Supersaturated Designs Research Articles

Screening Active Effects in Supersaturated Designs with Binary Response via Control Charts

Supersaturated designs are designs in which the number of factors exceeds the run size; consequently, there are not enough degrees of freedom to estimate all the main effects. The goal here is to identify the dominant factors that constitute a small proportion of the overall set of factors, according to the assumption of effect sparsity. The analysis of such designs constitutes a challenging task and, even though many methods have been proposed in the literature assuming a normal response, only few works attempted to address the case of non‐normal responses. In this paper, we propose a method for screening out the most important features in supersaturated designs assuming a Bernoulli distributed response. This new approach is based on an effective chart in Statistical Process Control, the cumulative sum control chart, combined with an information theoretic measure, and it is referred as the MIC algorithm. We judge the value of MIC through comparisons with three existing approaches suggested in the literature: the least absolute shrinkage and selection operator penalization method, and two feature selection algorithms, the Conditional Mutual Information Maximization and the minimal‐redundancy‐maximal‐relevance. The simulation study reveals that the proposed method can be considered an advantageous method because of its extremely good performance in terms of statistical power. Copyright © 2017 John Wiley & Sons, Ltd.

A three-stage variable selection method for supersaturated designs

ABSTRACTA supersaturated design (SSD) is a design whose run size is not enough for estimating all main effects. Such a design is commonly used in screening experiments to screen active effects based on the effect sparsity principle. Traditional approaches, such as the ordinary stepwise regression and the best subset variable selection, may not be appropriate in this situation. In this article, a new variable selection method is proposed based on the idea of staged dimensionality reduction. Simulations and several real data studies indicate that the newly proposed method is more effective than the existing data analysis methods.

Construction of Super-Saturated Designs

A design which is having more number of factors than the number of design points is called a Super-saturated Design. In this paper, an attempt is made to propose two new series of constructions of super-saturated designs using mutual orthogonal Latin squares and balanced incomplete block designs and the methods are illustrated with suitable examples.

A New Method for the Analysis of Supersaturated Designs with Discrete Data

Supersaturated designs are factorial designs in which the number of potential effects is greater than the run size. They are commonly used in screening experiments, with the aim of identifying the dominant active factors with low cost. However, an important research field, which is poorly developed, is the analysis of such designs with non-normal response. In this article, we develop a variable selection strategy, through the modification of the PageRank algorithm, which is commonly used in the Google search engine for ranking Webpages. The proposed method incorporates an appropriate information theoretical measure into this algorithm and as a result, it can be efficiently used for factor screening. A noteworthy advantage of this procedure is that it allows the use of supersaturated designs for analyzing discrete data and therefore a generalized linear model is assumed. As it is depicted via a thorough simulation study, in which the Type I and Type II error rates are computed for a wide range of underlying models and designs, the presented approach can be considered quite advantageous and effective.

Use of the common components and specific weights analysis to interpret supersaturated designs

We propose a new method to treat supersaturated designs which relies on the multivariate CCSWA (Common Components and Specific Weights Analysis), or ComDim, method. This method can be applied through two approaches. The first considers this method as a novel procedure to analyse supersaturated designs. This can reveal influential factors, and correctly identify the active factors even when the results do not fully comply with the standard hypotheses (sparsity). The second approach extracts the information common to the results obtained using a wide range of methods to analyse the designs. This could make it possible to overcome the errors specific to each of these methods.

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.

Constrained Estimators and Consistency of a Regression Model on a Lexis Diagram

This article considers a regression model on a Lexis diagram of an a × p table with a single response in each cell following a distribution in the exponential family. A regression model on the fixed effects of a rows, p columns, and a + p − 1 diagonals induces a singular design matrix and yields multiple estimators, leading to parameter identifiability problem in age–period–cohort analysis in social sciences, demography, and epidemiology, where assessment of secular trend in age, period, and birth cohort of social events (e.g., violence) and diseases (e.g., cancer) is of interest. Similar problems also exist in other settings, such as in supersaturated designs. In this article, we study the finite sample properties of the multiple estimators, propose a penalized profile likelihood method to study the consistency and asymptotic bias, and demonstrate the results through simulations and data analysis. As a by-product, the identifiability problem is addressed with consistent estimation for model parameters and secular trend. We conclude that consistent estimation can be identified through estimable function and asymptotics studies in regressions with a singular design. Our method provides a novel approach to studying asymptotics of multiple estimators with a diverging number of nuisance parameters.

