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Covariance matrix estimation, a classical statistical topic, poses significant challenges when the sample size is comparable to or smaller than the number of features. In this paper, we frame covariance matrix estimation as a compound decision problem and apply an optimal decision rule to estimate covariance parameters. To approximate this rule, we introduce an algorithm that integrates jackknife techniques with machine learning regression methods. This algorithm exhibits adaptability across diverse scenarios without relying on assumptions about data distribution. Simulation results and gene network inference from an RNA-seq experiment in mice demonstrate that our approach either matches or surpasses several state-of-the-art methods, Comment: 13 pages, 3 figures

Wide heterogeneity exists in cancer patients' survival, ranging from a few months to several decades. To accurately predict clinical outcomes, it is vital to build an accurate predictive model that relates patients' molecular profiles with patients' survival. With complex relationships between survival and high-dimensional molecular predictors, it is challenging to conduct non-parametric modeling and irrelevant predictors removing simultaneously. In this paper, we build a kernel Cox proportional hazards semi-parametric model and propose a novel regularized garrotized kernel machine (RegGKM) method to fit the model. We use the kernel machine method to describe the complex relationship between survival and predictors, while automatically removing irrelevant parametric and non-parametric predictors through a LASSO penalty. An efficient high-dimensional algorithm is developed for the proposed method. Comparison with other competing methods in simulation shows that the proposed method always has better predictive accuracy. We apply this method to analyze a multiple myeloma dataset and predict patients' death burden based on their gene expressions. Our results can help classify patients into groups with different death risks, facilitating treatment for better clinical outcomes.

Biomarkers are often measured in bulk to diagnose patients, monitor patient conditions, and research novel drug pathways. The measurement of these biomarkers often suffers from detection limits that result in missing and untrustworthy measurements. Frequently, missing biomarkers are imputed so that down-stream analysis can be conducted with modern statistical methods that cannot normally handle data subject to informative censoring. This work develops an empirical Bayes $g$-modeling method for imputing and denoising biomarker measurements. We establish superior estimation properties compared to popular methods in simulations and demonstrate the utility of the estimated biomarker measurements for down-stream analysis.

The simultaneous estimation of multiple unknown parameters lies at heart of a broad class of important problems across science and technology. Currently, the state-of-the-art performance in the such problems is achieved by nonparametric empirical Bayes methods. However, these approaches still suffer from two major issues. First, they solve a frequentist problem but do so by following Bayesian reasoning, posing a philosophical dilemma that has contributed to somewhat uneasy attitudes toward empirical Bayes methodology. Second, their computation relies on certain density estimates that become extremely unreliable in some complex simultaneous estimation problems. In this paper, we study these issues in the context of the canonical Gaussian sequence problem. We propose an entirely frequentist alternative to nonparametric empirical Bayes methods by establishing a connection between simultaneous estimation and penalized nonparametric regression. We use flexible regularization strategies, such as shape constraints, to derive accurate estimators without appealing to Bayesian arguments. We prove that our estimators achieve asymptotically optimal regret and show that they are competitive with or can outperform nonparametric empirical Bayes methods in simulations and an analysis of spatially resolved gene expression data.

We propose a new prediction method for multivariate linear regression problems where the number of features is less than the sample size but the number of outcomes is extremely large. Many popular procedures, such as penalized regression procedures, require parameter tuning that is computationally untenable in such large-scale problems. We take a different approach, motivated by ideas from simultaneous estimation problems, that performs linear shrinkage on ordinary least squares parameter estimates. Our approach is extremely computationally efficient and tuning-free. We show that it can asymptotically outperform ordinary least squares without any structural assumptions on the true regression coefficients and illustrate its good performance in simulations and an analysis of single-cell RNA-seq data.

