A Method for Analyzing Supersaturated Designs with a Block Orthogonal Structure

Supersaturated designs is a large class of factorial designs which can be used for screening out the important factors from a large set of potentially active variables. The huge advantage of these designs is that they reduce the experimental cost drastically, but their critical disadvantage is the confounding involved in the statistical analysis. In this article, we propose a method for analyzing data using a specific type of supersaturated designs. This method heavily uses the special block orthogonal structure of the supersaturated designs given by Tang and Wu (1997 Tang , B. , Wu , C. F. J. ( 1997 ). A method for constructing supersaturated designs and its Es ²-optimality . Canadian J. Statist. 25 : 191 – 201 .[Crossref], [Web of Science ®] , [Google Scholar]). Also, we compare our method with several known statistical analysis methods by using some of the existing supersaturated designs. The comparison is performed by some simulating experiments and the Type I and Type II error rates are calculated. The results are presented in tables and the discussion to follow.     

The effects of nonnormality on the analysis of supersaturated designs: a comparison of stepwise,SCAD and permutation test methods

Supersaturated designs (SSDs) are useful in examining many factors with a restricted number of experimental units. Many analysis methods have been proposed to analyse data from SSDs, with some methods performing better than others when data are normally distributed. It is possible that data sets violate assumptions of standard analysis methods used to analyse data from SSDs, and to date the performance of these analysis methods have not been evaluated using nonnormally distributed data sets. We conducted a simulation study with normally and nonnormally distributed data sets to compare the identification rates, power and coverage of the true models using a permutation test, the stepwise procedure and the smoothly clipped absolute deviation (SCAD) method. Results showed that at the level of significance α=0.01, the identification rates of the true models of the three methods were comparable; however at α=0.05, both the permutation test and stepwise procedures had considerably lower identification rates than SCAD. For most cases, the three methods produced high power and coverage. The experimentwise error rates (EER) were close to the nominal level (11.36%) for the stepwise method, while they were somewhat higher for the permutation test. The EER for the SCAD method were extremely high (84–87%) for the normal and t-distributions, as well as for data with outlier.     

A three-stage variable selection method for supersaturated designs

A 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.     

A method for constructing supersaturated designs and its Es2 optimality

A lower bound for the Es² value of an arbitrary supersaturated design is derived. A general method for constructing supersaturated designs is proposed and shown to produce designs with n runs and m = k(n — 1) factors that achieve the lower bound for Es² and are thus optimal with respect to the Es² criterion. Within the class of designs given by the construction method, further discrimination can be made by minimizing the pairwise correlations and using the generalized D and A criteria proposed by Wu (1993). Efficient designs of 12, 16, 20 and 24 runs are constructed by following this approach.     

A new variable selection method for uniform designs

As an important class of space-filling designs, uniform designs (UDs) choose a set of points over a certain domain such that these points are uniformly scattered, under a specific discrepancy measure. They have been applied successfully in many industrial and scientific experiments since they appeared in 1980. A noteworthy and practical advantage is their ability to investigate a large number of high-level factors simultaneously with a fairly economical set of experimental runs. As a result, UDs can be properly used as experimental plans that are intended to derive the significant factors from a list of many potential ones. To this end, a new screening procedure is introduced via penalized least squares. A simulation study is conducted to support the proposed method, which reveals that it can be considered quite promising and expedient, as judged in terms of Type I and Type II error rates.     

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

Supersaturated 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.     

A method for screening active effects in supersaturated designs

A supersaturated design (SSD) is a design whose run size is not enough for estimating all the main effects. The goal in conducting such a design is to identify, presumably only a few, relatively dominant active effects with a cost as low as possible. However, data analysis of such designs remains primitive: traditional approaches are not appropriate in such a situation and several methods which were proposed in the literature in recent years are effective when used to analyze two-level SSDs. In this paper, we introduce a variable selection procedure, called the PLSVS method, to screen active effects in mixed-level SSDs based on the variable importance in projection which is an important concept in the partial least-squares regression. Simulation studies show that this procedure is effective.     