Multifaceted Evaluation of 2-Level Supersaturated Designs

Multifaceted Evaluation of 2-Level Supersaturated Designs

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.

Computer-aided unbalanced supersaturated designs involving interactions

Supersaturated designs (SSDs) are defined as fractional factorial designs whose experimental run size is smaller than the number of main effects to be estimated. While most of the literature on SSDs has focused only on main effects designs, the construction and analysis of such designs involving interactions has not been developed to a great extent. In this paper, we propose a backward elimination design-driven optimization (BEDDO) method, with one main goal in mind, to eliminate the factors which are identified to be fully aliased or highly partially aliased with each other in the design. Under the proposed BEDDO method, we implement and combine correlation-based statistical measures taken from classical test theory and design of experiments field, and we also present an optimality criterion which is a modified form of Cronbach's alpha coefficient. In this way, we provide a new class of computer-aided unbalanced SSDs involving interactions, that derive directly from BEDDO optimization.

Searching for Powerful Supersaturated Designs

An important property of any experimental design is its power, defined roughly as its ability to detect active factors. For supersaturated designs, power is even more critical. We consider several popular supersaturated design construction criteria in the literature, propose several of our own, and perform a simulation study to evaluate them in terms of power. We use two analysis methods—forward selection and the Dantzig selector—and find that, although the Dantzig selector clearly outperforms forward selection, there is no clear winner among the design construction criteria.

Optimal Supersaturated Designs

We consider screening experiments where an investigator wishes to study many factors using fewer observations. Our focus is on experiments with two-level factors and a main effects model with intercept. Since the number of parameters is larger than the number of observations, traditional methods of inference and design are unavailable. In 1959, Box suggested the use of supersaturated designs and in 1962, Booth and Cox introduced measures for efficiency of these designs including E(s2), which is the average of squares of the off-diagonal entries of the information matrix, ignoring the intercept. For a design to be E(s2)-optimal, the main effect of every factor must be orthogonal to the intercept (factors are balanced), and among all designs that satisfy this condition, it should minimize E(s2). This is a natural approach since it identifies the most nearly orthogonal design, and orthogonal designs enjoy many desirable properties including efficient parameter estimation. Factor balance in an E(s2)-optimal design has the consequence that the intercept is the most precisely estimated parameter. We introduce and study UE(s2)-optimality, which is essentially the same as E(s2)-optimality, except that we do not insist on factor balance. We also provide a method of construction. We introduce a second criterion from a traditional design optimality theory viewpoint. We use minimization of bias as our estimation criterion, and minimization of the variance of the minimum bias estimator as the design optimality criterion. Using D-optimality as the specific design optimality criterion, we introduce D-optimal supersaturated designs. We show that D-optimal supersaturated designs can be constructed from D-optimal chemical balance weighing designs obtained by Galil and Kiefer (1980, 1982), Cheng (1980) and other authors. It turns out that, except when the number of observations and the number of factors are in a certain range, an UE(s2)-optimal design is also a D-optimal supersaturated design. Moreover, these designs have an interesting connection to Bayes optimal designs. When the prior variance is large enough, a D-optimal supersaturated design is Bayes D-optimal and when the prior variance is small enough, an UE(s2)-optimal design is Bayes D-optimal. While E(s2)-optimal designs yield precise intercept estimates, our study indicates that UE(s2)-optimal designs generally produce more efficient estimates for the main effects of the factors. Based on theoretical properties and the study of examples, we recommend UE(s2)-optimal designs for screening experiments.

Extended mixed-level supersaturated designs

This paper considers the study of the optimality of the extended design generated by adding few runs to an existing E(χ2)-optimal mixed-level supersaturated design. This paper covers the work of Gupta et al. (2010, 2012) on extended two-level and s-level supersaturated designs as two special cases. A lower bound to E(χ2) has been obtained for the proposed designs. A small example is presented here attaining the lower bound.