Capturing patterns of variation present in a dataset is important in exploratory data analysis and unsupervised learning. Contrastive dimension reduction methods, such as contrastive principal component analysis (cPCA), find patterns unique to a target dataset of interest by contrasting with a carefully chosen background dataset representing unwanted or uninteresting variation. However, such methods typically require a tuning parameter that governs the level of contrast, and it is unclear how to choose this parameter objectively. Furthermore, it is frequently of interest to contrast against multiple backgrounds, which is difficult to accomplish with existing methods. We propose unique component analysis (UCA), a tuning-free method that identifies low-dimensional representations of a target dataset relative to one or more comparison datasets. It is computationally efficient even with large numbers of features. We show in several experiments that UCA with a single background dataset achieves similar results compared to cPCA with various tuning parameters, and that UCA with multiple individual background datasets is superior to both cPCA with any single background data and cPCA with a pooled background dataset.

Covariance matrix estimation is a fundamental statistical task in many applications, but the sample covariance matrix is sub-optimal when the sample size is comparable to or less than the number of features. Such high-dimensional settings are common in modern genomics, where covariance matrix estimation is frequently employed as a method for inferring gene networks. To achieve estimation accuracy in these settings, existing methods typically either assume that the population covariance matrix has some particular structure, for example sparsity, or apply shrinkage to better estimate the population eigenvalues. In this paper, we study a new approach to estimating high-dimensional covariance matrices. We first frame covariance matrix estimation as a compound decision problem. This motivates defining a class of decision rules and using a nonparametric empirical Bayes g-modeling approach to estimate the optimal rule in the class. Simulation results and gene network inference in an RNA-seq experiment in mouse show that our approach is comparable to or can outperform a number of state-of-the-art proposals., Comment: 20 pages, 4 figures. Biometrics (2022)

Mediation analysis is difficult when the number of potential mediators is larger than the sample size. In this paper we propose new inference procedures for the indirect effect in the presence of high-dimensional mediators for linear mediation models. We develop methods for both incomplete mediation, where a direct effect may exist, as well as complete mediation, where the direct effect is known to be absent. We prove consistency and asymptotic normality of our indirect effect estimators. Under complete mediation, where the indirect effect is equivalent to the total effect, we further prove that our approach gives a more powerful test compared to directly testing for the total effect. We confirm our theoretical results in simulations, as well as in an integrative analysis of gene expression and genotype data from a pharmacogenomic study of drug response. We present a novel analysis of gene sets to understand the molecular mechanisms of drug response, and also identify a genome-wide significant noncoding genetic variant that cannot be detected using standard analysis methods., Comment: To appear in Biometrika

The integrative analysis of multiple datasets is an important strategy in data analysis. It is increasingly popular in genomics, which enjoys a wealth of publicly available datasets that can be compared, contrasted, and combined in order to extract novel scientific insights. This paper studies a stylized example of data integration for a classical statistical problem: leveraging side information to estimate a vector of normal means. This task is formulated as a compound decision problem, an oracle integrative decision rule is derived, and a data-driven estimate of this rule based on minimizing an unbiased estimate of its risk is proposed. The data-driven rule is shown to asymptotically achieve the minimum possible risk among all separable decision rules, and it can outperform existing methods in numerical properties. The proposed procedure leads naturally to an integrative high-dimensional classification procedure, which is illustrated by combining data from two independent gene expression profiling studies., Comment: accepted by Statistica Sinica

Genome-wide association studies (GWA studies or GWAS) investigate the relationships between genetic variants such as single-nucleotide polymorphisms (SNPs) and individual traits. Recently, incorporating biological priors together with machine learning methods in GWA studies has attracted increasing attention. However, in real-world, nucleotide-level bio-priors have not been well-studied to date. Alternatively, studies at gene-level, for example, protein--protein interactions and pathways, are more rigorous and legitimate, and it is potentially beneficial to utilize such gene-level priors in GWAS. In this paper, we proposed a novel two-level structured sparse model, called Sparse Group Lasso with Group-level Graph structure (SGLGG), for GWAS. It can be considered as a sparse group Lasso along with a group-level graph Lasso. Essentially, SGLGG penalizes the nucleotide-level sparsity as well as takes advantages of gene-level priors (both gene groups and networks), to identifying phenotype-associated risk SNPs. We employ the alternating direction method of multipliers algorithm to optimize the proposed model. Our experiments on the Alzheimer's Disease Neuroimaging Initiative whole genome sequence data and neuroimage data demonstrate the effectiveness of SGLGG. As a regression model, it is competitive to the state-of-the-arts sparse models; as a variable selection method, SGLGG is promising for identifying Alzheimer's disease-related risk SNPs.

