Classical design structure of orthogonal designs with six to eight factors and sixteen runs

Most two‐level fractional factorial designs used in practice involve independent or fully confounded effects (so‐called regular designs). For example, for 16 runs and 6 factors, the classical resolution IV design with defining relation I = ABCE = BCDF = ADEF has become the de facto gold standard. Recent work has indicated that non‐regular orthogonal designs could be preferable in some circumstances. Inhibiting a wider usage of these non‐regular designs seems to be a combination of inertia/status quo and perhaps the general resistance and suspicion to designs that are computer generated to achieve ‘X–Y–Z’ optimality. In this paper each of the orthogonal non‐isomorphic two‐level, 16 run designs with 6, 7, or 8 factors (both regular and non‐regular) are shown to have a classical‐type construction with a 2⁴ or a replicated 2³ starting point. Additional factor columns are defined either using the familiar one‐term column generators or generators using weighted sums of effects. Copyright © 2010 John Wiley & Sons, Ltd.     

Constructing Two-Level Designs by Concatenation of Strength-3 Orthogonal Arrays

Two-level orthogonal arrays of N runs, k factors, and a strength of 3 provide suitable fractional factorial designs in situations where many of the main effects are expected to be active, as well as some two-factor interactions. If they consist of N/2 mirror image pairs, these designs are fold-over designs. They are called even and provide at most N/2 ? 1 degrees of freedom to estimate interactions. For k < N/3 factors, there exist strength-3 designs that are not fold-over designs. They are called even-odd designs and they provide many more degrees of freedom to estimate interactions. For N ? 48, attractive even-odd designs can be extracted from complete catalogs of strength-3 orthogonal arrays. However, for larger run sizes, no complete catalogs exist. To construct even-odd designs with N > 48, we develop an algorithm for an optimal concatenation of strength-3 designs involving N/2 runs. Our approach involves column permutations of one of the concatenated designs, as well as sign switches of the elements of one or more columns of that design. We illustrate the potential of the algorithm by generating two-level even-odd designs with 64 and 128 runs involving up to 33 factors, because this allows a comparison with benchmark designs from the literature. With a few exceptions, our even-odd designs outperform or are competitive with the benchmark designs in terms of the aliasing of two-factor interactions and in terms of the available degrees of freedom to estimate two-factor interactions. Supplementary materials for the article are available online.     

Elementary Statistics

Fractional two-level factorial designs are often used in the early stages of an investigation to screen for important factors. Traditionally, 2 ^n-k fractional factorial designs of resolution III, IV, or V have been used for this purpose. When the investigator is able to specify the set of nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design with fewer runs than a standard textbook design by searching within a wider class of designs called parallel-flats designs. The run sizes in this class of designs do not necessarily need to be powers of 2. We discuss an algorithm for constructing orthogonal parallel-flats designs to meet user specifications. Several examples illustrate the use of the algorithm.     

21st century screening experiments: What,why, and how

ABSTRACT

The primary aim of screening experiments is to identify the active factors; that is, those having the largest effects on the response of interest. Large factor effects can be either main effects, two-factor interactions (2FIs), or even strong curvature effects. Because the number of runs in a screening experiment is generally on the order of the number of factors, the designs rely heavily on the factor or effect sparsity assumption. That is, practitioners performing such experiments must be willing to assume that only a small fraction of the factors or effects are active.

Traditional screening designs such as regular fractional factorial and Plackett-Burman designs employ factors at two levels only. Though they have orthogonal linear main effects, such designs cannot uniquely identify factors with strong curvature effects.

Definitive screening designs (DSDs) have many desirable properties that make them appealing alternatives to other screening design methods. They are orthogonal for the main effects. In addition, main effects are orthogonal to all second-order effects and second-order effects are not confounded with each other. In addition, quadratic effects of every factor are estimable. For more than five factors, a DSD projects onto any three factors so that a full quadratic model in those three factors is estimable with reasonable efficiency. As a result, when three or fewer factors turn out to be important, follow-up optimization experiments may not be necessary.

All this begs the question, “Are DSDs really as good as they are advertised to be?” This article addresses this question with an even-handed comparison of the various screening approaches. It also considers the sparsity assumption common to all screening designs and provides some guidance for quantifying what effect sparsity means for both traditional screening designs and DSDs.     

