arXiv subjects

Patrick Breheny

Publications and source records attributed to Patrick Breheny.

15 recordsLinked to original sources

arXiv2025

Penalized mixed models to adjust for batch effects and unobserved confounding in high dimensional regression

Confounding can lead to spurious associations. Typically, one must observe confounders in order to adjust for them, but in high-dimensional settings, recent research has shown that it becomes possible to adjust even for unobserved confounders. The methods for carrying out these adjustments, however, have not been thoroughly investigated. In this study, we derive a confounding framework in which the signal, bias, and variability can be cleanly partitioned and quantified, thereby enabling simulations in which one varies the bias-to-signal ratio while holding the signal-to-noise ratio fixed. Using this construction, we demonstrate the impact of the amount and complexity of unobserved confounding on the performance of competing methods, including the LASSO, principal components LASSO (PC-LASSO), and penalized linear mixed models (PLMMs). We identify scenarios in which each method outperforms the others and find that overall, PLMM is the most robust approach.

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arXiv2025

Alternative Likelihood Approximations for High-Dimensional Intervals for Lasso

Classical frequentist approaches to inference for the lasso emphasize exact coverage for each feature, which requires debiasing and severs the connection between confidence intervals and the original lasso estimates. To address this, in earlier work we introduced the idea of average coverage, allowing for biased intervals that align with the lasso point estimates, and proposed the Relaxed Lasso Posterior (RL-P) intervals, which leverage the Bayesian interpretation of the lasso penalty as a Laplace prior together with a Normal likelihood conditional on the selected features. While RL-P achieves approximate average coverage, its intervals need not contain the lasso estimates. In this work, we propose alternative constructions based on different likelihood approximations to the full high-dimensional likelihood, yielding intervals that remain centered on the lasso estimates while still achieving average coverage. Our results continue to demonstrate that intentionally biased intervals provide a principled and practically useful framework for inference in high-dimensional regression.

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arXiv2025

A New Perspective on High Dimensional Confidence Intervals

Classically, confidence intervals are required to have consistent coverage across all values of the parameter. However, this will inevitably break down if the underlying estimation procedure is biased. For this reason, many efforts have focused on debiased versions of the lasso for interval construction. In the process of debiasing, however, the connection to the original estimates are often obscured. In this work, we offer a different perspective focused on average coverage in contrast to individual coverage. This perspective results in confidence intervals that better reflect the original assumptions, as opposed to debiased intervals, which often do not even contain the original lasso estimates. To this end we propose a method based on the Relaxed Lasso that gives approximately correct average coverage and compare this to debiased methods which attempt to produce correct individual coverage. With this new definition of coverage we also briefly revisit the bootstrap, which Chatterjee and Lahiri (2010) showed was inconsistent for lasso, but find that it fails even under this alternative coverage definition.

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arXiv2019

Cross validation approaches for penalized Cox regression

Cross validation is commonly used for selecting tuning parameters in penalized regression, but its use in penalized Cox regression models has received relatively little attention in the literature. Due to its partial likelihood construction, carrying out cross validation for Cox models is not straightforward, and there are several potential approaches for implementation. Here, we propose two new cross-validation methods for Cox regression and compare them to approaches that have been proposed elsewhere. Our proposed approach of cross-validating the linear predictors seems to offer an attractive balance of performance and numerical stability. We illustrate these advantages using simulated data as well as using them to analyze data from a high-dimensional study of survival in lung cancer patients.

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arXiv2018

Feature-specific inference for penalized regression using local false discovery rates

Penalized regression methods, most notably the lasso, are a popular approach to analyzing high-dimensional data. An attractive property of the lasso is that it naturally performs variable selection. An important area of concern, however, is the reliability of these variable selections. Motivated by local false discovery rate methodology from the large-scale hypothesis testing literature, we propose a method for calculating a local false discovery rate for each variable under consideration by the lasso model. These rates can be used to assess the reliability of an individual feature, or to estimate the model's overall false discovery rate. The method can be used for all values of $\lambda$. This is particularly useful for models with a few highly significant features but a high overall Fdr, which are a relatively common occurrence when using cross validation to select $\lambda$. It is also flexible enough to be applied to many varieties of penalized likelihoods including GLM and Cox models, and a variety of penalties, including MCP and SCAD. We demonstrate the validity of this approach and contrast it with other inferential methods for penalized regression as well as with local false discovery rates for univariate hypothesis tests. Finally, we show the practical utility of our method by applying it to two case studies involving high dimensional genetic data.

