Optimal criteria and equivalence for nonregular fractional factorial designs

By considering the so-called algebraic orthogonality among experimental runs, this paper introduces a new criterion, minimum inner-product moment (MIPM), for general asymmetrical designs, and shows that MIPM is equivalent to the minimum moment aberration (MMA) criterion for the natural weights. Furthermore, the relationship between the generalized minimum aberration (GMA) and some model-dependent efficiency criteria is investigated by using the complex contrasts. Thus, two new justifications of GMA criterion is given from the points of view of orthogonality among experimental runs and design efficiency. They are generalizations of the related results of Butler (2003) and Cheng, Deng, and Tang (2002) for two-level factorial designs.     

Eight-run two-level factorial designs under dependence

Dependent observations commonly arise in factorial experiments. Apart from main-effects two-level designs formed by the Cheng & Steinberg reverse foldover algorithm, which are known to be very efficient designs under dependence using the D-criterion, little is known about other designs, models and criteria, and the range of possible behaviour. In this paper, we investigate in detail 8-run two-level designs. Received February 1998     

Maximal Rank Minimum Aberration Foldover Plans for 2<Superscript><Emphasis Type="Italic">m</Emphasis>-<Emphasis Type="Italic">k</Emphasis></Superscript> Fractional Factorial Designs

Two level regular fractional factorial designs are often used in industry as screening designs to help identify early on in an experimental process those experimental or system variables which have significant effects on the process being studied. In a recent paper, Li and Lin (2003) suggested a strategy for constructing optimal follow up designs using the well known foldover technique and the minimum aberration criterion. In this paper, we extend the results of Li and Lin (2003) by giving an alternate technique for constructing optimal follow up designs using the foldover technique in conjunction with the maximal rank–minimum aberration criterion suggested in Jacroux (2003).     

Mixed-level fractional factorial split-plot designs containing clear effects

Mixed-level designs are widely used in the practical experiments. When the levels of some factors are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are often used. This paper investigates the sufficient and necessary conditions for a ${2^{(n_{1}+n_{2})-(k_1+k_2)}4_s^{1}}$ FFSP design with resolution III or IV to have various clear factorial effects, including two types of main effects and three types of two-factor interaction components. The structures of such designs are shown and illustrated with examples.     

Connection between uniformity and minimum moment aberration

The issue of uniformity in symmetrical fractional factorial designs is studied in this paper. The so-called discrete discrepancy is employed as a measure of uniformity. In this paper we give linkages between uniformity measured by the discrete discrepancy and minimum moment aberration, which provide a significant statistical justification of the discrete discrepancy.     

Construction of mixed-level supersaturated design

Supersaturated design is a form of fractional factorial design in which the number of columns is greater than the number of experimental runs. Construction methods of supersaturated design have been mainly focused on two levels cases. Much practical experience, however, indicates that two-level may sometimes be inadequate. This paper proposed a construction method of mixed-level supersaturated designs consisting of two-level and three-level columns. The χ² statistic is used for a measure of dependency of the design columns. The dependency properties for the newly constructed designs are derived and discussed. It is shown that these new designs have low dependencies and thus can be useful for practical uses.     

On simplifying the calculations leading to designs with general minimum lower-order confounding

Motivated by the effect hierarchy principle, Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect number pattern (AENP) for regular fractional factorial designs and based on the new pattern proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal $2^{n-m}$ designs. Zhang et al. (Stat Sinica 18:1689–1705, 2008) proved that most existing criteria can be obtained by functions of the AENP. In this paper we propose a simple method for the calculation of AENP. The method is much easier than before since the calculation only makes use of the design matrix. All 128-run GMC designs with the number of factors ranging from 8 to 32 are provided for practical use.     

Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs

Recently, Xu and Wu (2001) presented generalized minimum aberration criterion for comparing and selecting general fractional factorial designs. This criterion is defined using a set of _u(D) values, called J-characteristics by us. In this paper, we find a set of linear equations that relate the set of design points to that of J-characteristics, which implies that a factorial design is uniquely determined by its J-characteristics once the orthonormal contrasts are designated. Thereto, a projection justification of generalized minimum aberration is established. All of these conclusions generalize the results for two-level symmetrical factorial designs in Tang (2001).Acknowledgements The authors are grateful to the editor, the associate editor and the referees for their valuable comments. This paper is supported by NNSF of P.R.China grant No. 10171051. and RFDP grant No. 1999005512.     

Generalized Wordtype Pattern for Nonregular Factorial Designs with Multiple Groups of Factors

This paper introduces the generalized wordtype pattern of a nonregular fractional factorial design and considers its connection with the distance distribution. Based on the corresponding relationship between a fractional factorial design and a code, we develop a consulting design theory for fractional factorial designs with two groups of factors. It works for regular and nonregular designs and covers the previous results as special cases. As an illustration, it is further applied to the selection of optimal two-level single arrays derived from Hadamard Matrices.     

Asymmetrical Factorial Designs Containing Clear Effects

The asymmetrical or mixed-level factorial design is a kind of important design in practice. There is a natural problem on how to choose an optimal (s ²)s ⁿ design for the practical need, where s is any prime or prime power. This paper considers the clear effects criterion for selecting good designs. It answers the questions of when an (s ²)s ⁿ design with fixed number of runs contains clear two-factor interaction (in brief 2fi) components and when it contains clear main effects or clear 2fis. It further gives the complete classification of (s ²)s ⁿ designs according to the clear 2fi components, main effects and 2fis they have.     

