Three Parallel Flats Designs for Two-level Factorial Experiments

This paper investigates the properties of the class of three parallel flats designs for two-level factorial experiments. It shows that the designs constructed from this class of designs can have a very simple correlation structure. The correlation of any pair of best linear unbiased estimators of factorial effects is 0, ⅓ or ¼. Furthermore, the designs obtained also have high D-efficiency. Finally, a class of designs is generated with run-size N = 12 to illustrate the use of the theorem.     

Catalogue of Group Structures for Three-level Fractional Factorial Designs

Taguchi (1959) introduced the concept of split-unit design to sort the factors into different groups depending upon the difficulties involved in changing the levels of factors. Li et al. (1991) renamed it as split-plot design. Chen et al. (1993) have given a catalogue of small designs for two- and three-level fractional factorial designs pertaining to a single type of factors. Aggarwal et al. (1997) have given a catalogue of group structure for two-level fractional factorial designs developed under the concept of split-plot design. In this paper, an algorithm has been developed for generating group structure and possible allocations for various 3^n-k fractional factorial designs.     

Catalogue of group structures for two-level fractional factorial designs

SUMMARY Taguchi introduced the concept of split-unit design to sort factors into different groups with respect to difficulties involved in changing the levels of factors. Li et al. have developed all possible group structures for eight factors in an L16 orthogonal array for resolution IV with split-plot design. Chen et al. have searched for a best design, according to the various criteria for two-level fractional factorial design and have presented a catalogue. In this paper, we have developed an algorithm for generating group structure and possible allocations for various 2n- k fractional factorial designs that correspond to the designs given by Chen et al.     

Catalogue of small runs nonregular designs from hadamard matrices with generalized minimum aberration

Fries and Hunter ( 1980 ) proposed the Minimum Aberration criterion (MA) for selecting regular designs. The regular designs with MA are msot commonly used because they are considered as the best designs. How ever, as pointed out by Chen, Sun and Wu ( 1993 ), there are situations that other designs may better meet the design need. Therefore, they catalogued some two-level and three-level fractional factorial regular designs with small (16,27,32,64) runs. For nonregular designs, such as the ones taken from Hadamard matrices, the MA criterion is not appUcable. Deng and Tang ( 1999 ) introduced Generalized Minimum Aberration Criterion (GMA) as a natural extension to the MA criterion. Similar to the case in the regular designs, other designs may better meet practical need, In this paper, we use the GMA criterion to give a catalogue of nonregular designs with smaU (16,20,24) runs.     

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.     

Supersaturated designs with high searching probability

A supersaturated design is essentially a fractional factorial design whose number of experimental variables is greater than or equal to its number of experimental runs. Under the effect sparsity assumption, a supersaturated design can be very cost-effective. In this paper, our prime objective is to compare the existing two-level supersaturated designs for the noisy case through the probability of correct searching—a powerful criterion proposed by Shirakura et al. [1996. Searching probabilities for nonzeroeffects in search designs for the noisy case. Ann. Statist. 24, 2560–2568]. An algorithm is proposed to construct supersaturated designs with high probability of correct searching. Examples are given for illustration.     

Blocking in two-level non-regular fractional factorial designs

In this paper we consider screening experiments where a two-level fractional factorial design is to be used to identify significant factors in an experimental process and where the runs in the experiment are to occur in blocks of equal size. A simple method based on the foldover technique is given for constructing resolution IV orthogonal and non-orthogonal blocked designs and examples are given to illustrate the process.     

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 .     

Partial duplication in two-level fractional factorial designs

Most fractional factorial designs have no replicated points and thus do not provide an estimate for pure error. The construction methods for orthogonal main-effect plan in the literature usually do not produce designs with duplicate points. However, it is possible to combine four fractions to provide a set of duplicate points without sacrificing the orthogonality of main effects. This paper proposes two techniques of this idea to produce designs with replicate points in two-level fractional factorial designs.     

Irregular Four-level Response Surface Designs

Four-level response surface designs based on regular two-level fractional factorial designs were introduced by Edmondson (1991). Here, the methods are extended to include designs based on irregular two-level fractional factorials. These designs allow orthogonal blocking and require fewer experimental units than the regular designs.     

A survey and evaluation of methods for determination of combinatorial equivalence of factorial designs

Equivalent factorial designs have identical statistical properties for estimation of factorial contrasts and for model fitting. Non-equivalent designs, however, may have the same statistical properties under one particular model but different properties under a different model. In this paper, we describe known methods for the determination of equivalence or non-equivalence of two-level factorial designs, whether they be regular factorial designs, non-regular orthogonal arrays, or have no particular structure. In addition, we evaluate a number of potential fast screening methods for detecting non-equivalence of designs. Although the paper concentrates mainly on symmetric designs with factors at two levels, we also evaluate methods of determining combinatorial equivalence and non-equivalence of three-level designs and indicate extensions to larger numbers of levels and to asymmetric designs.     

On sum composition of fractional factorial designs

Combinatorial extension and composition methods have been extensively used in the construction of block designs. One of the composition methods, namely the direct product or Kronecker product method was utilized by Chakravarti [1956] to produce certain types of fractional factorial designs. The present paper shows how the direct sum operation can be utilized in obtaining from initial fractional factorial designs for two separate symmetrical factorials a fractional factorial design for the corresponding asymmetrical factorial. Specifically, we provide some results which are useful in the construction of non-singular fractional factorial designs via the direct sum composition method. In addition a modified direct sum method is discussed and the consequences of imposing orthogonality are explored.     