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arXiv2017

Marginal false discovery rate control for likelihood-based penalized regression models

The popularity of penalized regression in high-dimensional data analysis has led to a demand for new inferential tools for these models. False discovery rate control is widely used in high-dimensional hypothesis testing, but has only recently been considered in the context of penalized regression. Almost all of this work, however, has focused on lasso-penalized linear regression. In this paper, we derive a general method for controlling the marginal false discovery rate that can be applied to any penalized likelihood-based model, such as logistic regression and Cox regression. Our approach is fast, flexible and can be used with a variety of penalty functions including lasso, elastic net, MCP, and MNet. We derive theoretical results under which the proposed method is valid, and use simulation studies to demonstrate that the approach is reasonably robust, albeit slightly conservative, when these assumptions are violated. Despite being conservative, we show that our method often offers more power to select causally important features than existing approaches. Finally, the practical utility of the method is demonstrated on gene expression data sets with binary and time-to-event outcomes.

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arXiv2017

Hybrid safe-strong rules for efficient optimization in lasso-type problems

The lasso model has been widely used for model selection in data mining, machine learning, and high-dimensional statistical analysis. However, with the ultrahigh-dimensional, large-scale data sets now collected in many real-world applications, it is important to develop algorithms to solve the lasso that efficiently scale up to problems of this size. Discarding features from certain steps of the algorithm is a powerful technique for increasing efficiency and addressing the Big Data challenge. In this paper, we propose a family of hybrid safe-strong rules (HSSR) which incorporate safe screening rules into the sequential strong rule (SSR) to remove unnecessary computational burden. In particular, we present two instances of HSSR, namely SSR-Dome and SSR-BEDPP, for the standard lasso problem. We further extend SSR-BEDPP to the elastic net and group lasso problems to demonstrate the generalizability of the hybrid screening idea. Extensive numerical experiments with synthetic and real data sets are conducted for both the standard lasso and the group lasso problems. Results show that our proposed hybrid rules can substantially outperform existing state-of-the-art rules.

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arXiv2017

The biglasso Package: A Memory- and Computation-Efficient Solver for Lasso Model Fitting with Big Data in R

Penalized regression models such as the lasso have been extensively applied to analyzing high-dimensional data sets. However, due to memory limitations, existing R packages like glmnet and ncvreg are not capable of fitting lasso-type models for ultrahigh-dimensional, multi-gigabyte data sets that are increasingly seen in many areas such as genetics, genomics, biomedical imaging, and high-frequency finance. In this research, we implement an R package called biglasso that tackles this challenge. biglasso utilizes memory-mapped files to store the massive data on the disk, only reading data into memory when necessary during model fitting, and is thus able to handle out-of-core computation seamlessly. Moreover, it's equipped with newly proposed, more efficient feature screening rules, which substantially accelerate the computation. Benchmarking experiments show that our biglasso package, as compared to existing popular ones like glmnet, is much more memory- and computation-efficient. We further analyze a 31 GB real data set on a laptop with only 16 GB RAM to demonstrate the out-of-core computation capability of biglasso in analyzing massive data sets that cannot be accommodated by existing R packages.

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arXiv2016

Marginal false discovery rates for penalized regression models

Penalized regression methods are an attractive tool for high-dimensional data analysis, but their widespread adoption has been hampered by the difficulty of applying inferential tools. In particular, the question "How reliable is the selection of those features?" has proved difficult to address. In part, this difficulty arises from defining false discoveries in the classical, fully conditional sense, which is possible in low dimensions but does not scale well to high-dimensional settings. Here, we consider the analysis of marginal false discovery rates for penalized regression methods. Restricting attention to the marginal FDR permits straightforward estimation of the number of selections that would likely have occurred by chance alone, and therefore provides a useful summary of selection reliability. Theoretical analysis and simulation studies demonstrate that this approach is quite accurate when the correlation among predictors is mild, and only slightly conservative when the correlation is stronger. Finally, the practical utility of the proposed method and its considerable advantages over other approaches are illustrated using gene expression data from The Cancer Genome Atlas and GWAS data from the Myocardial Applied Genomics Network.

