Two-level Supersaturated Designs Research Articles

Selection of Two-Level Supersaturated Designs for Main Effects Models

An extensive literature is available on design selection criteria and analysis techniques for two-level supersaturated designs. The most notable design selection criteria are the popular -criterion, -criterion, and more recently, the var()-criterion, while the most notable analysis technique is the Gauss-Dantzig Selector. It has been observed that while the Gauss-Dantzig Selector is often the preferred analysis technique, differences in the screening performance of different designs are not captured well by any of the common design selection criteria. In addition, none of the criteria have any direct connection to the Gauss-Dantzig Selector. We develop two new design selection criteria inspired by large sample desiderata of the Gauss-Dantzig Selector. Then, using a multi-objective Pareto-based coordinate exchange algorithm, we find Pareto efficient designs. The obtained Pareto efficient designs perform better in about 85% of the considered cases as screening designs than the var()-optimal designs, especially when the true signs of effects are known. For the remaining 15% of the cases as well as for the unknown effect signs, the Pareto efficient designs perform at par with the var()-optimal designs.

R-optimal two-level supersaturated designs

At the primary stage of most of scientific investigations, supersaturated designs (SSDs) are extremely cost-effective to scientists and engineers. Booth and Cox [Booth KHV, Cox DR. Some systematic supersaturated designs. Technometrics. 1962;4:489–495] proposed as a criterion for comparing several SSDs. It is to be remarked that fails to compare many of competing SSDs. To overcome this drawback of the measure proposed by Booth and Cox [Booth KHV, Cox DR. Some systematic supersaturated designs. Technometrics. 1962;4:489–495], here we propose R-optimality as a measure for comparing SSDs. Moreover, the proposed measure unifies the existing measures for column balanced SSDs.

Projection properties of two-level supersaturated designs constructed from Hadamard designs using Lin’s method

In the initial stages of experimentation, many factors are examined for a possible significant influence on the response of interest. After such screening, the design used is projected into the significant factors and further evaluation of their effects is performed using the projection design. It is therefore interesting to evaluate the projection properties of screening designs since such an evaluation is extremely useful in selecting the best design for experimentation. In this paper, we examine two-level supersaturated screening designs that are constructed following Lin’s method. Some theoretical results are given and a detailed evaluation of supersaturated designs with up to 12 runs is performed.

Formula omitted]- and [formula omitted]-optimal supersaturated designs

Supersaturated designs are useful for factor screening experiments under the factor sparsity assumption that only a small number of factors are active. The popular E(s2)-criterion for choosing two-level supersaturated designs minimizes the sum of squares of theentries of the information matrix over the designs in which the two levels of each factor appear equal number of times. Jones and Majumdar (2014) proposed the UE(s2)-criterion which is essentially the same as the E(s2)-criterion except that the requirement of factor-level-balance is dropped. Removing this constraint makes UE(s2)-optimal designs easy to construct, but it also produces many UE(s2)-optimal designs with diverse performances. It is necessary to choose better designs from them, especially those with good lower-dimensional projection properties. While E(s2)-optimal designs tend to have better lower-dimensional projections than arbitrary UE(s2)-optimal designs, they are usually very difficult to construct. We propose a secondary criterion and provide simple and systematic constructions of superior UE(s2)-optimal designs having good projection properties. We also derive conditions under which E(s2)-optimal designs are UE(s2)-optimal as well and identify several families of designs that are optimal under both criteria.

Optimizing Two-Level Supersaturated Designs Using Swarm Intelligence Techniques

Supersaturated designs (SSDs) are often used to reduce the number of experimental runs in screening experiments with a large number of factors. As more factors are used in the study, the search for an optimal SSD becomes increasingly challenging because of the large number of feasible selection of factor level settings. This article tackles this discrete optimization problem via an algorithm based on swarm intelligence. Using the commonly used E(s2) criterion as an illustrative example, we propose an algorithm to find E(s2)-optimal SSDs by showing that they attain the theoretical lower bounds found in previous literature. We show that our algorithm consistently produces SSDs that are at least as efficient as those from the traditional CP exchange method in terms of computational effort, frequency of finding the E(s2)-optimal SSD, and also has good potential for finding D3-, D4-, and D5-optimal SSDs. Supplementary materials for this article are available online.

