Sparse partial least squares regression for simultaneous dimension reduction and variable selection

Summary.  Partial least squares regression has been an alternative to ordinary least squares for handling multicollinearity in several areas of scientific research since the 1960s. It has recently gained much attention in the analysis of high dimensional genomic data. We show that known asymptotic consistency of the partial least squares estimator for a univariate response does not hold with the very large p and small n paradigm. We derive a similar result for a multivariate response regression with partial least squares. We then propose a sparse partial least squares formulation which aims simultaneously to achieve good predictive performance and variable selection by producing sparse linear combinations of the original predictors. We provide an efficient implementation of sparse partial least squares regression and compare it with well-known variable selection and dimension reduction approaches via simulation experiments. We illustrate the practical utility of sparse partial least squares regression in a joint analysis of gene expression and genomewide binding data.     

On distribution-weighted partial least squares with diverging number of highly correlated predictors

Summary.  Because highly correlated data arise from many scientific fields, we investigate parameter estimation in a semiparametric regression model with diverging number of predictors that are highly correlated. For this, we first develop a distribution-weighted least squares estimator that can recover directions in the central subspace, then use the distribution-weighted least squares estimator as a seed vector and project it onto a Krylov space by partial least squares to avoid computing the inverse of the covariance of predictors. Thus, distrbution-weighted partial least squares can handle the cases with high dimensional and highly correlated predictors. Furthermore, we also suggest an iterative algorithm for obtaining a better initial value before implementing partial least squares. For theoretical investigation, we obtain strong consistency and asymptotic normality when the dimension p of predictors is of convergence rate O { n ^1/2/ log ( n )} and o ( n ^1/3) respectively where n is the sample size. When there are no other constraints on the covariance of predictors, the rates n ^1/2 and n ^1/3 are optimal. We also propose a Bayesian information criterion type of criterion to estimate the dimension of the Krylov space in the partial least squares procedure. Illustrative examples with a real data set and comprehensive simulations demonstrate that the method is robust to non-ellipticity and works well even in 'small n –large p ' problems.     

Partial least squares Cox regression for genome-wide data

Most methods for survival prediction from high-dimensional genomic data combine the Cox proportional hazards model with some technique of dimension reduction, such as partial least squares regression (PLS). Applying PLS to the Cox model is not entirely straightforward, and multiple approaches have been proposed. The method of Park et al. (Bioinformatics 18(Suppl. 1):S120–S127, 2002) uses a reformulation of the Cox likelihood to a Poisson type likelihood, thereby enabling estimation by iteratively reweighted partial least squares for generalized linear models. We propose a modification of the method of park et al. (2002) such that estimates of the baseline hazard and the gene effects are obtained in separate steps. The resulting method has several advantages over the method of park et al. (2002) and other existing Cox PLS approaches, as it allows for estimation of survival probabilities for new patients, enables a less memory-demanding estimation procedure, and allows for incorporation of lower-dimensional non-genomic variables like disease grade and tumor thickness. We also propose to combine our Cox PLS method with an initial gene selection step in which genes are ordered by their Cox score and only the highest-ranking k% of the genes are retained, obtaining a so-called supervised partial least squares regression method. In simulations, both the unsupervised and the supervised version outperform other Cox PLS methods.     

Dimensionality reduction approach to multivariate prediction

The authors consider dimensionality reduction methods used for prediction, such as reduced rank regression, principal component regression and partial least squares. They show how it is possible to obtain intermediate solutions by estimating simultaneously the latent variables for the predictors and for the responses. They obtain a continuum of solutions that goes from reduced rank regression to principal component regression via maximum likelihood and least squares estimation. Different solutions are compared using simulated and real data.     

Prediction of response values in linear regression models from replicated experiments

This paper considers the problem of prediction in a linear regression model when data sets are available from replicated experiments. Pooling the data sets for the estimation of regression parameters, we present three predictors — one arising from the least squares method and two stemming from Stein-rule method. Efficiency properties of these predictors are discussed when they are used to predict actual and average values of response variable within/outside the sample. Received: November 17, 1999; revised version: August 10, 2000     

Fitting linear regression models to censored data by least squares and maximum likelihood methods

Several approaches have been suggested for fitting linear regression models to censored data. These include Cox's propor­tional hazard models based on quasi-likelihoods. Methods of fitting based on least squares and maximum likelihoods have also been proposed. The methods proposed so far all require special purpose optimization routines. We describe an approach here which requires only a modified standard least squares routine.

We present methods for fitting a linear regression model to censored data by least squares and method of maximum likelihood. In the least squares method, the censored values are replaced by their expectations, and the residual sum of squares is minimized. Several variants are suggested in the ways in which the expect­ation is calculated. A parametric (assuming a normal error model) and two non-parametric approaches are described. We also present a method for solving the maximum likelihood equations in the estimation of the regression parameters in the censored regression situation. It is shown that the solutions can be obtained by a recursive algorithm which needs only a least squares routine for optimization. The suggested procesures gain considerably in computational officiency. The Stanford Heart Transplant data is used to illustrate the various methods.     

