Journal of Multivariate Analysis

Case-controls studies are popular epidemiological designs for detecting gene-environment interactions in the etiology of complex diseases, where the genetic susceptibility and environmental exposures may often be reasonably assumed independent in the source population. Various papers have presented analytical methods exploiting gene-environment independence to achieve better efficiency, all of which require either a rare disease assumption or a distributional assumption on the genetic variables. We relax both assumptions...

Forward regression, a classical variable screening method, has been widely used for model building when the number of covariates is relatively low. However, forward regression is seldom used in high-dimensional settings because of the cumbersome computation and unknown theoretical properties. Some recent works have shown that forward regression, coupled with an extended Bayesian information criterion (EBIC)-based stopping rule, can consistently identify all relevant predictors in high-dimensional linear regression settings...

Linear discriminant analysis (LDA) is a well-known classification technique that enjoyed great success in practical applications. Despite its effectiveness for traditional low-dimensional problems, extensions of LDA are necessary in order to classify high-dimensional data. Many variants of LDA have been proposed in the literature. However, most of these methods do not fully incorporate the structure information among predictors when such information is available. In this paper, we introduce a new high-dimensional LDA technique, namely graph-based sparse LDA (GSLDA), that utilizes the graph structure among the features...

The varying-coefficient Cox model is flexible and useful for modeling the dynamic changes of regression coefficients in survival analysis. In this paper, we study feature screening for varying-coefficient Cox models in ultrahigh-dimensional covariates. The proposed screening procedure is based on the joint partial likelihood of all predictors, thus different from marginal screening procedures available in the literature. In order to carry out the new procedure, we propose an effective algorithm and establish its ascent property...

By optimizing index functions against different outcomes, we propose a multivariate single-index model (SIM) for development of medical indices that simultaneously work with multiple outcomes. Fitting of a multivariate SIM is not fundamentally different from fitting a univariate SIM, as the former can be written as a sum of multiple univariate SIMs with appropriate indicator functions. What have not been carefully studied are the theoretical properties of the parameter estimators. Because of the lack of asymptotic results, no formal inference procedure has been made available for multivariate SIMs...

Covariate measurement error is a common problem. Improper treatment of measurement errors may affect the quality of estimation and the accuracy of inference. Extensive literature exists on homoscedastic measurement error models, but little research exists on heteroscedastic measurement. In this paper, we consider a general parametric regression model allowing for a covariate measured with heteroscedastic error. We allow both the variance function of the measurement errors and the conditional density function of the error-prone covariate given the error-free covariates to be completely unspecified...

We consider the problem of high-dimensional classification between two groups with unequal covariance matrices. Rather than estimating the full quadratic discriminant rule, we propose to perform simultaneous variable selection and linear dimension reduction on the original data, with the subsequent application of quadratic discriminant analysis on the reduced space. In contrast to quadratic discriminant analysis, the proposed framework doesn't require the estimation of precision matrices; it scales linearly with the number of measurements, making it especially attractive for the use on high-dimensional datasets...

With its important biological implications, modeling the associations of gene expression (GE) and copy number variation (CNV) has been extensively conducted. Such analysis is challenging because of the high data dimensionality, lack of knowledge regulating CNVs for a specific GE, different behaviors of the cis -acting and trans -acting CNVs, possible long-tailed distributions and contamination of GE measurements, and correlations between CNVs. The existing methods fail to address one or more of these challenges...

This paper studies the asymptotic properties of a sparse linear regression estimator, referred to as broken adaptive ridge (BAR) estimator, resulting from an L 0 -based iteratively reweighted L 2 penalization algorithm using the ridge estimator as its initial value. We show that the BAR estimator is consistent for variable selection and has an oracle property for parameter estimation. Moreover, we show that the BAR estimator possesses a grouping effect: highly correlated covariates are naturally grouped together, which is a desirable property not known for other oracle variable selection methods...

The estimation of continuous treatment effect functions using observational data often requires parametric specification of the effect curves, the conditional distributions of outcomes and treatment assignments given multi-dimensional covariates. While nonparametric extensions are possible, they typically suffer from the curse of dimensionality. Dimension reduction is often inevitable and we propose a sufficient dimension reduction framework to balance parsimony and flexibility. The joint central subspace can be estimated at a n 1/2 -rate without fixing its dimension in advance, and the treatment effect function is estimated by averaging local estimates of a reduced dimension...

