs such that no two factors are aliases. The minimum moment aberration criterion proposed by Xu (2003) is used to judge the optimality of the designs. The minimum moment aberration criterion is equivalent to the popular generalized minimum aberration criterion proposed by Xu and Wu (2001), but the minimum moment criterion is simpler to formulate and employ computationally. Some optimal designs are constructed by using generalized Hadamard matrices.     

Analytic connections between a double design and its original design in terms of different optimality criteria

In recent years, there has been increasing interest in the study of double designs. Various popular optimality criteria have been proposed from different principles for design construction and comparison, such as E(s²), generalized minimum aberration (GMA), minimum moment aberration (MMA), and minimum projection uniformity (MPU). In this article, these criteria are reviewed, and analytic connections between a double design and its original design in terms of these criteria are investigated. These connections are suitable for general original two-level factorial design, whether regular or non regular. In addition, these results provide strong insight into the relationship between double design and original design from different viewpoints.     

Designing fractional factorial split-plot experiments with few whole-plot factors

Summary.  When it is impractical to perform the experimental runs of a fractional factorial design in a completely random order, restrictions on the randomization can be imposed. The resulting design is said to have a split-plot, or nested, error structure. Similarly to fractional factorials, fractional factorial split-plot designs can be ranked by using the aberration criterion. Techniques that generate the required designs systematically presuppose unreplicated settings of the whole-plot factors. We use a cheese-making experiment to demonstrate the practical relevance of designs with replicated settings of these factors. We create such designs by splitting the whole plots according to one or more subplot effects. We develop a systematic method to generate the required designs and we use the method to create a table of designs that is likely to be useful in practice.     

Generalized minimum aberration mixed-level orthogonal arrays: A general approach based on sequential integer quadratically constrained quadratic programming

Orthogonal fractional factorial designs and in particular orthogonal arrays (OAs) are frequently used in many fields of application, including medicine, engineering, and agriculture. In this article, we present a methodology and an algorithm to find an OA, of given size and strength, which satisfies the generalized minimum aberration criterion. The methodology is based on the joint use of polynomial counting functions, complex coding of levels, and algorithms for quadratic optimization and puts no restriction on the number of levels of each factor.     

Switching-one-column follow-up experiments for Plackett-Burman designs

Industrial experiments are frequently performed sequentially using two-level fractional factorial designs. In this context, a common strategy for the design of follow-up experiments is to switch the signs in one column. It is well known that this strategy, when applied to two-level fractional factorial resolution III designs, will clear the main effect, for which the switch was performed, from any confounding with any other two-factor interactions and will also clear all the two-factor interactions between that factor and the other main effects from any confounding with other two-factor interactions. In this article, we extend this result and show that this strategy applies to any orthogonal two-level resolution III design and therefore specifically to any two-level Plackett- Burman design .     

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.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal designs.     

On the Use of Semi-folding in Regular Blocked Two-level Factorial Designs

In this article, we consider experimental situations where a blocked regular two-level fractional factorial initial design is used. We investigate the use of the semi-fold technique as a follow-up strategy for de-aliasing effects that are confounded in the initial design as well as an alternative method for constructing blocked fractional factorial designs. A construction method is suggested based on the full foldover technique and sufficient conditions are obtained when the semi-fold yields as many estimable effects as the full foldover.