Comparison of prediction methods for multicollinear data

In this paper we discuss the partial least squares (PLS) prediction method. The method is compared to the predictor based on principal component regression (PCR). Both theoretical considerations and computations on artificial and real data are presented.     

Selection of Predictors in Distance-Based Regression

Distance-based regression is a prediction method consisting of two steps: from distances between observations we obtain latent variables which, in turn, are the regressors in an ordinary least squares linear model. Distances are computed from actually observed predictors by means of a suitable dissimilarity function. Being generally nonlinearly related with the response, their selection by the usual F tests is unavailable. In this article, we propose a solution to this predictor selection problem by defining generalized test statistics and adapting a nonparametric bootstrap method to estimate their p-values. We include a numerical example with automobile insurance data.     

In-Sample Inference and Forecasting in Misspecified Factor Models

This article considers in-sample prediction and out-of-sample forecasting in regressions with many exogenous predictors. We consider four dimension-reduction devices: principal components, ridge, Landweber Fridman, and partial least squares. We derive rates of convergence for two representative models: an ill-posed model and an approximate factor model. The theory is developed for a large cross-section and a large time-series. As all these methods depend on a tuning parameter to be selected, we also propose data-driven selection methods based on cross-validation and establish their optimality. Monte Carlo simulations and an empirical application to forecasting inflation and output growth in the U.S. show that data-reduction methods outperform conventional methods in several relevant settings, and might effectively guard against instabilities in predictors’ forecasting ability.     

Multivariate Calibration — Direct and Indirect Regression Methodology

This paper tries first to introduce and motivate the methodology of multivariate calibration. Next a review is given, mostly avoiding technicalities, of the somewhat messy theory of the subject. Two approaches are distinguished: the estimation approach (controlled calibration) and the prediction approach (natural calibration). Among problems discussed are the choice of estimator, the choice of confidence region, methodology for handling situations with more variables than observations, near-collinearities (with counter-measures like ridge type regression, principal components regression, partial least squares regression and continuum regression), pretreatment of data, and cross-validation vs true prediction. Examples discussed in detail concern estimation of the age of a rhinoceros from its horn lengths (low-dimensional), and nitrate prediction in waste-water from high-dimensional spectroscopic measurements.     

Data augmentation,frequentist estimation,and the Bayesian analysis of multinomial logit models

This article describes a convenient method of selecting Metropolis– Hastings proposal distributions for multinomial logit models. There are two key ideas involved. The first is that multinomial logit models have a latent variable representation similar to that exploited by Albert and Chib (J Am Stat Assoc 88:669–679, 1993) for probit regression. Augmenting the latent variables replaces the multinomial logit likelihood function with the complete data likelihood for a linear model with extreme value errors. While no conjugate prior is available for this model, a least squares estimate of the parameters is easily obtained. The asymptotic sampling distribution of the least squares estimate is Gaussian with known variance. The second key idea in this paper is to generate a Metropolis–Hastings proposal distribution by conditioning on the estimator instead of the full data set. The resulting sampler has many of the benefits of so-called tailored or approximation Metropolis–Hastings samplers. However, because the proposal distributions are available in closed form they can be implemented without numerical methods for exploring the posterior distribution. The algorithm converges geometrically ergodically, its computational burden is minor, and it requires minimal user input. Improvements to the sampler’s mixing rate are investigated. The algorithm is also applied to partial credit models describing ordinal item response data from the 1998 National Assessment of Educational Progress. Its application to hierarchical models and Poisson regression are briefly discussed.     

Forecasting functional time series

We propose forecasting functional time series using weighted functional principal component regression and weighted functional partial least squares regression. These approaches allow for smooth functions, assign higher weights to more recent data, and provide a modeling scheme that is easily adapted to allow for constraints and other information. We illustrate our approaches using age-specific French female mortality rates from 1816 to 2006 and age-specific Australian fertility rates from 1921 to 2006, and show that these weighted methods improve forecast accuracy in comparison to their unweighted counterparts. We also propose two new bootstrap methods to construct prediction intervals, and evaluate and compare their empirical coverage probabilities.     

Nonlinear Censored Regression Using Synthetic Data

Abstract.  The problem of estimating a nonlinear regression model, when the dependent variable is randomly censored, is considered. The parameter of the model is estimated by least squares using synthetic data. Consistency and asymptotic normality of the least squares estimators are derived. The proofs are based on a novel approach that uses i.i.d. representations of synthetic data through Kaplan–Meier integrals. The asymptotic results are supported by a small simulation study.     

Support vector regression based residual control charts

Control charts for residuals, based on the regression model, require a robust fitting technique for minimizing the error resulting from the fitted model. However, in the multivariate case, when the number of variables is high and data become complex, traditional fitting techniques, such as ordinary least squares (OLS), lose efficiency. In this paper, support vector regression (SVR) is used to construct robust control charts for residuals, called SVR-chart. This choice is based on the fact that the SVR is designed to minimize the structural error whereas other techniques minimize the empirical error. An application shows that SVR methods gives competitive results in comparison with the OLS and the partial least squares method, in terms of standard deviation of the error prediction and the standard error of performance. A sensitivity study is conducted to evaluate the SVR-chart performance based on the average run length (ARL) and showed that the SVR-chart has the best ARL behaviour in comparison with the other residuals control charts.     