Two-Level Designs to Estimate All Main Effects and Two-Factor Interactions

We study the design of two-level experiments with N runs and n factors large enough to estimate the interaction model, which contains all the main effects and all the two-factor interactions. Yet, an effect hierarchy assumption suggests that main effect estimation should be given more prominence than the estimation of two-factor interactions. Orthogonal arrays (OAs) favor main effect estimation. However, complete enumeration becomes infeasible for cases relevant for practitioners. We develop a partial enumeration procedure for these cases and we establish upper bounds on the D-efficiency for the interaction model based on arrays that have not been generated by the partial enumeration. We also propose an optimal design procedure that favors main effect estimation. Designs created with this procedure have smaller D-efficiencies for the interaction model than D-optimal designs, but standard errors for the main effects in this model are improved. Generated OAs for 7–10 factors and 32–72 runs are smaller or have a higher D-efficiency than the smallest OAs from the literature. Designs obtained with the new optimal design procedure or strength-3 OAs (which have main effects that are not correlated with two-factor interactions) are recommended if main effects unbiased by possible two-factor interactions are of primary interest. D-optimal designs are recommended if interactions are of primary interest. Supplementary materials for this article are available online.     

Constructing General Orthogonal Fractional Factorial Split-Plot Designs

While the orthogonal design of split-plot fractional factorial experiments has received much attention already, there are still major voids in the literature. First, designs with one or more factors acting at more than two levels have not yet been considered. Second, published work on nonregular fractional factorial split-plot designs was either based only on Plackett–Burman designs, or on small nonregular designs with limited numbers of factors. In this article, we present a novel approach to designing general orthogonal fractional factorial split-plot designs. One key feature of our approach is that it can be used to construct two-level designs as well as designs involving one or more factors with more than two levels. Moreover, the approach can be used to create two-level designs that match or outperform alternative designs in the literature, and to create two-level designs that cannot be constructed using existing methodology. Our new approach involves the use of integer linear programming and mixed integer linear programming, and, for large design problems, it combines integer linear programming with variable neighborhood search. We demonstrate the usefulness of our approach by constructing two-level split-plot designs of 16–96 runs, an 81-run three-level split-plot design, and a 48-run mixed-level split-plot design. Supplementary materials for this article are available online.     

Analysis and efficient 2k − 1 designs for experiments in blocks of size two

Two‐level factorial designs in blocks of size two are useful in a variety of experimental settings, including microarray experiments. Replication is typically used to allow estimation of the relevant effects, but when the number of factors is large this common practice can result in designs with a prohibitively large number of runs. One alternative is to use a design with fewer runs that allows estimation of both main effects and two‐factor interactions. Such designs are available in full factorial experiments, though they may still require a great many runs. In this article, we develop fractional factorial design in blocks of size two when the number of factors is less than nine, using just half of the runs needed for the designs given by Kerr (J Qual. Tech. 2006; 38 :309–318). Two approaches, the orthogonal array approach and the generator approach, are utilized to construct our designs. Analysis of the resulting experimental data from the suggested design is also given. Copyright © 2011 John Wiley & Sons, Ltd.     

The general balance metric for mixed‐level fractional factorial designs

Mixed‐level designs are employed when factors with different numbers of levels are involved. Practitioners use mixed‐level fractional factorial designs as the total number of runs of the full factorial increases rapidly as the number of factors and/or the number of factor levels increases. One important decision is to determine which fractional designs should be chosen. A new criterion, the general balance metric (GBM), is proposed to evaluate and compare mixed‐level fractional factorial designs. The GBM measures the degree of balance for both main effects and interaction effects. This criterion is tied to, and dominates orthogonality criteria as well as traditional minimum aberration criteria. Furthermore, the proposal is easy to use and has practical interpretations. As part of the GBM, the concept of resolution is generalized and the confounding structure of mixed‐level fractional factorial designs is also revealed. Moreover, the metric can also be used for the purpose of design augmentation. Examples are provided to compare this approach with existing criteria. Copyright © 2008 John Wiley & Sons, Ltd.     

Applied Statistics: Analysis of Variance and Regression

A new robust criterion for experimental designs is proposed. For a given design, the criterion is to minimize, over all possible run orders, the absolute value of the change of the variance function due to possible correlation between the observations. The resulting design is robust against possible auto-correlation among the observations in the sense that confidence-interval coverage levels are maintained accurately. Applications are given for 2^k factorial designs and one mixed-level fractional experiment, and robust runs are obtained. Computational strategies are also discussed, and a simulated annealing algorithm is developed to search for robust designs.     