Screening of factors influencing the extraction of gelatin from the skin of cuttlefish using supersaturated design

Supersaturated design (SSD) was used for screening the key parameters influencing gelatin extraction yield from cuttlefish (Sepia officinalis) skin. Results indicated that among a list of 17 factors only five parameters, namely, alkali (NaOH) concentration, acid reagent (acetic acid), enzyme, thermal treatment temperature and centrifugation time, were factors influencing gelatin yield. The optimal conditions for gelatin extraction were found to be: pretreatment with NaOH 0.03M for 1h; treatment with pepsin for 24h at 4°C in acetic acid 100mM; extraction for 14h at 40°C. The yield of gelatin extraction was 54.6%. Cuttlefish skin gelatin (CSG) contained protein as the major compound (90.95%) and low fat (0.3%) and ash (0.05%) contents. The physico-chemical properties of the CSG were characterized and compared with those of bovine gelatin (BG). The result of textural properties showed that hardness, elasticity and cohesiveness of CSG were lower than those of BG. Further, the gel strength of CSG (192.01g) was lower than that of BG (259.65g), possibly due to lower imino-acid content. The functional properties, including emulsion activity index and foam stability were similar to those of BG. The CSG showed stronger ability of apple juice clarification, than BG without affecting its nutritional values.

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.

A Penalized Wrapper Method for Screening Main Effects and Interactions in Supersaturated Designs

Supersaturated designs (SSDs) are defined as fractional factorial designs whose experimental run size is smaller than the number of main effects to be estimated. The main goal using the class of SSDs is to identify the important effects efficiently, that is, at a minimal computational cost and time. Several methods for analyzing SSDs have been proposed in recent literature. While most of the literature on SSDs has focused on main effects models, the analysis of such designs involving models with interactions has not been developed to a great extent. In this paper, we attempt to relate several penalty and loss functions with support vector machines, with one main goal in mind, screening active effects in SSDs. In this spirit, we propose a penalized wrapper screening method for identifying in one stage the important main effects and two‐factor interactions of two‐level SSDs, by assuming generalized linear models. We also carry out simulation studies and a real data analysis to assess the performance of the proposed screening procedure, showing that the proposed method works satisfactorily. Copyright © 2014 John Wiley & Sons, Ltd.

Supersaturated plans for variable selection in large databases

Over the last decades, the collection and storage of data has become massive with the advance of technology and variable selection has become a fundamental tool to large dimensional statistical modelling problems. In this study we implement data mining techniques, metaheuristics and use experimental designs in databases in order to determine the most relevant variables for classification in regression problems in cases where observations and labels of a large database are available. We propose a database-driven scheme for the encryption of specific fields of a database in order to select an optimal supersaturated design consisting of the variables of a large database which have been found to influence significantly the response outcome. The proposed design selection approach is quite promising, since we are able to retrieve an optimal supersaturated plan using a very small percentage of the available runs, a fact that makes the statistical analysis of a large database computationally feasible and affordable.

Supersaturated plans for variable selection in large databases

Over the last decades, the collection and storage of data has become massive with the advance of technology and variable selection has become a fundamental tool to large dimensional statistical modelling problems. In this study we implement data mining techniques, metaheuristics and use experimental designs in databases in order to determine the most relevant variables for classification in regression problems in cases where observations and labels of a large database are available. We propose a database-driven scheme for the encryption of specific fields of a database in order to select an optimal supersaturated design consisting of the variables of a large database which have been found to influence significantly the response outcome. The proposed design selection approach is quite promising, since we are able to retrieve an optimal supersaturated plan using a very small percentage of the available runs, a fact that makes the statistical analysis of a large database computationally feasible and affordable.

Screening active factors in supersaturated designs

Identification of active factors in supersaturated designs (SSDs) has been the subject of much recent study. Although several methods have been previously proposed, a solution to the problem beyond one or two active factors still seems to be unsatisfactory. The smoothly clipped absolute deviation (SCAD) penalty function for variable selection has nice theoretical properties, but due to its nonconvex nature, it poses computational issues in model fitting. As a result, so far it has not shown much promise for SSDs. Another issue regarding its inefficiency, particularly for SSDs, has been the method used for choosing the SCAD sparsity tuning parameter. The selection of the SCAD sparsity tuning parameter using the AIC and BIC information criteria, generalized cross-validation, and a recently proposed method based on the norm of the error in the solution of systems of linear equations are investigated. This is performed in conjunction with a recently developed more efficient algorithm for implementing the SCAD penalty. The small sample bias-corrected cAIC is found to yield a model size closer to the true model size. Results of the numerical study and real data analyses reveal that the SCAD is a valuable tool for identifying active factors in SSDs.

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.