The partially linear model (PLM) is a useful semiparametric extension of the linear model that has been well studied in the statistical literature. This paper proposes a variable selection procedure for the PLM with ultrahigh dimensional predictors. The proposed method is different from the existing penalized least squares procedure in that it relies on partial correlation between the partial residuals of the response and the predictors. We systematically study the theoretical properties of the proposed procedure and prove its model consistency property...

Principal Component Analysis (PCA) is a classical method for reducing the dimensionality of data by projecting them onto a subspace that captures most of their variation. Effective use of PCA in modern applications requires understanding its performance for data that are both high-dimensional and heteroscedastic. This paper analyzes the statistical performance of PCA in this setting, i.e., for high-dimensional data drawn from a low-dimensional subspace and degraded by heteroscedastic noise. We provide simplified expressions for the asymptotic PCA recovery of the underlying subspace, subspace amplitudes and subspace coefficients; the expressions enable both easy and efficient calculation and reasoning about the performance of PCA...

Variable selection plays a fundamental role in high-dimensional data analysis. Various methods have been developed for variable selection in recent years. Well-known examples are forward stepwise regression (FSR) and least angle regression (LARS), among others. These methods typically add variables into the model one by one. For such selection procedures, it is crucial to find a stopping criterion that controls model complexity. One of the most commonly used techniques to this end is cross-validation (CV) which, in spite of its popularity, has two major drawbacks: expensive computational cost and lack of statistical interpretation...

Large-margin classifiers are powerful techniques for classification problems. Although binary large-margin classifiers are heavily studied, multicategory problems are more complicated and challenging. A common approach is to construct k different decision functions for a k -class problem with a sum-to-zero constraint. However, such a constraint can be inefficient. Moreover, many large-margin classifiers can be sensitive to outliers in the training sample. In this article, we use the angle-based classification framework to avoid the explicit sum-to-zero constraint, and we propose two adaptively weighted large-margin classification techniques...

It is frequently of interest to jointly analyze two paired sequences of multiple tests. This paper studies the problem of detecting whether there are more pairs of tests that are significant in both sequences than would be expected by chance. The asymptotic detection boundary is derived in terms of parameters such as the sparsity of non-null cases in each sequence, the effect sizes of the signals, and the magnitude of the dependence between the two sequences. A new test for detecting weak dependence is also proposed, shown to be asymptotically adaptively optimal, studied in simulations, and applied to study genetic pleiotropy in 10 pediatric autoimmune diseases...

Many modern statistical problems can be cast in the framework of multivariate regression, where the main task is to make statistical inference for a possibly sparse and low-rank coefficient matrix. The low-rank structure in the coefficient matrix is of intrinsic multivariate nature, which, when combined with sparsity, can further lift dimension reduction, conduct variable selection, and facilitate model interpretation. Using a Bayesian approach, we develop a unified sparse and low-rank multivariate regression method to both estimate the coefficient matrix and obtain its credible region for making inference...

In the era of big data, integrative analyses that pool data from different sources are now extensively conducted in order to improve performance. Among many interesting applications, genomics research is an area where integrative methods become popular tools to identify prognostic biomarkers for various diseases. In this paper, we propose such a framework for pathway and gene identification. Our method employs a hierarchical decomposition on genes' effects followed by a proper regularization to identify important pathways and genes across multiple studies...

Exploring resting-state brain functional connectivity of autism spectrum disorders (ASD) using functional magnetic resonance imaging (fMRI) data has become a popular topic over the past few years. The data in a standard brain template consist of over 170,000 voxel specific points in time for each human subject. Such an ultra-high dimensionality makes the voxel-level functional connectivity analysis (involving four billion voxel pairs) both statistically and computationally inefficient. In this work, we introduce a new framework to identify the functional brain network at the anatomical region level for each individual...

Missing data occur frequently in a wide range of applications. In this paper, we consider estimation of high-dimensional covariance matrices in the presence of missing observations under a general missing completely at random model in the sense that the missingness is not dependent on the values of the data. Based on incomplete data, estimators for bandable and sparse covariance matrices are proposed and their theoretical and numerical properties are investigated. Minimax rates of convergence are established under the spectral norm loss and the proposed estimators are shown to be rate-optimal under mild regularity conditions...

Joint models for a wide class of response variables and longitudinal measurements consist on a mixed-effects model to fit longitudinal trajectories whose random effects enter as covariates in a generalized linear model for the primary response. They provide a useful way to assess association between these two kinds of data, which in clinical studies are often collected jointly on a series of individuals and may help understanding, for instance, the mechanisms of recovery of a certain disease or the efficacy of a given therapy...