Past Editors and Editorial Collaborators 1976

D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.     

Making full use of taguchi's orthogonal arrays

Taguchi¹ has provided 18 orthogonal arrays which have been widely touted as useful frameworks for planning experiments. Thirteen of these are ‘saturated designs’, that is, they are appropriate for investigating (N - 1) factors in N runs, thus using the full capacity of the design. Here, the other five ‘non-saturated’ designs are discussed. By creating additional, orthogonal columns which provide estimates of interaction effects, we can essentially wring out some additional information over and above that suggested by Taguchi, without additional cost. In particular, if only the linear effect is of interest for any specific factor, one can accommodate more factors than the number suggested by Taguchi. An example is given for illustration.     

Semifolding 2 k–P Designs

This article addresses the varied possibilities for following a two-level fractional factorial with another fractional factorial half the size of the original experiment. Although follow-up fractions of the same size as an original experiment are common practice, in many situations a smaller followup experiment will sufIice. Peter John coined the term “semifolding” to describe using half of a foldover design. Existing literature does include brief mention and examples of semifolding but no thorough development of this follow-up strategy. After a quick examination of the estimation details for semlfoldmg the 2^{4 – 1} _IV design, we focus on following 16-run fractions with a semifold design of eight runs. Two such examples are considered—one in which the initial fraction ia resolution IV, the other resolution III. A general result is proven for semifolding 2 ^{k – p} _IV designs. Conducting full foldover designs in two blocks is also recommended.     

Application of Uniform Design in the Formation of Cement Mixtures

As products and processes become more and more complex, there is an increasing need in the industry to perform experiments with a large number of factors and a large number of levels for each factor. For such experiments, application of traditional designs such as factorial designs or orthogonal arrays is impractical because of the large number of runs required. As an alternative, a type of design, called the uniform design, can be used to solve such problems. The uniform design has been intensively studied by theoreticians for several decades and has many successful examples of application in industry. In this article, we report a successful application of uniform design in product formation in the cement manufacturing industry. Specifically, we investigate the effects of additives on bleeding and compressive strength of a cement mixture. This example illustrates how an experiment of 16 runs was performed to study three factors with 16 levels, 8 levels, and 8 levels, respectively.     

Orthogonal Main-Effect Plans Permitting Estimation of All Two-Factor Interactions for the 2n3m Factorial Series of Designs

If we assume no higher order interactions for the 2ⁿ3^m factorial series of designs, then relaxing the restrictions concerning equal frequency for the factors and complete orthogonality for each estimate permits considerable savings in the number of runs required to estimate all the main effects and two-factor interactions. Three construction techniques are discussed which yield designs providing orthogonal estimates of all the main effects and allowing estimation of all the two-factor interactions. These techniques are: (i) collapsing of factors in symmetrical fractionated 3^m–p designs, (ii) conjoining fractionated designs, and (iii) combinations of (i) and (ii). Collapsing factors in a design either maintains or increases the resolution of the original design, but does not decrease it. Plans are presented for certain values of (n, m) as examples of the construction techniques. Systematic methods of analysis are also discussed.     

Estimating the error of an effect in unreplicated 2-level fractional factorial designs

Saturated fractional factorial experimental designs and orthogonal main effect plans are extremely valuable tools in quality engineering. However, one problem with these designs is that there are no replicate runs to be used for estimating experimental error. This note develops an estimator of the experimental error based on the hypothesis that not all factor effects will be non-zero. A joint Bayesian prior distribution is presented for the experimental error variance of an effect, σ², and the probability that each effect is non-zero. From this prior distribution a posterior marginal distribution for σ² is derived along with a direct estimate of σ². This method is compared with the traditional methods of estimating σ² in unreplicated designs through a numerical example.     

