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.     

An isomorphism check for two-level fractional factorial designs

Two fractional factorial designs are isomorphic if one can be obtained from the other by reordering the treatment combinations, relabelling the factor levels and relabelling the factors. By defining a word-pattern matrix, we are able to create a new isomorphism check which is much faster than existing checks for certain situations. We combine this with a new, extremely fast, sufficient condition for non-isomorphism to avoid checking certain cases. We then create a faster search algorithm by combining the Bingham and Sitter [1999. Minimum aberration fractional factorial split-plot designs. Technometrics 41, 62–70] search algorithm, the isomorphism check algorithm of Clark and Dean [2001. Equivalence of fractional factorial designs. Statist. Sinica 11, 537–547] with our proposed isomorphism check. The algorithm is used to extend the known set of existing non-isomorphic 128-run two-level regular designs with resolution ?4$?4$ to situations with 12, 13, 14, 15 and 16 factors, 256- and 512-run designs with resolution ?5$?5$ and ?17$?17$ factors and 1024-run even designs with resolution ?6$?6$ and ?18$?18$ factors.     

Constructing non-regular robust parameter designs

In recent years there has been considerable attention paid to robust parameter design as a strategy for variance reduction. Of particular concern is the selection of a good experimental plan in light of the two different types of factors in the experiment (control and noise) and the asymmetric manner in which effects of the same order are treated. Recent work has focussed on the selection of regular fractional factorial designs in this setting. In this article, we consider the construction and selection of optimal non-regular experiment plans for robust parameter design. Our approach defines the word-length pattern for non-regular fractional factorial designs with two different types of factors which allows for the choice of optimal design to emphasize the estimation of the effects of interest. We use this new word-length pattern to rank non-regular robust parameter designs. We show that one can easily find minimum aberration robust parameter designs from existing orthogonal arrays. The methodology is demonstrated by finding optimal assignments for control and noise factors for 12, 16 and 20-run orthogonal arrays.     

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.     

Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O(2^p)+O(3k³)+O(2^k) operations to check two 2^k−p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.     

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.     

D‐optimal minimax fractional factorial designs

The D‐optimal minimax criterion is proposed to construct fractional factorial designs. The resulting designs are very efficient, and robust against misspecification of the effects in the linear model. The criterion was first proposed by Wilmut & Zhou (2011); their work is limited to two‐level factorial designs, however. In this paper we extend this criterion to designs with factors having any levels (including mixed levels) and explore several important properties of this criterion. Theoretical results are obtained for construction of fractional factorial designs in general. This minimax criterion is not only scale invariant, but also invariant under level permutations. Moreover, it can be applied to any run size. This is an advantage over some other existing criteria. The Canadian Journal of Statistics 41: 325–340; 2013 © 2013 Statistical Society of Canada     

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.     

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 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.     

Supersaturated design including an orthogonal base

Recently, many supersaturated designs have been proposed. A supersaturated design is a fractional factorial design in which the number of factors is greater than the number of experimental runs. The main thrust of the previous studies has been to generate more columns while avoiding large values of squared inner products among all design columns. These designs would be appropriate if the probability for each factor being active is uniformly distributed. When factors can be partitioned into two groups, namely, with high and low probabilities of each factor being active, it is desirable to maintain orthogonality among columns to be assigned to the factors in the high-probability group. We discuss a supersaturated design including an orthogonal base which is suitable for this common situation. Mathematical results on the existence of the supersaturated designs are shown, and the construction of supersaturated designs is presented. We next discuss some properties of the proposed supersaturated designs based on the squared inner products.     

Construction of balanced incomplete block designs with partial cycles of blocks

When considering systems statisfying balanced incomplete block design structure, one is interested only in distinct configurations in the sense of isomorphism. We have determined additional non-isomorphic solutions to designs with previously established solutions through consideration of automorphism groups. This effort utilizes a classic technique of direct construction, namely, symmetrically repeated differences, modified slightly when constucting designs with partial cycles of blocks. To determine te base blocks, we utilize the computer to expedite some formidable enumeration     

Indicator function based on complex contrasts and its application in general factorial designs

A factorial design can be uniquely determined by an indicator function which is constructed by means of orthogonal contrasts. Since the orthogonal contrasts are not unique, invariant measures are preferred. However, some particular orthogonal contrasts may express more information about designs than the others and be worth our attention. In this paper, a kind of indicator function based on orthogonal complex contrasts is introduced to represent general factorial designs and its significance on projection designs is presented. Based on this function, a generalized resolution and a new aberration criterion are developed to rank combinatorially non-isomorphic designs with prime levels. Some results and comparison are provided by means of examples.     

Equivalence of factorial designs with qualitative and quantitative factors

Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed.     

On equivalence of fractional factorial designs based on singular value decomposition

The singular value decomposition of a real matrix always exists and is essentially unique. Based on the singular value decomposition of the design matrices of two general 2-level fractional factorial designs, new necessary and sufficient conditions for the determination of combinatorial equivalence or non-equivalence of the corresponding designs are derived. Equivalent fractional 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. Results extend to designs with factors at larger number of levels.     

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.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

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.     

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.     

Balanced arrays and main-effect plans from regular group-divisible designs

A method of constructing balanced arrays of strength two and corresponding main-effect fractional factorial plans from some regular group-divisible designs is presented. Main-effect fractional factorial plans for three-level factorial experiments are constructed. The plans have reasonably high efficiencies for different single degree of freedom effects.