A study on discriminant analysis techniques applied to multivariate lognormal data

The purpose of this paper is to examine the multiple group (>2) discrimination problem in which the group sizes are unequal and the variables used in the classification are correlated with skewed distributions. Using statistical simulation based on data from a clinical study, we compare the performances, in terms of misclassification rates, of nine statistical discrimination methods. These methods are linear and quadratic discriminant analysis applied to untransformed data, rank transformed data, and inverse normal scores data, as well as fixed kernel discriminant analysis, variable kernel discriminant analysis, and variable kernel discriminant analysis applied to inverse normal scores data. It is found that the parametric methods with transformed data generally outperform the other methods, and the parametric methods applied to inverse normal scores usually outperform the parametric methods applied to rank transformed data. Although the kernel methods often have very biased estimates, the variable kernel method applied to inverse normal scores data provides considerable improvement in terms of total nonerror rate.     

ON SELECTING VARIABLES AND ASSESSING THEIR PERFORMANCE IN LINEAR DISCRIMINANT ANALYSIS

Linear discriminant analysis between two populations is considered in this paper. Error rate is reviewed as a criterion for selection of variables, and a stepwise procedure is outlined that selects variables on the basis of empirical estimates of error. Problems with assessment of the selected variables are highlighted. A leave-one-out method is proposed for estimating the true error rate of the selected variables, or alternatively of the selection procedure itself. Monte Carlo simulations, of multivariate binary as well as multivariate normal data, demonstrate the feasibility of the proposed method and indicate its much greater accuracy relative to that of other available methods.     

A comparison of the post selection error rate behaviour of the normal linear and quadratic discriminant rules

The normal linear discriminant rule (NLDR) and the normal quadratic discriminant rule (NQDR) are popular classifiers when working with normal populations. Several papers in the literature have been devoted to a comparison of these rules with respect to classification performance. An aspect which has, however, not received any attention is the effect of an initial variable selection step on the relative performance of these classification rules. Cross model validation variabie selection has been found to perform well in the linear case, and can be extended to the quadratic case. We report the results of a simulation study comparing the NLDR and the NQDR with respect to the post variable selection classification performance. It is of interest that the NQDR generally benefits from an initial variable selection step. We also comment briefly on the problem of estimating the post selection error rates of the two rules.     

Multistage sampling for latent variable models

I consider the design of multistage sampling schemes for epidemiologic studies involving latent variable models, with surrogate measurements of the latent variables on a subset of subjects. Such models arise in various situations: when detailed exposure measurements are combined with variables that can be used to assign exposures to unmeasured subjects; when biomarkers are obtained to assess an unobserved pathophysiologic process; or when additional information is to be obtained on confounding or modifying variables. In such situations, it may be possible to stratify the subsample on data available for all subjects in the main study, such as outcomes, exposure predictors, or geographic locations. Three circumstances where analytic calculations of the optimal design are possible are considered: (i) when all variables are binary; (ii) when all are normally distributed; and (iii) when the latent variable and its measurement are normally distributed, but the outcome is binary. In each of these cases, it is often possible to considerably improve the cost efficiency of the design by appropriate selection of the sampling fractions. More complex situations arise when the data are spatially distributed: the spatial correlation can be exploited to improve exposure assignment for unmeasured locations using available measurements on neighboring locations; some approaches for informative selection of the measurement sample using location and/or exposure predictor data are considered.     

Bandwidth selection procedures tor kernel density estimates

A crucial problem in kernel density estimates of a probability density function is the selection of the bandwidth. The aim of this study is to propose a procedure for selecting both fixed and variable bandwidths. The present study also addresses the question of how different variable bandwidth kernel estimators perform in comparison with each other and to the fixed type of bandwidth estimators. The appropriate algorithms for implementation of the proposed method are given along with a numerical simulation.The numerical results serve as a guide to determine which bandwidth selection method is most appropriate for a given type of estimator over a vide class of probability density functions, Also, we obtain a numerical comparison of the different types of kernel estimators under various types of bandwidths.     

Efficient Bandwidth Selection in Non-parametric Regression

In this paper we use non-parametric local polynomial methods to estimate the regression function, m ( x ). Y may be a binary or continuous response variable, and X is continuous with non-uniform density. The main contributions of this paper are the weak convergence of a bandwidth process for kernels of order (0, k ), k =2 ^j , j ≥1 and the proposal of a local data-driven bandwidth selection method which is particularly beneficial for the case when X is not distributed uniformly. This selection method minimizes estimates of the asymptotic MSE and estimates the bias portion in an innovative way which relies on the order of the kernel and not estimation of m ²( x ) directly. We show that utilization of this method results in the achievement of the optimal asymptotic MSE by the estimator, i.e. the method is efficient. Simulation studies are provided which illustrate the method for both binary and continuous response cases.     

