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.     

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 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.     

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.     

Variable selection for semiparametric varying coefficient partially linear model based on modal regression with missing data

Abstract

In this article, we focus on the variable selection for semiparametric varying coefficient partially linear model with response missing at random. Variable selection is proposed based on modal regression, where the non parametric functions are approximated by B-spline basis. The proposed procedure uses SCAD penalty to realize variable selection of parametric and nonparametric components simultaneously. Furthermore, we establish the consistency, the sparse property and asymptotic normality of the resulting estimators. The penalty estimation parameters value of the proposed method is calculated by EM algorithm. Simulation studies are carried out to assess the finite sample performance of the proposed variable selection procedure.     

A Variable Selection Method for Analyzing Supersaturated Designs

Supersaturated designs are 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 several types of supersaturated designs. Modifications of widely used information criteria are given and applied to the variable selection procedure for the identification of the active factors. The effectiveness of the proposed method is depicted via simulated experiments and comparisons.     

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.     

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.     

Random effects selection in generalized linear mixed models via shrinkage penalty function

In this paper, we discuss the selection of random effects within the framework of generalized linear mixed models (GLMMs). Based on a reparametrization of the covariance matrix of random effects in terms of modified Cholesky decomposition, we propose to add a shrinkage penalty term to the penalized quasi-likelihood (PQL) function of the variance components for selecting effective random effects. The shrinkage penalty term is taken as a function of the variance of random effects, initiated by the fact that if the variance is zero then the corresponding variable is no longer random (with probability one). The proposed method takes the advantage of a convenient computation for the PQL estimation and appealing properties for certain shrinkage penalty functions such as LASSO and SCAD. We propose to use a backfitting algorithm to estimate the fixed effects and variance components in GLMMs, which also selects effective random effects simultaneously. Simulation studies show that the proposed approach performs quite well in selecting effective random effects in GLMMs. Real data analysis is made using the proposed approach, too.     

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.     

Projection Properties of Certain Three-Level Main Effect Plans with Quantitative Factors

ABSTRACT

Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a “projection design”. In this article, we have identified all the geometric non isomorphic projection designs of an OA(27,13,3,2), an OA(18,7,3,2) and an OA(36,13,3,2) into k = 3,4, and 5 factors when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D-efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model.     

Analysis of a supersaturated design using Entropy Prior Complexity for binary responses via generalized linear models

A supersaturated design is a factorial design in which the number of effects to be estimated is greater than the available number of experimental runs. It is used in many experiments for screening purposes, i.e., for studying a large number of factors and then identifying the active ones. The goal with such a design is to identify just a few of the factors under consideration, that have dominant effects and to do this at minimum cost. While most of the literature on supersaturated designs has focused on the construction of designs and their optimality, the data analysis of such designs remains still at an early stage. In this paper, we incorporate the parameter model complexity into the supersaturated design analysis process, by assuming generalized linear models for a Bernoulli response, for analyzing main effects designs and discovering simultaneously the effects that are significant.     

Factor screening in nonregular two-level designs based on projection-based variable selection

In this paper, we focus on the problem of factor screening in nonregular two-level designs through gradually reducing the number of possible sets of active factors. We are particularly concerned with situations when three or four factors are active. Our proposed method works through examining fits of projection models, where variable selection techniques are used to reduce the number of terms. To examine the reliability of the methods in combination with such techniques, a panel of models consisting of three or four active factors with data generated from the 12-run and the 20-run Plackett–Burman (PB) design is used. The dependence of the procedure on the amount of noise, the number of active factors and the number of experimental factors is also investigated. For designs with few runs such as the 12-run PB design, variable selection should be done with care and default procedures in computer software may not be reliable to which we suggest improvements. A real example is included to show how we propose factor screening can be done in practice.     

正则化Beta回归及其应用

随着计算机的飞速发展,极大地便利了数据的获取和存储,很多企业积累了大量的数据,同时数据的维度也越来越高,噪声变量越来越多,因此在建模分析时面临的重要问题之一就是从高维的变量中筛选出少数的重要变量。针对因变量取值为(0,1)区间的比例数据提出了正则化Beta回归,研究了在LASSO、SCAD和MCP三种惩罚方法下的极大似然估计及其渐进性质。统计模拟表明MCP的方法会优于SCAD和LASSO,并且随着样本量的增大,SCAD的方法也将优于LASSO。最后,将该方法应用到中国上市公司股息率的影响因素研究中。     