The performance of multistage group screening designs

This article deals with multistage group screening in which group-factors contain the same number of factors. A usual assumption of this procedure is that the directions of possible effects are known. In practice, however, this assumption i s often unreasonable. This paper examines, in the case of no errors in observations, the performance of multistage group screening when this assumption is false . This enails consideration of cancellation effects within group-factors.     

A general construction of E(fNOD)-optimal multi-level supersaturated designs

The present paper deals with E(f_NOD)-optimal multi-level supersaturated designs. We present a new technique for the construction of supplementary difference sets. Based on the new supplementary difference sets, we also provide E(f_NOD)-optimal multi-level supersaturated designs with a large number of columns when compared with other designs. Moreover, these designs retain the equal occurrence property.     

A method for analyzing supersaturated designs inspired by control charts

The identification of active effects in supersaturated designs (SSDs) constitutes a problem of considerable interest to both scientists and engineers. The complicated structure of the design matrix renders the analysis of such designs a complicated issue. Although several methods have been proposed so far, a solution to the problem beyond one or two active factors seems to be inadequate. This article presents a heuristic approach for analyzing SSDs using the cumulative sum control chart (CUSUM) under a sure independence screening approach. Simulations are used to investigate the performance of the method comparing the proposed method with other well-known methods from the literature. The results establish the powerfulness of the proposed methodology.     

A lower bound based smoothed quasi-Newton algorithm for group bridge penalized regression

In this paper, we propose a lower bound based smoothed quasi-Newton algorithm for computing the solution paths of the group bridge estimator in linear regression models. Our method is based on the quasi-Newton algorithm with a smoothed group bridge penalty in combination with a novel data-driven thresholding rule for the regression coefficients. This rule is derived based on a necessary KKT condition of the group bridge optimization problem. It is easy to implement and can be used to eliminate groups with zero coefficients. Thus, it reduces the dimension of the optimization problem. The proposed algorithm removes the restriction of groupwise orthogonal condition needed in coordinate descent and LARS algorithms for group variable selection. Numerical results show that the proposed algorithm outperforms the coordinate descent based algorithms in both efficiency and accuracy.     

An optimality criterion for supersaturated designs with quantitative factors

A supersaturated design (SSD) is a factorial design in which the degrees of freedom for all its main effects exceed the total number of distinct factorial level-combinations (runs) of the design. Designs with quantitative factors, in which level permutation within one or more factors could result in different geometrical structures, are very different from designs with nominal ones which have been treated as traditional designs. In this paper, a new criterion is proposed for SSDs with quantitative factors. Comparison and analysis for this new criterion are made. It is shown that the proposed criterion has a high efficiency in discriminating geometrically nonisomorphic designs and an advantage in computation.     

Optimal designs for statistical analysis with Zernike polynomials

The Zernike polynomials arise in several applications such as optical metrology or image analysis on a circular domain. In the present paper, we determine optimal designs for regression models which are represented by expansions in terms of Zernike polynomials. We consider two estimation methods for the coefficients in these models and determine the corresponding optimal designs. The first one is the classical least squares method and Φ _p -optimal designs in the sense of Kiefer [Kiefer, J., 1974, General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2 849–879.] are derived, which minimize an appropriate functional of the covariance matrix of the least squares estimator. It is demonstrated that optimal designs with respect to Kiefer's Φ _p -criteria (p>?∞) are essentially unique and concentrate observations on certain circles in the experimental domain. E-optimal designs have the same structure but it is shown in several examples that these optimal designs are not necessarily uniquely determined. The second method is based on the direct estimation of the Fourier coefficients in the expansion of the expected response in terms of Zernike polynomials and optimal designs minimizing the trace of the covariance matrix of the corresponding estimator are determined. The designs are also compared with the uniform designs on a grid, which is commonly used in this context.     