Kernel naive Bayes discrimination for high‐dimensional pattern recognition

Kernel discriminant analysis translates the original classification problem into feature space and solves the problem with dimension and sample size interchanged. In high‐dimension low sample size (HDLSS) settings, this reduces the ‘dimension’ to that of the sample size. For HDLSS two‐class problems we modify Mika's kernel Fisher discriminant function which – in general – remains ill‐posed even in a kernel setting; see Mika et al. (1999). We propose a kernel naive Bayes discriminant function and its smoothed version, using first‐ and second‐degree polynomial kernels. For fixed sample size and increasing dimension, we present asymptotic expressions for the kernel discriminant functions, discriminant directions and for the error probability of our kernel discriminant functions. The theoretical calculations are complemented by simulations which show the convergence of the estimators to the population quantities as the dimension grows. We illustrate the performance of the new discriminant rules, which are easy to implement, on real HDLSS data. For such data, our results clearly demonstrate the superior performance of the new discriminant rules, and especially their smoothed versions, over Mika's kernel Fisher version, and typically also over the commonly used naive Bayes discriminant rule.     

ESTIMATION FOR THE GENERAL SAMPLE SELECTION MODELS

Consider a general regression model with an arbitrary and unknown link function and a stochastic selection variable that determines whether the outcome variable is observable or missing. The paper proposes U-statistics that are based on kernel functions as estimators for the directions of the parameter vectors in the link function and the selection equation, and shows that these estimators are consistent and asymptotically normal.     

Finite Mixture of Generalized Semiparametric Models: Variable Selection via Penalized Estimation

Selection of the important variables is one of the most important model selection problems in statistical applications. In this article, we address variable selection in finite mixture of generalized semiparametric models. To overcome computational burden, we introduce a class of variable selection procedures for finite mixture of generalized semiparametric models using penalized approach for variable selection. Estimation of nonparametric component will be done via multivariate kernel regression. It is shown that the new method is consistent for variable selection and the performance of proposed method will be assessed via simulation.     

Variational discriminant analysis with variable selection

A fast Bayesian method that seamlessly fuses classification and hypothesis testing via discriminant analysis is developed. Building upon the original discriminant analysis classifier, modelling components are added to identify discriminative variables. A combination of cake priors and a novel form of variational Bayes we call reverse collapsed variational Bayes gives rise to variable selection that can be directly posed as a multiple hypothesis testing approach using likelihood ratio statistics. Some theoretical arguments are presented showing that Chernoff-consistency (asymptotically zero type I and type II error) is maintained across all hypotheses. We apply our method on some publicly available genomics datasets and show that our method performs well in practice for its computational cost. An R package VaDA has also been made available on Github.     

Functional Discriminant Coordinates

In this article we propose a new method of construction of discriminant coordinates and their kernel variant based on the regularization (ridge regression). Moreover, we compare the case of discriminant coordinates, functional discriminant coordinates and the kernel version of functional discriminant coordinates on 20 data sets from a wide variety of application domains using values of the criterion of goodness and statistical tests. Our experiments show that the kernel variant of discriminant coordinates provides significantly more accurate results on the examined data sets.     

Robust variable selection for the varying coefficient model based on composite L 1–L 2 regression

The varying coefficient model (VCM) is an important generalization of the linear regression model and many existing estimation procedures for VCM were built on L ₂ loss, which is popular for its mathematical beauty but is not robust to non-normal errors and outliers. In this paper, we address the problem of both robustness and efficiency of estimation and variable selection procedure based on the convex combined loss of L ₁ and L ₂ instead of only quadratic loss for VCM. By using local linear modeling method, the asymptotic normality of estimation is driven and a useful selection method is proposed for the weight of composite L ₁ and L ₂. Then the variable selection procedure is given by combining local kernel smoothing with adaptive group LASSO. With appropriate selection of tuning parameters by Bayesian information criterion (BIC) the theoretical properties of the new procedure, including consistency in variable selection and the oracle property in estimation, are established. The finite sample performance of the new method is investigated through simulation studies and the analysis of body fat data. Numerical studies show that the new method is better than or at least as well as the least square-based method in terms of both robustness and efficiency for variable selection.     

Variable Selection and Updating In Model-Based Discriminant Analysis for High Dimensional Data with Food Authenticity Applications

Food authenticity studies are concerned with determining if food samples have been correctly labelled or not. Discriminant analysis methods are an integral part of the methodology for food authentication. Motivated by food authenticity applications, a model-based discriminant analysis method that includes variable selection is presented. The discriminant analysis model is fitted in a semi-supervised manner using both labeled and unlabeled data. The method is shown to give excellent classification performance on several high-dimensional multiclass food authenticity datasets with more variables than observations. The variables selected by the proposed method provide information about which variables are meaningful for classification purposes. A headlong search strategy for variable selection is shown to be efficient in terms of computation and achieves excellent classification performance. In applications to several food authenticity datasets, our proposed method outperformed default implementations of Random Forests, AdaBoost, transductive SVMs and Bayesian Multinomial Regression by substantial margins.     

