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.     

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.     

Robust design of a polysilicon deposition process using split-plot analysis

The technique used for the analysis of experimental data must be appropriate for the design and treatment structures of the experiment; failure to take this into account can produce misleading results. This paper illustrates how split-plot designs can be used for the analysis of robust design experiments. In particular, the polysilicon deposition process data presented by Phadke¹ is analysed, and comparisons are made between the split-plot analysis of the raw data and the analysis conducted using signal-to-noise ratios. In addition, we demonstrate how the split-plot analysis provides information about interactions between control and noise factors, and how interaction plots can be used to assess the performance of the control factors across the levels of the noise factors. This information is particularly important to select the settings of the control factors that minimize the variation in the response induced by the noise factors.     

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.     

A METHOD TO IDENTIFY DISPERSION EFFECTS FROM UNREPLICATED MULTILEVEL EXPERIMENTS

The use of experimental design to identify dispersion effects is an effective tool for improving quality. In this article the problem of identifying such effects from unreplicated experiments is studied. We focus on a method developed for two-level 2^k-p experiments and discuss how the method can be generalized to a wider range of experimental designs. In particular, we show how dispersion effects can be identified from non-geometric Plackett–Burman designs and from experiments with more than two factor levels. In contrast to related methods our approach provides a test statistic with a well-known reference distribution. Three actual experiments are used to illustrate the method and to make comparisons to related methods. © 1997 John Wiley & Sons, Ltd.     

Improved Split-Plot and Multistratum Designs

Many industrial experiments involve some factors whose levels are harder to set than others. The best way to deal with these is to plan the experiment carefully as a split-plot, or more generally a multistratum, design. Several different approaches for constructing split-plot type response surface designs have been proposed in the literature since 2001, which has allowed experimenters to make better use of their resources by using more efficient designs than the classical balanced ones. One of these approaches, the stratum-by-stratum strategy has been shown to produce designs that are less efficient than locally D-optimal designs. An improved stratum-by-stratum algorithm is given, which, though more computationally intensive than the old one, makes better use of the advantages of this approach, that is, it can be used for any structure and does not depend on prior estimates of the variance components. This is shown to be almost as good as the locally optimal designs in terms of their own criteria and more robust across a range of criteria. Supplementary materials for this article are available online.     

Constructing Tables of Minimum Aberration Pn–m Designs

Fries and Hunter (1980) presented a practical algorithm for selecting standard 2 ^n–m fractional factorial designs based on a criterion they called “minimum aberration.” In this article some simple results are presented that enable the Fries–Hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p ≥ 2 is prime. Examples of minimum aberration 2 ^n–m designs with resolution R ≥ 4 are given for m, n – m < 9. A matrix is given for generating 3 ^n–m designs with m, n – m ≤ 6, which have, or nearly have, minimum aberration.     

Orthogonal Main-Effect 2 n 3 m Designs and Two-Factor Interaction Aliasing

Methods are presented for the determination of the alias matrix of two-factor interactions for the orthogonal main-effect 2 ⁿ 3 ^m plans catalogued by Addelman and Kempthorne. This catalogue includes Placket-Burman designs and designs obtained by replacement in 2 ^n–p plans or collapsing in 3 ^m–m plans. Systematic methods are included to facilitate the data computations. For standard r ^n–p factorial designs, techniques are given to determine a set of live factors, a generating set of linear sum congruences and the alias matrix. Additional orthogonal main-effect 2 ⁿ 3 ^m designs are constructed to supplement the Addelman-Kempthorne catalogue of designs.     

Split-Plot and Multi-Stratum Designs for Statistical Inference

It is increasingly recognized that many industrial and engineering experiments use split-plot or other multi-stratum structures. Much recent work has concentrated on finding optimum, or near-optimum, designs for estimating the fixed effects parameters in multi-stratum designs. However, often inference, such as hypothesis testing or interval estimation, will also be required and for inference to be unbiased in the presence of model uncertainty requires pure error estimates of the variance components. Most optimal designs provide few, if any, pure error degrees of freedom. Gilmour and Trinca (2012 Gilmour, S. G., and Trinca, L. A. (2012), “Optimum Design of Experiments for Statistical Inference” (with discussion), Applied Statistics, 61, 345–401.[Crossref], [Web of Science ®] , [Google Scholar]) introduced design optimality criteria for inference in the context of completely randomized and block designs. Here these criteria are used stratum-by-stratum to obtain multi-stratum designs. It is shown that these designs have better properties for performing inference than standard optimum designs. Compound criteria, which combine the inference criteria with traditional point estimation criteria, are also used and the designs obtained are shown to compromise between point estimation and inference. Designs are obtained for two real split-plot experiments and an illustrative split–split-plot structure. Supplementary materials for this article are available online.     