Combined array experiment design

Experiment plans formed by combining two or more designs, such as orthogonal arrays primarily with 2- and 3-level factors, creating multi-level arrays with subsets of different strength are proposed for computer experiments to conduct sensitivity analysis. Specific illustrations are designs for 5-level factors with fewer runs than generally required for 5-level orthogonal arrays of strength 2 or more. At least 5 levels for each input are desired to allow for runs at a nominal value, 2-values either side of nominal but within a normal, anticipated range, and two, more extreme values either side of nominal. This number of levels allows for a broader range of input combinations to test the input combinations where a simulation code operates. Five-level factors also allow the possibility of up to fourth-order polynomial models for fitting simulation results, at least in one dimension. By having subsets of runs with more than strength 2, interaction effects may also be considered. The resulting designs have a “checker-board” pattern in lower-dimensional projections, in contrast to grid projection that occurs with orthogonal arrays. Space-filling properties are also considered as a basis for experiment design assessment.     

Near Minimal Resolution IV Designs for the sn Factorial Experiment

A fractional factorial design is of resolution IV if all main effects are estimable in the presence of two-factor interactions. For the sⁿ factorial experiment such a design requires at least N = s[(s – I)n – (s – 2)] runs. In this paper a series of resolution IV designs are given for the s” factorial, s = p ^α where p is prime, in N = s(s – I)n runs. A special case of the construction method produces a series of generalized foldover designs for the sⁿ experiment, s ≥ 3 and n ≥ 3, in N = s(s – I)n + s runs. These foldover designs permit estimation of the general mean in addition to all main effects and provide s degrees of freedom for estimating error. A section on analysis is included.     

Discussion: Posterior Probabilities and Experimental Designs for Discriminating Between Regression Models

The treatment-design portion of fractionated two-level split-plot designs is associated with a subset of the 2 ^n–k fractional factorial designs. The concept of aberration is then extended to these splitplot designs to compare designs. Two methods are presented for constructing two-level minimumaberration split-plot designs, along with examples. An extensive catalog of such designs is tabulated. Extensions to prime-level designs and relations to inner-outer arrays are also presented.     

Taguchi’s Orthogonal Arrays Are Classical Designs of Experiments

Taguchi’s catalog of orthogonal arrays is based on the mathematical theory of factorial designs and difference sets developed by R. C. Bose and his associates. These arrays evolved as extensions of factorial designs and latin squares. This paper (1) describes the structure and constructions of Taguchi’s orthogonal arrays, (2) illustrates their fractional factorial nature, and (3) points out that Taguchi’s catalog can be expanded to include orthogonal arrays developed since 1960.     

Graphical and computer-aided approaches to plan experiments

Design of experiments is a quality technology to achieve product excellence, that is to achieve high quality at low cost. It is a tool to optimize product and process designs, to accelerate the development cycle, to reduce development costs, to improve the transition of products from R & D to manufacturing and to troubleshoot manufacturing problems effectively. It has been successfully, but sporadically, used in the United States. More recently, it has been identified as a major technological reason for the success of Japan in producing high-quality products at low cost. In the United States, the need for increased competitiveness and the emphasis on quality improvement demands a widespread use of design of experiments by engineers, scientists and quality professionals. In the past, such widespread use has been hampered by a lack of proper training and a lack of availability of tools to easily implement design of experiments in industry. Three steps are essential, and are being taken, to change this situation dramatically. First, simple graphical methods, to design and analyse experiments, need to be developed, particularly when the necessary microcomputer resources are not available. Secondly, engineers, scientists and quality professionals must have access to microcomputer-based software for design and analysis of experiments.¹ Availability of such software would allow users to concentrate on the important scientific and engineering aspects of the problem by computerizing the necessary statistical expertise. Finally, since a majority of the current workforce is expected to be working in the year 2000, a massive training effort, based upon simple graphical methods and appropriate computer software, is necessary.² The purpose of this paper is to describe a methodology based upon a new graphical method called interaction graphs and other previously known techniques, to simplify the correct design of practically important fractional factorial experiments. The essential problem in designing a fractional factorial experiment is first stated. The interaction graph for a 16-trial fractional factorial design is given to illustrate how the graphical procedure can be easily used to design a two-level fractional factorial experiment. Other previously known techniques are described to easily modify the two-level fractional factorial designs to create mixed multi-level designs. Interaction graphs for other practically useful fractional factorial designs are provided. A computer package called CADE (computer aided design of experiments), which automatically generates the appropriate fractional factorial designs based upon user specifications of factors, levels and interactions and conducts complete analyses of the designed experiments is briefly described.¹ Finally, the graphical method is compared with other available methods for designing fractional factorial experiments.