Constructing optimal projection designs

ABSTRACT

The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the experimenters do not know which factors turn out to be important before conducting the experiment. Therefore, designs with good projections are desirable for factor screening. In this regard, significant questions are arising such as whether the optimal full-dimensional designs have good projections onto low dimensions? How experimenters can measure the goodness of a full-dimensional design by focusing on all of its projections?, and are there linkages between the optimality of a full-dimensional design and the optimality of its projections? Through theoretical justifications, this paper tries to provide answers to these interesting questions by investigating the construction of optimal (average) projection designs for screening either nominal or quantitative factors. The main results show that: based on the aberration and orthogonality criteria the full-dimensional design is optimal if and only if it is optimal projection design; the full-dimensional design is optimal via the aberration and orthogonality if and only if it is uniform projection design; there is no guarantee that a uniform full-dimensional design is optimal projection design via any criterion; the projection design is optimal via the aberration, orthogonality and uniformity criteria if it is optimal via any criterion of them; and the saturated orthogonal designs have the same average projection performance.     

Block thresholding wavelet regression using SCAD penalty

This paper concerns wavelet regression using a block thresholding procedure. Block thresholding methods utilize neighboring wavelet coefficients information to increase estimation accuracy. We propose to construct a data-driven block thresholding procedure using the smoothly clipped absolute deviation (SCAD) penalty. A simulation study demonstrates competitive finite sample performance of the proposed estimator compared to existing methods. We also show that the proposed estimator achieves optimal convergence rates in Besov spaces.     

A unified class of penalties with the capability of producing a differentiable alternative to l1 norm penalty

Abstract

This article presents a class of novel penalties that are defined under a unified framework, which includes lasso, SCAD and ridge as special cases, and novel functions, such as the asymmetric quantile check function. The proposed class of penalties is capable of producing alternative differentiable penalties to lasso. We mainly focus on this case and show its desirable properties, propose an efficient algorithm for the parameter estimation and prove the theoretical properties of the resulting estimators. Moreover, we exploit the differentiability of the penalty function by deriving a novel Generalized Information Criterion (GIC) for model selection. The method is implemented in the R package DLASSO freely available from CRAN, http://CRAN.R-project.org/package=DLASSO.     

Robust estimation and variable selection for varying-coefficient single-index models based on modal regression

ABSTRACT

In this paper, we propose a new efficient and robust penalized estimating procedure for varying-coefficient single-index models based on modal regression and basis function approximations. The proposed procedure simultaneously solves two types of problems: separation of varying and constant effects and selection of variables with non zero coefficients for both non parametric and index components using three smoothly clipped absolute deviation (SCAD) penalties. With appropriate selection of the tuning parameters, the new method possesses the consistency in variable selection and the separation of varying and constant coefficients. In addition, the estimators of varying coefficients possess the optimal convergence rate and the estimators of constant coefficients and index parameters have the oracle property. Finally, we investigate the finite sample performance of the proposed method through a simulation study and real data analysis.     

Quadratic Approximation via the SCAD Penalty with a Diverging Number of Parameters

The high-dimensional data arises in diverse fields of sciences, engineering and humanities. Variable selection plays an important role in dealing with high dimensional statistical modelling. In this article, we study the variable selection of quadratic approximation via the smoothly clipped absolute deviation (SCAD) penalty with a diverging number of parameters. We provide a unified method to select variables and estimate parameters for various of high dimensional models. Under appropriate conditions and with a proper regularization parameter, we show that the estimator has consistency and sparsity, and the estimators of nonzero coefficients enjoy the asymptotic normality as they would have if the zero coefficients were known in advance. In addition, under some mild conditions, we can obtain the global solution of the penalized objective function with the SCAD penalty. Numerical studies and a real data analysis are carried out to confirm the performance of the proposed method.     

Paths Following Algorithm for Penalized Logistic Regression Using SCAD and MCP

In this article, we develop a generalized penalized linear unbiased selection (GPLUS) algorithm. The GPLUS is designed to compute the paths of penalized logistic regression based on the smoothly clipped absolute deviation (SCAD) and the minimax concave penalties (MCP). The main idea of the GPLUS is to compute possibly multiple local minimizers at individual penalty levels by continuously tracing the minimizers at different penalty levels. We demonstrate the feasibility of the proposed algorithm in logistic and linear regression. The simulation results favor the SCAD and MCP’s selection accuracy encompassing a suitable range of penalty levels.