Genetic risk prediction is an important component of individualized medicine, but prediction accuracies remain low for many complex diseases. A fundamental limitation is the sample sizes of the studies on which the prediction algorithms are trained. One way to increase the effective sample size is to integrate information from previously existing studies. However, it can be difficult to find existing data that examine the target disease of interest, especially if that disease is rare or poorly studied. Furthermore, individual-level genotype data from these auxiliary studies are typically difficult to obtain. This paper proposes a new approach to integrative genetic risk prediction of complex diseases with binary phenotypes. It accommodates possible heterogeneity in the genetic etiologies of the target and auxiliary diseases using a tuning parameter-free nonparametric empirical Bayes procedure, and can be trained using only auxiliary summary statistics. Simulation studies show that the proposed method can provide superior predictive accuracy relative to non-integrative as well as integrative classifiers. The method is applied to a recent study of pediatric autoimmune diseases, where it substantially reduces prediction error for certain target/auxiliary disease combinations. The proposed method is implemented in the R package ssa.

It is frequently of interest to jointly analyze multiple sequences of multiple tests in order to identify simultaneous signals, defined as features tested in multiple studies whose test statistics are non-null in each. In many problems, however, the null distributions of the test statistics may be complicated or even unknown, and there do not currently exist any procedures that can be employed in these cases. This paper proposes a new nonparametric procedure that can identify simultaneous signals across multiple studies even without knowing the null distributions of the test statistics. The method is shown to asymptotically control the false discovery rate, and in simulations had excellent power and error control. In an analysis of gene expression and histone acetylation patterns in the brains of mice exposed to a conspecific intruder, it identified genes that were both differentially expressed and next to differentially accessible chromatin. The proposed method is available in the R package github.com/sdzhao/ssa.

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 detection boundary is derived in terms of parameters such as the sparsity of the latent significance indicators of the tests in each sequence, the effect sizes of the non-null tests, 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 of genetic pleiotropy in 10 pediatric autoimmune diseases.

Nonparametric empirical Bayes methods provide a flexible and attractive approach to high-dimensional data analysis. One particularly elegant empirical Bayes methodology, involving the Kiefer-Wolfowitz nonparametric maximum likelihood estimator (NPMLE) for mixture models, has been known for decades. However, implementation and theoretical analysis of the Kiefer-Wolfowitz NPMLE are notoriously difficult. A fast algorithm was recently proposed that makes NPMLE-based procedures feasible for use in large-scale problems, but the algorithm calculates only an approximation to the NPMLE. In this paper we make two contributions. First, we provide upper bounds on the convergence rate of the approximate NPMLE's statistical error, which have the same order as the best known bounds for the true NPMLE. This suggests that the approximate NPMLE is just as effective as the true NPMLE for statistical applications. Second, we illustrate the promise of NPMLE procedures in a high-dimensional binary classification problem. We propose a new procedure and show that it vastly outperforms existing methods in experiments with simulated data. In real data analyses involving cancer survival and gene expression data, we show that it is very competitive with several recently proposed methods for regularized linear discriminant analysis, another popular approach to high-dimensional classification.

As the number of possible predictors generated by high-throughput experiments continues to increase, methods are needed to quickly screen out unimportant covariates. Model-based screening methods have been proposed and theoretically justified, but only for a few specific models. Model-free screening methods have also recently been studied, but can have lower power to detect important covariates. In this paper we propose EEScreen, a screening procedure that can be used with any model that can be fit using estimating equations, and provide unified results on its finite-sample screening performance. EEScreen thus generalizes many recently proposed model-based and model-free screening procedures. We also propose iEEScreen, an iterative version of EEScreen, and show that it is closely related to a recently studied boosting method for estimating equations. We show via simulations for two different estimating equations that EEScreen and iEEScreen are useful and flexible screening procedures, and demonstrate our methods on data from a multiple myeloma study.