Sensitivity analysis and related analyses: A review of some statistical techniques

This paper reviews five related types of analysis, namely (i) sensitivity or what-if analysis, (ii) uncertainty or risk analysis, (iii) screening, (iv) validation, and (v) optimization. The main questions are: when should which type of analysis be applied; which statistical techniques may then be used? This paper claims that the proper sequence to follow in the evaluation of simulation models is as follows. 1) Validation, in which the availability of data on the real system determines which type of statistical technique to use for validation. 2) Screening: in the simulation‘s pilot phase the really important inputs can be identified through a novel technique, called sequential bifurcation, which uses aggregation and sequential experimentation. 3) Sensitivity analysis: the really important inputs should be subjected to a more detailed analysis, which includes interactions between these inputs; relevant statistical techniques are design of experiments (DOE) and regression analysis. 4) Uncertainty analysis: the important environmental inputs may have values that are not precisely known, so the uncertainties of the model outputs that result from the uncertainties in these model inputs should be quantified; relevant techniques are the Monte Carlo method and Latin hypercube sampling. 5) Optimization: the policy variables should be controlled; a relevant technique is Response Surface Methodology (RSM), which combines DOE, regression analysis, and steepest-ascent hill-climbing. The recommended sequence implies that sensitivity analysis procede uncertainty analysis. Several case studies for each phase are briefly discussed in this paper.     

A general method of constructing E(s2)-optimal supersaturated designs

There has been much recent interest in supersaturated designs and their application in factor screening experiments. Supersaturated designs have mainly been constructed by using the E ( s ²)-optimality criterion originally proposed by Booth and Cox in 1962. However, until now E ( s ²)-optimal designs have only been established with certainty for n experimental runs when the number of factors m is a multiple of n-1 , and in adjacent cases where m = q ( n -1) + r (| r | 2, q an integer). A method of constructing E ( s ²)-optimal designs is presented which allows a reasonably complete solution to be found for various numbers of runs n including n ,=8 12, 16, 20, 24, 32, 40, 48, 64.     

Restricted minimax robust designs for misspecified regression models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression.     

Optimal selection and ordering of columns in supersaturated designs

Two methods to select columns for assigning factors to work on supersaturated designs are proposed. The focus of interest is the degree of non-orthogonality between the selected columns. One method is the exhaustive enumeration of selections of p columns from all k columns to find the exact optimality, while the other is intended to find an approximate solution by applying techniques used in the corresponding analysis, aiming for ease of use as well as a reduction in the large computing time required for large k with the first method. Numerical illustrations for several typical design matrices reveal that the resulting “approximately” optimal assignments of factors to their columns are exactly optimal for any p. Ordering the columns in E(s²)-optimal designs results in promising new findings including a large number of E(s²)-optimal designs.     

Parametric scalp mapping and inference via spatially smooth linear models for mismatch negativity studies

Mismatch negativity (MMN) is a neurophysiological tool that can be used to investigate various facets of comprehension. Subjects are presented with different stimuli to elicit the MMN response, which is derived from electroencephalography (EEG) signals recorded at electrodes across the brain. We propose a methodology to extend single electrode analyses of MMN data by generating smooth scalp maps of estimated experimental effects. It is shown that penalized least squares estimates of effect maps can be produced using a two step procedure involving (a) ANOVA at each electrode and (b) spatial smoothing across electrodes. A Fisher von-Mises kernel is used for smoothing scalp maps with cross-validated bandwidth selection. The methodology is applied to a case control study involving aphasics (language disordered individuals). Analysis of residuals shows possible heteroscedasticity and non-Gaussian tail behavior. For robust inference, a semiparametric multivariate approach is proposed to determine the significance of parametric maps. A variety of global and regional test statistics are developed to investigate the significance of spatial patterns in treatment effects. The methodology is seen to confirm previous findings from single electrode analysis and identifies some new significant spatial patterns of difference between controls and aphasics.     

Minimax A-, c-, and I-optimal regression designs for models with heteroscedastic errors

It is well known that it is difficult to construct minimax optimal designs. Furthermore, since in practice we never know the true error variance, it is important to allow small deviations and construct robust optimal designs. We investigate a class of minimax optimal regression designs for models with heteroscedastic errors that are robust against possible misspecification of the error variance. Commonly used A-, c-, and I-optimality criteria are included in this class of minimax optimal designs. Several theoretical results are obtained, including a necessary condition and a reflection symmetry for these minimax optimal designs. In this article, we focus mainly on linear models and assume that an approximate error variance function is available. However, we also briefly discuss how the methodology works for nonlinear models. We then propose an effective algorithm to solve challenging nonconvex optimization problems to find minimax designs on discrete design spaces. Examples are given to illustrate minimax optimal designs and their properties.