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arXiv2015

Overlapping group logistic regression with applications to genetic pathway selection

Discovering important genes that account for the phenotype of interest has long been challenging in genomewide expression analysis. Analyses such as Gene Set Enrichment Analysis (GSEA) that incorporate pathway information have become widespread in hypothesis testing, but pathway-based approaches have been largely absent from regression methods due to the challenges of dealing with overlapping pathways and the resulting lack of available software. The R package grpreg is widely used to fit group lasso and other group-penalized regression models; in this study, we develop an extension, grpregOverlap, to allow for overlapping group structure using the latent variable approach proposed by Jacob et al. (2009). We compare this approach to the ordinary lasso and to GSEA using both simulated and real data. We find that incorporation of prior pathway information substantially improves the accuracy of gene expression classifiers, and we shed light on several ways in which hypothesis-testing approaches such as GSEA differ from regression approaches with respect to the analysis of pathway data.

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arXiv2014

Strong rules for nonconvex penalties and their implications for efficient algorithms in high-dimensional regression

We consider approaches for improving the efficiency of algorithms for fitting nonconvex penalized regression models such as SCAD and MCP in high dimensions. In particular, we develop rules for discarding variables during cyclic coordinate descent. This dimension reduction leads to a substantial improvement in the speed of these algorithms for high-dimensional problems. The rules we propose here eliminate a substantial fraction of the variables from the coordinate descent algorithm. Violations are quite rare, especially in the locally convex region of the solution path, and furthermore, may be easily detected and corrected by checking the Karush-Kuhn-Tucker conditions. We extend these rules to generalized linear models, as well as to other nonconvex penalties such as the $\ell_2$-stabilized Mnet penalty, group MCP, and group SCAD. We explore three variants of the coordinate decent algorithm that incorporate these rules and study the efficiency of these algorithms in fitting models to both simulated data and on real data from a genome-wide association study.

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arXiv2013

A Selective Review of Group Selection in High-Dimensional Models

Grouping structures arise naturally in many statistical modeling problems. Several methods have been proposed for variable selection that respect grouping structure in variables. Examples include the group LASSO and several concave group selection methods. In this article, we give a selective review of group selection concerning methodological developments, theoretical properties and computational algorithms. We pay particular attention to group selection methods involving concave penalties. We address both group selection and bi-level selection methods. We describe several applications of these methods in nonparametric additive models, semiparametric regression, seemingly unrelated regressions, genomic data analysis and genome wide association studies. We also highlight some issues that require further study.

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arXiv2012

Kernel-based aggregation of marker-level genetic association tests involving copy-number variation

Genetic association tests involving copy-number variants (CNVs) are complicated by the fact that CNVs span multiple markers at which measurements are taken. The power of an association test at a single marker is typically low, and it is desirable to pool information across the markers spanned by the CNV. However, CNV boundaries are not known in advance, and the best way to proceed with this pooling is unclear. In this article, we propose a kernel-based method for aggregation of marker-level tests and explore several aspects of its implementation. In addition, we explore some of the theoretical aspects of marker-level test aggregation, proposing a permutation-based approach that preserves the family-wise error rate of the testing procedure, while demonstrating that several simpler alternatives fail to do so. The empirical power of the approach is studied in a number of simulations constructed from real data involving a pharmacogenomic study of gemcitabine, and compares favorably with several competing approaches.

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arXiv2012

Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors

Penalized regression is an attractive framework for variable selection problems. Often, variables possess a grouping structure, and the relevant selection problem is that of selecting groups, not individual variables. The group lasso has been proposed as a way of extending the ideas of the lasso to the problem of group selection. Nonconvex penalties such as SCAD and MCP have been proposed and shown to have several advantages over the lasso; these penalties may also be extended to the group selection problem, giving rise to group SCAD and group MCP methods. Here, we describe algorithms for fitting these models stably and efficiently. In addition, we present simulation results and real data examples comparing and contrasting the statistical properties of these methods.

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arXiv2011

Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection

A number of variable selection methods have been proposed involving nonconvex penalty functions. These methods, which include the smoothly clipped absolute deviation (SCAD) penalty and the minimax concave penalty (MCP), have been demonstrated to have attractive theoretical properties, but model fitting is not a straightforward task, and the resulting solutions may be unstable. Here, we demonstrate the potential of coordinate descent algorithms for fitting these models, establishing theoretical convergence properties and demonstrating that they are significantly faster than competing approaches. In addition, we demonstrate the utility of convexity diagnostics to determine regions of the parameter space in which the objective function is locally convex, even though the penalty is not. Our simulation study and data examples indicate that nonconvex penalties like MCP and SCAD are worthwhile alternatives to the lasso in many applications. In particular, our numerical results suggest that MCP is the preferred approach among the three methods.

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