Discrete discrepancy in factorial designs

Discrepancy measure can be utilized as a uniformity measure for comparing factorial designs. A so-called discrete discrepancy has been used to evaluate the uniformity of factorials. In this paper we give linkages among uniformity measured by the discrete discrepancy, generalized minimum aberration, minimum moment aberration and uniformity measured by the centered L₂-discrepancy/the wrap-around L₂-discrepancy. These close linkages provide a significant justification for the discrete discrepancy used to measure uniformity of factorial designs.     

On a property of orthogonal arrays and optimal blocking of fractional factorial plans

It is shown that fractional factorial plans represented by orthogonal arrays of strength three are universally optimal under a model that includes the mean, all main effects and all two-factor interactions between a specified factor and each of the other factors. Thus, such plans exhibit a kind of model robustness in being universally optimal under two different models. Procedures for obtaining universally optimal block designs for fractional factorial plans represented by orthogonal arrays are also discussed. Acknowledgements. The authors wish to thank the referees for making several useful comments on a previous version.     

A note on connections among criteria for asymmetrical factorials

Various optimal criteria have been proposed to rank asymmetrical fractional factorial designs. Among them, the generalized minimum aberration and the minimum moment aberration criteria are the most popular ones and have received much attention. Recently, Liu et al. (Stat Sin 16:1285–1297, 2006) proposed the minimum χ ² criterion in terms of level-combinations. In this paper, the equivalency of the generalized minimum aberration and the minimum χ ² criteria is reported, which not only provides another justification for each other but also develops some theoretical results for designs with generalized minimum aberration and some lower bounds for the generalized wordlength pattern. Besides, an analytic relationship between generalized minimum aberration and minimum moment aberration is obtained for asymmetrical fractional factorial designs.     

Optimal mixed-level supersaturated design

A supersaturated design is essentially a fractional factorial in which the number of potential effects is greater than the number of runs. In this paper, E(f _NOD ) criterion is employed for comparing supersaturated designs from the viewpoint of orthogonality and uniformity, and a lower bound of E(f _NOD ) which can serve as a benchmark of design optimality is obtained. It is shown that the existing E(s ²) and ave ² criteria (for two- and three-level supersaturated designs respectively) are in fact special cases of this criterion. Furthermore, a construction method for mixed-level supersaturated designs is proposed and some properties of the resulting designs are investigated. Key words:Discrepancy; Hamming distance; Orthogonal array; Supersaturated design; Uniformity; U-type design. 2000 Mathematics Subject Classifications62K15, 62K05, 62K99. Corresponding author.     

Multi-level and mixed-level <Emphasis Type="Italic">k</Emphasis>-circulant supersaturated designs

Supersaturated designs (SSDs) constitute an important class of fractional factorial designs that could be extremely useful in factor screening experiments. Most of the existing studies have focused on balanced designs. This paper provides a new lower bound for the \(E(f_{NOD})\)-optimality measure of SSDs with general run sizes. This bound is a generalization of existing bounds since it is applicable to both balanced and unbalanced designs. Optimal multi and mixed-level, balanced and nearly balanced SSDs are constructed by applying a k-circulant type methodology. Necessary and sufficient conditions are introduced for the generator vectors, in order to pre-ensure the optimality of the constructed k-circulant SSDs. The provided lower bounds were used to measure the efficiency of the generated designs. The presented methodology leads to a number of new families of improved SSDs, providing tools for directly constructing optimal or nearly-optimal k-circulant designs by just checking the corresponding generator vector.     

Nonregular designs from Hadamard matrices and their estimation capacity

Deng and Tang (1999) proposed the generalized minimum aberration (GMA) criterion to assess fractional factorial designs, and a design with GMA is often regarded as the best. However, there exist situations where some other designs may suit practical needs better. In this article, we propose an algorithm to sequentially examine designs obtained from Hadamard matrices under estimation capacity (EC) and provide designs with maximum or high EC for various combinations of run-size and number-of-factors. The usefulness of maximum or high EC designs is discussed.2000 Mathematics Subject Classification: 62K15, 05B20.Acknowledgements The research of Yingfu Li is supported by a Faculty Research Support Fund through the School of Science and Computer Engineering, University of Houston - Clear Lake. The authors are very grateful to the editor and two referees for their helpful comments that have led to the improvement of the paper.     

On the construction of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">s</Emphasis><Superscript>2</Superscript>)-optimal supersaturated designs

Supersaturated designs are an important class of factorial designs in which the number of factors is larger than the number of runs. These designs supply an economical method to perform and analyze industrial experiments. In this paper, we consider generalized Legendre pairs and their corresponding matrices to construct E(s ²)-optimal two-level supersaturated designs suitable for screening experiments. Also, we provide some general theorems which supply several infinite families of E(s ²)-optimal two-level supersaturated designs of various sizes.      

Lower bounds of various discrepancies on combined designs

The foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this paper, some lower bounds of various discrepancies of combined designs, such as centered L ₂-discrepancy, symmetric L ₂-discrepancy and wrap-around L ₂-discrepancy, under a general foldover plan are obtained, which can be used as a benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

Small asymmetric fractional factorial plans for main effects and specified two-factor interactions

Fractional factorial plans represented by orthogonal arrays of strength two are known to be optimal in a very strong sense under a model that includes the mean and all the main effects, when all interactions are assumed to be absent. When a fractional factorial plan given by an orthogonal array of strength two is not saturated, one might think of entertaining some two-factor interactions also in the model. In such a situation, it is of interest to examine which of the two-factor interactions can be estimated via a plan represented by an orthogonal array, as also to study the overall efficiency of the plan when some interactions are in the model alongwith the mean and all main effects. In this paper, an attempt has been made to examine these issues by considering some practically useful plans for asymmetric (mixed level) factorials with small number of runs.