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.     

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.     

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.     

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.     

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.     

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.     

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.     

Partially replicated two-level fractional factorial designs via semifoldover

Most fractional factorial designs have no replicated points and thus do not provide a reliable estimate for experimental error. The objective of this paper is to study the issue of partially replicated two-level fractional factorial (FF) designs, thereby allowing for the unbiased estimation of the experimental error while maintaining the orthogonality of the main effects. Through the tool of indicator function and the idea of semifoldover, we propose two simple and effective techniques to produce designs with partially replicated points in general two-level FF designs, whether they are regular or not. The related properties of constructed partially replicated designs are investigated. Our results indicate that partially replicated FF are competitive in practice.     

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.     

On the use of partial confounding for the construction of alternative regular two-level blocked fractional factorial designs

In this paper, we consider experimental situations in which a regular fractional factorial design is to be used to study the effects of m two-level factors using n=2^m−k experimental units arranged in 2^p blocks of size 2^m−k−p. In such situations, two-factor interactions are often confounded with blocks and complete information is lost on these two-factor interactions. Here we consider the use of the foldover technique in conjunction with combining designs having different blocking schemes to produce alternative partially confounded blocked fractional factorial designs that have more estimable two-factor interactions or a higher estimation capacity or both than their traditional counterparts.     

Results for two-level fractional factorial designs of resolution IV or more

The main theorem of this paper shows that foldover designs are the only (regular or nonregular) two-level factorial designs of resolution IV (strength 3) or more for n   runs and n/3?m?n/2$n/3?m?n/2$ factors. This theorem is a generalization of a coding theory result of Davydov and Tombak [1990. Quasiperfect linear binary codes with distance 4 and complete caps in projective geometry. Problems Inform. Transmission 25, 265–275] which, under translation, effectively states that foldover (or even) designs are the only regular two-level factorial designs of resolution IV or more for n   runs and 5n/16?m?n/2$5n/16?m?n/2$ factors. This paper also contains other theorems including an alternative proof of Davydov and Tombak's result.     

Defining equations for two-level factorial designs

Defining equations are introduced in the context of two-level factorial designs and they are shown to provide a concise specification of both regular and nonregular designs. The equations are used to find orthogonal arrays of high strength and some optimal designs. The latter optimal designs are formed in a new way by augmenting notional orthogonal arrays which are allowed to have some runs with a negative number of replicates before augmentation. Defining equations are also shown to be useful when the factorial design is blocked.     

Simultaneous estimation of location and dispersion in two-level fractional factorial designs

The reduction of variation is one of the obvious goals in quality improvement. The identification of factors aff ecting the dispersion is a step towards this goal. In this paper, the problem of estimating location effects and dispersion eff ects simultaneously in unreplicated factorial experiments is considered. By making a one-to-one transformation of the response variables, the study of the quadratic functions becomes clearer. The transformation also gives a natural motivation to the model of the variances of the original variables. The covariances of the transformed responses appear as parameters in the variances of the original variables. Results of Hadamard products are used for deriving these covariances. The method of estimating dispersion effects is shown in two illustrations. In a 24 factorial design, the essential covariance matrix of the transformed variables is also presented. The method is also illustrated in a 25-1 fractional design with a model which is saturated in this context.     

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.     

Classification of efficient two-level fractional factorial designs of resolution IV or more

Two-level fractional factorial designs that are efficient in terms of aberration or other aliasing properties are classified into four types of designs of resolution IV or more: the half-fraction designs, the even designs, the five-column designs and the join designs. The designs are shown to have concise grid representations which provide simple interpretations of their aliasing structure. New efficient 128-run designs are presented and blocking of the designs is considered.     

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.     

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.     

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.