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.     

Some Results on Fractional Factorial Split-Plot Designs with Multi-Level Factors

Fractional factorial split-plot (FFSP) designs have received much attention in recent years. In this article, the matrix representation for FFSP designs with multi-level factors is first developed, which is an extension of the one proposed by Bingham and Sitter (1999b Bingham , D. , Sitter , R. R. ( 1999b ). Some theoretical results for fractional factorial split-plot designs . Ann. Statist. 27 : 1240 – 1255 . [Google Scholar]) for the two-level case. Based on this representation, periodicity results of maximum resolution and minimum aberration for such designs are derived. Differences between FFSP designs with multi-level factors and those with two-level factors are highlighted.     

On some optimal fractional 2m factorial designs of resolution V

This paper presents the trace of the covariance matrix of the estimates of effects based on a fractional 2^m factorial (2^m-FF) design T of resolution V for the following two cases: One is the case where T is constructed by adding some restricted assemblies to an orthogonal array. The other is one where T is constructed by removing some restricted assemblies from an orthogonal array of index unity. In the class of 2^m-FF designs of resolution V considered here, optimal designs with respect to the trace criterion, i.e. A-optimal, are presented for m = 4, 5, and 6 and for a range of practical values of N (the total number of assemblies). Some of them are better than the corresponding A-optimal designs in the class of balanced fractional 2^m factorial designs of resolution V obtained by Srivastava and Chopra (1971b) in such a sense that the trace of the covariance matrix of the estimates is small.     

Resolution v.2 search designs

A fractional factorial design is called a resolution V.2 plan if it is capable of estimating all main effects and two-factor interaction effects, plus two three-factor interaction effects, In this paper, a necessary and sufficient condition for such a resolution V.2 plan is given, Furthermore, a new class of two-level resolution V.2 designs is proposed, We prove that the proposed design always satisfies such a necessary and sufficient condition, A comparison of run size between designs of resolutions VII and V.2 is made, It is shown that run size for design of resolution V.2 is significantly smaller.     

Minimum aberration criterion for screening experiments at two levels from an entropy-based perspective

This paper presents a new criterion for selecting a two-level fractional factorial design. The theoretical underpinning for the criterion is the Shannon entropy. The criterion, which is referred to as the entropy-based minimum aberration criterion, has several advantages. The advantage of the entropy-based criterion over the classical minimum aberration criterion is that it utilizes a measure of uncertainty on the skewness of the distribution of word length patterns in the selection of the “best” design in a family of two-level fractional factorial plans. The criterion evades the trauma associated with the lack of prior knowledge on the important effects.     

On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs

Box and Meyer [1986. Dispersion effects from fractional designs. Technometrics 28(1), 19–27] were the first to consider identifying both location and dispersion effects from unreplicated two-level fractional factorial designs. Since the publication of their paper a number of different procedures (both iterative and non-iterative) have been proposed for estimating the location and dispersion effects. An overview and a critical analysis of most of these procedures is given by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405]. Under a linear structure for the dispersion effects, non-iterative estimation methods for the dispersion effects were proposed by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405], Liao and Iyer [2000. Optimal 2^n-p$2^{n-p}$ fractional factorial designs for dispersion effects under a location-dispersion model. Comm. Statist. Theory Methods 29(4), 823–835] and Wiklander [1998. A comparison of two estimators of dispersion effects. Comm. Statist. Theory Methods 27(4), 905–923] (see also Wiklander and Holm [2003. Dispersion effects in unreplicated factorial designs. Appl. Stochastic. Models Bus. Ind. 19(1), 13–30]). We prove that for two-level factorial designs the proposed estimators are different representations of a single estimator. The proof uses the framework of Seely [1970a. Linear spaces and unbiased estimation. Ann. Math. Statist. 41, 1725–1734], in which quadratic estimators are expressed as inner products of symmetric matrices.     

Efficiency of bailey's trigonometric levels in factorial experiments

Bailey has shown that choice of certain trigonometlk levels for factors in a symmetrical confounded factorial design is more efficient for quantitative treatments. This paper introduces certain incidence matrices associated with the flats of different pencils of such designs to obtain an explicit expression for the efficiency and also gives a simpler derivation of Bailey's results.     

Algorithm for determining whether various two-level fractional factorial split-plot row–column designs are non-isomorphic

The confounding and aliasing scheme for fractional factorial split-plot designs with the units within each wholeplot arranged in rows and columns is described and illustrated. Isomorphism for this design type is described, together with a procedure which considers extensions of the concepts of wordlength patterns and letter patterns that can be used to test isomorphism between designs. Using in part this isomorphism testing procedure, a construction algorithm that may be used to obtain a complete set of such non-isomorphic two-level designs is described. Software based on this construction algorithm was used to obtain a complete set of non-isomorphic designs for up to five wholeplot factors, five subplot factors and up to 64 runs, which is presented as a table of designs. To aid the experimenter in distinguishing between competing designs, the estimation capacity sequence for each design is presented.