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.     

Blocking of 3(2 n–k ) Designs

3(2 ^n–2) designs may be divided into two blocks, one of size 2 ^n–1 and the other of size 2 ^n–2 by blocking on one of the defining contrasts and into three blocks of size 2 ^n–2 by blocking on all three defining contrasts. Blocking on an effect which is not a defining contrast gives two blocks of 3(2 ^n–3) runs each. In this paper these methods are applied to 3(2 ^n–k ) designs with twelve or twenty-four points. The designs considered are the 3(2^4–2) and 3(2^5–2) designs with all main effects and all two factor interactions estimable (assuming that higher order interactions are negligible), and saturated main effect plans with twelve points and up to eleven factors.     

A comparison of two-level designs to estimate all main effects and two-factor interactions

We compare cost-efficient alternatives for the full factorial 2⁴ design, the regular 2^5-1 fractional factorial design, and the regular 2^6-1 fractional factorial design that can fit the model consisting of all the main effects as well as all the two-factor interactions. For 4 and 5 factors we examine orthogonal arrays with 12 and 20 runs, respectively. For 6 factors we consider orthogonal arrays with 24 as well as 28 runs. We consult complete catalogs of two-level orthogonal arrays to find the ones that provide the most efficient estimation of all the effects in the model. We compare these arrays with D-optimal designs found using a coordinate exchange algorithm. The D-optimal designs are always preferable to the most efficient orthogonal arrays for fitting the full model in all the factors.     

Saturated Fractions of 2 n and 3 n Factorial Designs

Saturated fractions of 2 ⁿ and 3 ⁿ factorial designs which permit the estimation of both main effects and first order interactions are described. A simple method of generating these particular designs is given. In addition to presenting the specific designs for n = 3, …, 10, tables of variances and relative efficiencies are included to assist the potential user in assessing the suitability of the described designs.     

A New Balanced Design for Eighteen Varieties

A new balanced incomplete block design for v = |8, b = 5|, r = 17, k = 6. λ = 5 is presented. The design is resolvable and can be split into useful partially balanced suhdesigns. When these designs are used for the 2 × 3² factorial experiment they all have factorial structure.     

Designing Experiments for Tolerancing Assembled Products

This article provides an outline of theory and methods for the experimental determination of tolerance limits for mating components of assembled products. The emphasis is on novel combinatorial problems of pre- and post-fracfionafion of certain products of two-level factorial designs. The cost of experimentation is discussed and used as a guide to allocating experimental runs. Several design examples are provided. The article also includes a comprehensive example of the experimental determination of tolerances for the components of a throttle handle for small outboard motors.     

Screening Designs for Poly-Factor Experimentation

A survey is given of the following types of screening designs: Incomplete 2 ^k designs, srlpersaturated and grollp-screening designs. These designs are compared with each other. Some new results for group-screening are derived.     

Two-Level Multifactor Designs for Detecting the Presence of Interactions

A design optimality criterion, tr (L)-optimality, is applied to the problem of designing two-level multifactor experiments to detect the presence of interactions among the controlled variables. We give rules for constructing tr (L)-optimal foldover designs and tr (L)-optimal fractional factorial designs. Some results are given on the power of these designs for testing the hypothesis that there are no two-factor interactions. Augmentation of the tr (L)-optimal designs produces designs that achieve a compromise between the criteria of D-optimality (for parameter estimation in a first-order model) and tr (L)-optimality (for detecting lack of fit). We give an example to demonstrate an application to the sensitivity analysis of a computer model.     

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.     

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.     

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.