Statistical Estimates for Modified Gravitation Model by Aggregated Data

A nonlinear regression model for forecasting of passenger flow between various spatial points (towns) is described. Unknown parameters are estimated using aggregated data when the information about a number of the departed passengers from each town is available only. For estimation, the least squares and maximum likelihood methods are used. Numerical examples are performed to illustrate the proposed approaches.     

SNP selection for predicting a quantitative trait

Molecular markers combined with powerful statistical tools have made it possible to detect and analyze multiple loci on the genome that are responsible for the phenotypic variation in quantitative traits. The objectives of the study presented in this paper are to identify a subset of single nucleotide polymorphism (SNP) markers that are associated with a particular trait and to construct a model that can best predict the value of the trait given the genotypic information of the SNPs using a three-step strategy. In the first step, a genome-wide association test is performed to screen SNPs that are associated with the quantitative trait of interest. SNPs with p-values of less than 5% are then analyzed in the second step. In the second step, a large number of randomly selected models, each consisting of a fixed number of randomly selected SNPs, are analyzed using the least angle regression method. This step will further remove redundant SNPs due to the complicated association among SNPs. A subset of SNPs that are shown to have a significant effect on the response trait more often than by chance are considered for the third step. In the third step, two alternative methods are considered: the least angle shrinkage and selection operation and sparse partial least squares regression. For both methods, the predictive ability of the fitted model is evaluated by an independent test set. The performance of the proposed method is illustrated by the analysis of a real data set on Canadian Holstein cattle.     

Iterated Partial Sum Sequences of Regression Residuals and Tests for Changepoints with Continuity Constraints

Iterated partial sum sequences of regression least squares residuals are defined and large sample properties of sequences of stochastic processes defined by these iterated partial sums are discussed. Also, finite sample properties of the iterated partial sum sequences are obtained. These include a property of least squares residuals of polynomial fits to equispaced data, namely the iterated partial sums sum to 0 provided that the order of iteration is not greater than the order of the polynomial, thus extending the well-known result that residuals sum to 0. Iterated partial sums are shown to play an important role in testing regression parameters for changes at unknown times under the constraint of continuity.     

Negligible interaction test for continuous predictors

Behavioral science researchers are often interested in whether there is negligible interaction among continuous predictors of an outcome variable. For example, a researcher might be interested in demonstrating that the effect of perfectionism on depression is very consistent across age. In this case, the researcher is interested in assessing whether the interaction between the predictors is too small to be meaningful. Unfortunately, most researchers address the above research question using a traditional association-based null hypothesis test (e.g. regression) where their goal is to fail to reject the null hypothesis of no interaction. Common problems with traditional tests are their sensitivity to sample size and their opposite (and hence inappropriate) hypothesis setup for finding a negligible interaction effect. In this study, we investigated a method for testing for negligible interaction between continuous predictors using unstandardized and standardized regression-based models and equivalence testing. A Monte Carlo study provides evidence for the effectiveness of the equivalence-based test relative to traditional approaches.KEYWORDS: Moderation, negligible interaction, linear models, interaction, multiple regression     

Sparse alternatives to ridge regression: a random effects approach

In a calibration of near-infrared (NIR) instrument, we regress some chemical compositions of interest as a function of their NIR spectra. In this process, we have two immediate challenges: first, the number of variables exceeds the number of observations and, second, the multicollinearity between variables are extremely high. To deal with the challenges, prediction models that produce sparse solutions have recently been proposed. The term ‘sparse’ means that some model parameters are zero estimated and the other parameters are estimated naturally away from zero. In effect, a variable selection is embedded in the model to potentially achieve a better prediction. Many studies have investigated sparse solutions for latent variable models, such as partial least squares and principal component regression, and for direct regression models such as ridge regression (RR). However, in the latter, it mainly involves an L₁ norm penalty to the objective function such as lasso regression. In this study, we investigate new sparse alternative models for RR within a random effects model framework, where we consider Cauchy and mixture-of-normals distributions on the random effects. The results indicate that the mixture-of-normals model produces a sparse solution with good prediction and better interpretation. We illustrate the methods using NIR spectra datasets from milk and corn specimens.     

Regression methods for high dimensional multicollinear data

To compare their performance on high dimensional data, several regression methods are applied to data sets in which the number of exploratory variables greatly exceeds the sample sizes. The methods are stepwise regression, principal components regression, two forms of latent root regression, partial least squares, and a new method developed here. The data are four sample sets for which near infrared reflectance spectra have been determined and the regression methods use the spectra to estimate the concentration of various chemical constituents, the latter having been determined by standard chemical analysis. Thirty-two regression equations are estimated using each method and their performances are evaluated using validation data sets. Although it is the most widely used, stepwise regression was decidedly poorer than the other methods considered. Differences between the latter were small with partial least squares performing slightly better than other methods under all criteria examined, albeit not by a statistically significant amount.