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299–309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497–516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.     

A hybrid bandwidth selection methodology for kernel density estimation

A bandwidth selection method that combines the concept of least-squares cross-validation and the plug-in approach is being introduced in connection with kernel density estimation. A simulation study reveals that this hybrid methodology outperforms some commonly used bandwidth selection rules. It is shown that the proposed approach can also be readily employed in the context of variable kernel density estimation. We conclude with two illustrative examples.     

Estimation and variable selection for generalised partially linear single-index models

In this paper, we study the problem of estimation and variable selection for generalised partially linear single-index models based on quasi-likelihood, extending existing studies on variable selection for partially linear single-index models to binary and count responses. To take into account the unit norm constraint of the index parameter, we use the ‘delete-one-component’ approach. The asymptotic normality of the estimates is demonstrated. Furthermore, the smoothly clipped absolute deviation penalty is added for variable selection of parameters both in the nonparametric part and the parametric part, and the oracle property of the variable selection procedure is shown. Finally, some simulation studies are carried out to illustrate the finite sample performance.     

A pragmatic dose‐finding approach using short‐term surrogate efficacy outcomes to evaluate binary efficacy and toxicity outcomes in phase I cancer clinical trials

There is a growing need for study designs that can evaluate efficacy and toxicity outcomes simultaneously in phase I or phase I/II cancer clinical trials. Many dose‐finding approaches have been proposed; however, most of these approaches assume binary efficacy and toxicity outcomes, such as dose‐limiting toxicity (DLT), and objective responses. DLTs are often defined for short time periods. In contrast, objective responses are often defined for longer periods because of practical limitations on confirmation and the criteria used to define ‘confirmation’. This means that studies have to be carried out for unacceptably long periods of time. Previous studies have not proposed a satisfactory solution to this specific problem. Furthermore, this problem may be a barrier for practitioners who want to implement notable previous dose‐finding approaches. To cope with this problem, we propose an approach using unconfirmed early responses as the surrogate efficacy outcome for the confirmed outcome. Because it is reasonable to expect moderate positive correlation between the two outcomes and the method replaces the surrogate outcome with the confirmed outcome once it becomes available, the proposed approach can reduce irrelevant dose selection and accumulation of bias. Moreover, it is also expected that it can significantly shorten study duration. Using simulation studies, we demonstrate the positive utility of the proposed approach and provide three variations of it, all of which can be easily implemented with modified likelihood functions and outcome variable definitions. Copyright © 2013 John Wiley & Sons, Ltd.     

Variable selection for mode regression

From the prediction viewpoint, mode regression is more attractive since it pay attention to the most probable value of response variable given regressors. On the other hand, high-dimensional data are very prevalent as the advance of the technology of collecting and storing data. Variable selection is an important strategy to deal with high-dimensional regression problem. This paper aims to propose a variable selection procedure for high-dimensional mode regression via combining nonparametric kernel estimation method with sparsity penalty tactics. We also establish the asymptotic properties under certain technical conditions. The effectiveness and flexibility of the proposed methods are further illustrated by numerical studies and the real data application.     

High dimensional variable selection with clustered data: an application of random multivariate survival forests for detection of outlier medical device components

In many medical studies patients are nested or clustered within doctor. With many explanatory variables, variable selection with clustered data can be challenging. We propose a method for variable selection based on random forest that addresses clustered data through stratified binary splits. Our motivating example involves the detection orthopedic device components from a large pool of candidates, where each patient belongs to a surgeon. Simulations compare the performance of survival forests grown using the stratified logrank statistic to conventional and robust logrank statistics, as well as a method to select variables using a threshold value based on a variable's empirical null distribution. The stratified logrank test performs superior to conventional and robust methods when data are generated to have cluster-specific effects, and when cluster sizes are sufficiently large, perform comparably to the splitting alternatives in the absence of cluster-specific effects. Thresholding was effective at distinguishing between important and unimportant variables.     

Logistic Regression and Discriminant Analysis by Ordinary Least Squares

If the observations for fitting a polytomous logistic regression model satisfy certain normality assumptions, the maximum likelihood estimates of the regression coefficients are the discriminant function estimates. This article shows that these estimates, their unbiased counterparts, and associated test statistics for variable selection can be calculated using ordinary least squares regression techniques, thereby providing a convenient method for fitting logistic regression models in the normal case. Evidence is given indicating that the discriminant function estimates and test statistics merit wider use in nonnormal cases, especially in exploratory work on large data sets.