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.     

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.     

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 Orthogonal Designs With Mixed Integer Linear Programming

We present a mixed integer linear programming approach to orthogonally block two-level, multilevel, and mixed-level orthogonal designs. The approach involves an exact optimization technique which guarantees an optimal solution. It can be applied to many problems where combinatorial methods for blocking orthogonal designs cannot be used. By means of 54-run and 64-run examples, we demonstrate that our approach outperforms two benchmark techniques in terms of the number of estimable two-factor interaction contrasts and in terms of the D-efficiency for models with main effects and some two-factor interaction contrasts. We demonstrate the generic nature of our approach by applying it to the most challenging instances in a catalog of all orthogonal designs of strength 3 with up to 81 runs as well as a small catalog of strength-4 designs. The approach can also be applied to strength-2 designs, but, for these cases, alternative methods described in the literature may perform equally well. Supplementary materials for this article are available online.     

Selecting an Orthogonal or Nonorthogonal Two-Level Design for Screening

This article presents a comparison of criteria used to characterize two-level designs for screening purposes. To articulate the relationships among criteria, we focus on 7-factor designs with 16–32 runs and 11-factor designs with 20–48 runs. Screening based on selected designs for each of the run sizes considered is studied with simulation using a forward selection procedure and the Dantzig selector. This article compares Bayesian D-optimal designs, designs created algorithmically to optimize estimation capacity over various model spaces, and orthogonal designs by estimation-based criteria and simulation. In this way, we furnish both general insights regarding various design approaches, as well as a guide to make a choice among a few final candidate designs. Supplementary materials for this article are available online.     

Reduced Designs of Resolution Five

The two level fractional factorial designs of resolution five enable the experimenter to estimate independently all main effects and two-factor interactions under the assumptions that higher order interaction effects are negligible. By relaxing, very slightly, the requirement that all two-factor interactions be estimable, or that all estimated effects be orthogonal, the number of runs required for many resolution five designs can be greatly reduced.     

Benefits and Fast Construction of Efficient Two-Level Foldover Designs

Recent work in two-level screening experiments has demonstrated the advantages of using small foldover designs, even when such designs are not orthogonal for the estimation of main effects (MEs). In this article, we provide further support for this argument and develop a fast algorithm for constructing efficient two-level foldover (EFD) designs. We show that these designs have equal or greater efficiency for estimating the ME model versus competitive designs in the literature and that our algorithmic approach allows the fast construction of designs with many more factors and/or runs. Our compromise algorithm allows the practitioner to choose among many designs making a trade-off between efficiency of the main effect estimates and correlation of the two-factor interactions (2FIs). Using our compromise approach, practitioners can decide just how much efficiency they are willing to sacrifice to avoid confounded 2FIs as well as lowering an omnibus measure of correlation among the 2FIs.     

Optimal Projective Three-Level Designs for Factor Screening and Interaction Detection

Orthogonal arrays (OAs) are widely used in industrial experiments for factor screening. Suppose that only a few of the factors in the experiments turn out to be important. An OA can be used not only for screening factors, but also for detecting interactions among a subset of active factors. In this article a set of optimality criteria is proposed to assess the performance of designs for factor screening, projection, and interaction detection, and a three-step approach is proposed to search for optimal designs. Combinatorial and algorithmic construction methods are proposed for generating new designs. Permutations of levels are used for improving the eligibility and estimation efficiency of the projected designs. The techniques are then applied to search for best three-level designs with 18 and 27 runs. Many new, efficient, and practically useful nonregular designs are found and their properties are discussed.     

Applied Statistics Algorithms

This article describes a family of Resolution IIIdesigns for which the usual advice regarding foldover—reversing all factors—is ill advised. The smallest such designs have 16 runs. For designs in this family, alternative foldover fractions not only increase the resolution to IV, but also separate some of the aliased two-factor interactions. Although Resolution IV designs obtained by reversing all factors provide fewer than half the degrees of freedom for estimating two-factor interactions, the recommended foldovers for designs in this class permit estimation of many more two-factor interactions. This result implies that designs in this class make attractive initial designs, even if they are not minimum aberration.     

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.     

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.     

Aliased informed model selection strategies for six-factor no-confounding designs

Nonregular designs are a preferable alternative to regular resolution IV designs because they avoid confounding two-factor interactions. As a result nonregular designs can estimate and identify a few active two-factor interactions. However, due to the sometimes complex alias structure of nonregular designs, standard screening strategies can fail to identify all active effects. In this paper, we explore a specific no-confounding six-factor 16-run nonregular design with orthogonal main effects. By utilizing our knowledge of the alias structure, we can inform the model selection process. Our aliased informed model selection (AIMS) strategy is a design-specific approach that we compare to three generic model selection methods; stepwise regression, Lasso, and the Dantzig selector. The AIMS approach substantially increases the power to detect active main effects and two-factor interactions versus the aforementioned generic methodologies.     

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 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.     

Using the Folded-Over 12-Run Plackett—Burman Design to Consider Interactions

This article demonstrates that the folded-over 12-run Plackett–Burman design is useful for considering up to 12 factors in 24 runs, even if one anticipates that some two-factor interactions may be significant. The properties of this design are investigated, and a sequential procedure for analyzing the data from such a design is proposed. The performance of the procedure is investigated through the analysis of real and constructed examples and through a small simulation study. Applications to other folded-over Plackett–Burman designs are also briefly discussed.     

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.     

Semifold plans for mixed‐level designs

This article presents a more efficient method for sequential augmentation of mixed‐level designs. The proposed approach reduces the optimal foldover plan of a mixed‐level design to a semifold plan by selecting half of the treatment combinations of the foldover fraction using exhaustive search and the criterion of general balance metric. The resulting design is a more economic run size augmented fraction that possesses good balance and orthogonality properties for main effects and two‐factor interactions. Three efficient arrays consisting of 20, 24 and 30 runs were selected for the analysis. Efficient arrays composed of a higher number of runs can be semifolded in a similar manner. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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.     

Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations

In planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects. In this article, we propose a graph-aided method to solve this problem for two-level experiments. First, we choose the defining relations for a 2 ^n–k design according to a goodness criterion such as the minimum aberration criterion. Then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two-factor interactions for the given defining relations. In each graph a vertex represents a factor and an edge represents the interaction between the two factors. For the experiment planner, the job is simple: Draw a graph representing the specified interactions and compare it with the list of graphs obtained previously. Our approach is a substantial improvement over Taguchi's linear graphs.