On the analysis of unbalanced two-level supersaturated designs via generalized linear models

ABSTRACTSupersaturated designs (SSDs) are factorial designs in which the number of experimental runs is smaller than the number of parameters to be estimated in the model. While most of the literature on SSDs has focused on balanced designs, the construction and analysis of unbalanced designs has not been developed to a great extent. Recent studies discuss the possible advantages of relaxing the balance requirement in construction or data analysis of SSDs, and that unbalanced designs compare favorably to balanced designs for several optimality criteria and for the way in which the data are analyzed. Moreover, the effect analysis framework of unbalanced SSDs until now is restricted to the central assumption that experimental data come from a linear model. In this article, we consider unbalanced SSDs for data analysis under the assumption of generalized linear models (GLMs), revealing that unbalanced SSDs perform well despite the unbalance property. The examination of Type I and Type II error rates through an extensive simulation study indicates that the proposed method works satisfactorily.

Supersaturated Designs With the Maximum Number of Factors for a Given Resolution-Rank

This article introduces new balanced two-level supersaturated designs with a large number of factors for a given resolution-rank, a criterion that directly assesses a supersaturated design’s ability to detect active factors. The search for supersaturated designs with the largest possible number of factors for a given resolution-rank is implemented by binary integer programming and exhaustive search that exploits design equivalence. We explore supersaturated designs with n = 6, 8, 10, and 12 runs. Six designs have been shown to have the maximum number of columns, and six new designs with improved resolution rank are found.

Construction of Efficient Multi-Level k-Circulant Supersaturated Designs

This article proposes an algorithm to construct efficient balanced multi-level k-circulant supersaturated designs with m factors and n runs. The algorithm generates efficient balanced multi-level k-circulant supersaturated designs very fast. Using the proposed algorithm many balanced multi-level supersaturated designs are constructed and cataloged. A list of many optimal and near optimal, multi-level supersaturated designs is also provided for m ≤ 60 and number of levels (q) ≤10. The algorithm can be used to generate two-level k-circulant supersaturated designs also and some large optimal two-level supersaturated designs are presented. An upper bound to the number of factors in a balanced multi-level supersaturated design such that no two columns are fully aliased is also provided.

Screening Strategies in the Presence of Interactions

Product and process improvement can involve a large number of factors that must be varied simultaneously. Understanding how factors interact is a key step in identifying those factors that have a substantial impact on the response. This article gives the first comprehensive assessment and comparison of screening strategies for interactions using two-level supersaturated designs, group screening, and a variety of data analysis methods including shrinkage regression and Bayesian methods. We develop novel methodology to allow application of Bayesian methods in two-stage group screening. Insights on using the strategies are provided through a variety of simulation scenarios and open issues are discussed. Supplementary materials are available online.

Finding MDS-Optimal Supersaturated Designs Using Computer Searches

Supersaturated designs can be evaluated using the minimal dependent sets (MDSs) of columns in the design matrix. This article describes an extensive computer search of balanced two-level supersaturated designs to find those that are MDS-optimal.

Construction of mixed-level supersaturated designs and comparison of their performance: Application to a gas chromatographic method

A supersaturated design is a design for which there are fewer runs than effects to be estimated. Although two-level supersaturated designs are becoming increasingly popular, mixed-level designs are scarcely used. Mixed-level designs are needed when the response is based on a polynomial response surface model or in situations where factors are nominal variables (with more than two modalities). The aim of this study is to explore the construction of mixed-level supersaturated designs and to evaluate their performance from the analysis of peppermint oil using a gas chromatographic method as application. This experimental setup requires the study of seven factors at two levels and five factors at three levels. Different building methods are tested from asymmetric or symmetric supersaturated designs. The mixed-level supersaturated designs obtained are compared from the point of view of a priori criteria with the aim of evaluating which criteria are better suited to judge the quality and fitness for purpose of these experimental designs. Finally, the results of the supersaturated designs are compared to the complete classical design.

An Algorithmic Construction ofE(s2)-Optimal Supersaturated Designs

Abstract A general method based on an effective algorithm for construction of E(s 2)-optimal, two-level supersaturated designs with the equal occurrence property, from supplementary difference sets is introduced. Supersaturated designs constructed in this way are E(s 2)-optimal. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of supersaturated designs with good properties.

Supersaturated designs: Are our results significant?

