Fractional Factorial Designs for the Detection of Interactions between Design and Noise Factors

In industrial experiments on both design (control) factors and noise factors aimed at improving the quality of manufactured products, designs are needed which afford independent estimation of all design×noise interactions in as few runs as possible, while allowing aliasing between those factorial effects of less interest. An algorithm for generating orthogonal fractional factorial designs of this type is described for factors at two levels. The generated designs are appropriate for experimenting on individual factors or for experimentation involving group screening of factors.     

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.     

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.     

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.     

An optimal selection of two-level regular single arrays for robust parameter experiments

Efforts have been made in the literature to find optimal single arrays which work best for the robust parameter experiments. However, examples show that in many cases the optimal designs obtained by the existing criteria cloud not attain the maximum number of clear interested effects for robust parameter experiments. In this paper, through a similar way of Zhang et al. (2008) (ZLZA, in short), an aliasing pattern to measure the confounding between the interested effects and other effects for the case of robust parameter designs is introduced. A new criterion for selecting optimal two-level regular single arrays is proposed. In the consideration of the criterion, two rank-orders of effects are suggested: one is based on the interest of experimenters and the other is under the usual effect hierarchy principle. The optimal designs are tabulated in the appendix.     

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     

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.     

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.     

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.     

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.     

Mixed two- and four-level fractional factorial split-plot designs with clear effects

Mixed-level designs have become 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. It is highly to know when a mixed-level FFSP design with resolution III or IV has clear effects. This paper investigates the conditions of a resolution III or IV FFSP design with both two-level and four-level factors 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.     

2(n1+n2)-(k1+k2) Fractional factorial split-plot designs containing clear effects

Whole-plot (WP) factors and sub-plot (SP) factors play different roles in fractional factorial split-plot (FFSP) designs. In this paper, we consider FFSP designs with resolution III or IV from the viewpoint of clear factorial effects, classify two-factor interactions (2FIs) into three types, and give sufficient and necessary conditions for the existence of FFSP designs containing various clear factorial effects, including two types of main effects and three types of 2FIs. The structures of these designs are also shown and illustrated with examples.     

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.     

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.     

Comparisons of search designs using search probabilities

Search designs are considered for searching and estimating one nonzero interaction from the two and three factor interactions under the search linear model. We compare three 12-run search designs D1, D2, and D3, and three 11-run search designs D4, D5, and D6, for a 2⁴ factorial experiment. Designs D2 and D3 are orthogonal arrays of strength 2, D1 and D4 are balanced arrays of full strength, D5 is a balanced array of strength 2, and D6 is obtained from D3 by deleting the duplicate run. Designs D4 and D5 are also obtained by deleting a run from D1 and D2, respectively. Balanced arrays and orthogonal arrays are commonly used factorial designs in scientific experiments. “Search probabilities” are calculated for the comparison of search designs. Three criteria based on search probabilities are presented to determine the design which is most likely to identify the nonzero interaction. The calculation of these search probabilities depends on an unknown parameter ρ which has a signal-to-noise ratio form. For a given value of ρ, Criteria I and II are newly proposed in this paper and Criteria III is given in Shirakura et al. (Ann. Statist. 24 (6) (1996) 2560). We generalize Criteria I–III for all values of ρ so that the comparison of search designs can be made without requiring a specific value of ρ. We have developed simplified methods for comparing designs under these three criteria for all values of ρ. We demonstrate, under all three criteria, that the balanced array D1 is more likely to identify the nonzero interaction than the orthogonal arrays D2 and D3, and the design D4 is more likely to identify the nonzero interaction than the designs D5 and D6.The methods of comparing designs developed in this paper are applicable to other factorial experiments for searching one nonzero interaction of any order.     

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 .     

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.     

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.     

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.     

New light on fractional 2m factorial designs

New fractional 2^m factorial designs obtained by assigning factors to fractions of m columns of new saturated two symbol orthogonal arrays which are not isomorphic to the usual ones are proposed. Contrary to the usual assignment, examples show that some main effects are not totally but partially confounded with several two-factor interactions. Moreover, the recovery of the former from such partial confounding is possible in some cases by eliminating the latter.