Two-level supersaturated designs (SSDs) are designs that examine more than n − 1 factors in n runs. Although SSD literature for both construction and analysis is plentiful, the dearth of actual applications suggests that SSDs are still an unproven tool. Whether using forward selection or all-subsets regression, it is easy to select simple models from SSDs that explain a very large percentage of the total variation. Hence, naive p -values can persuade the user that included factors are indeed active. We propose the use of a global model randomization test in conjunction with all-subsets (or a shrinkage method) to more appropriately select candidate models of interest. For settings where the large number of factors makes repeated use of all-subsets expensive, we propose a short-cut approximation for the p -values. Two state-of-the-art model selection methods that have received considerable attention in recent years, Least Angle Regression and the Dantzig Selector, were likewise supplemented with the global randomization test. Finally, we propose a randomization test for reducing the number of terms in candidate models with small global p -values. Randomization tests effectively emphasize the limitations of SSDs, especially those with a large factor to run size ratio.

Two-Level Supersaturated Designs: A Review

Abstract Supersaturated Designs (SSDs) are fractional factorial designs in which the run size is not enough to estimate the main effects of all the factors in the experiment Two-level SSDs have been studied extensively in the literature. The thrust of research has been on obtaining lower bounds to the value of E(s 2), a measure of departure from orthogonality, and constructing designs that attain these lower bounds. The focus of this paper is to review the literature on two-level SSDs.

Computer Aided Construction of Efficient Multi-Level Supersaturated Designs

Abstract Motivated by the computer search algorithms for constructing two-level supersaturated designs by Heavlin and Finnegan (1993), Li and Wu (1997), Nguyen (1996), Lejeune (2003) and Gupta, Parsad, Kole and Bhar (2008), this paper develops an algorithm to generate multi-level supersaturated designs. Popular E(fnod) and E(λ2) criterion have been used as a measure of non-orthogonality for the designs generated. The algorithm also ensures that no two columns in the designs generated are fully aliased. A catalogue of 120 optimal supersaturated designs for different number of factors m, design runs n, with 5 ≤ n < 16 runs, and different number of factor levels q, with 3 < q < 6, has been prepared. All the designs generated are fnod-optimal; some designs are λ 2-optimal too.

Supersaturated designs of projectivity P=3 or near P=3

Supersaturated designs offer a potentially useful way to investigate many factors in few experiments i.e. typical screening situations. Their design properties have mainly been evaluated based on their ability to identify and estimate main effects. Projective properties have received little attention. In this paper we show how to construct two-level supersaturated designs for 2( n−2) factors in n runs ( n a multiple of four) of projectivity P=3 or near projectivity P=3 from orthogonal non-regular two-level designs. The designs obtained also have favourable properties such as low maximum absolute value of the inner product between a main effect column and a two-factor interaction column and relatively few types of different projections onto subsets consisting of three factor columns.

An Overview of Two-Level Supersaturated Designs with Cyclic Structure

An overview is given of the link between the k-circulant method of construction of two-level supersaturated designs and construction methods based on cyclic incomplete block designs. It is shown that this link enables a simple formula for the Es2-efficiency of all such designs to be derived. Generators are given for Es2-optimal and near-optimal designs that extend the range of previously known designs or that have a smaller number of highly correlated column pairs.

A general construction of E(s 2)-optimal supersaturated designs via supplementary difference sets

Supplementary difference sets can be used to construct a variety of combinatorial designs. In this work, we construct supplementary difference sets that provide E(s 2)-optimal, two-level supersaturated designs with the equal occurrence property. Comparisons are made with previous works and it is shown that the proposed method gives promising results for the construction of supersaturated designs with good properties.

Two-level supersaturated designs for [formula omitted] runs and other cases

Two-level supersaturated designs are constructed for n = 2 k ( k ⩾ 5 ) runs and m factors where n + 3 ⩽ m ⩽ 5 ( n - 4 ) . The designs so formed are shown to have a maximum absolute correlation between factors of 1 4 and to be efficient in terms of E ( s 2 ) , particularly when the number of factors m is approximately double the number of runs n or greater. Thus, supersaturated designs with favourable properties are found for much higher numbers of runs than would be possible solely using algorithms.

New E(S2)-Optimal Supersaturated Designs Constructed From Incomplete Block Designs

We present a method for constructing two-level supersaturated designs (SSDs) from incomplete block designs. A lower bound of E(s2) that also covers the case of odd run sizes is given. This bound is attained by SSDs constructed from balanced incomplete block designs. We study SSDs that can be constructed from regular graph designs when balanced incomplete block designs do not exist. A computer search is conducted to find SSDs with 5 ≤ n ≤ 50 and n ≤ m ≤ 2n that can be constructed from regular graph designs, where m is the number of factors and n is the run size. Many SSDs derived from regular graph designs are optimal. The best E(s2)-optimal SSDs with respect to additional optimality criteria are tabulated. Some notes on the construction of saturated designs also are given.