Regularization for Cox’s proportional hazards model with NP-dimensionality

Bradić, Jelena; Fan, Jianqing; Jiang, Jiancheng

doi:10.1214/11-aos911

Cited by 121 publications

(115 citation statements)

References 38 publications

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“…Lasso (and similar) methods for particular counting processes such as Cox model or multiplicative Aalen intensity have also been derived for instance in [8] or [19].…”

Section: Lasso Criterion and Other Counting Processesmentioning

confidence: 99%

Concentration inequalities, counting processes and adaptive statistics

Reynaud-Bouret

2014

ESAIM: Proc.

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Abstract.Adaptive statistics for counting processes need particular concentration inequalities to define and calibrate the methods as well as to precise the theoretical performance of the statistical inference. The present article is a small (non exhaustive) review of existing concentration inequalities that are useful in this context. Résumé. Les statistiques adaptatives pour les processus de comptage nécessitent des inégalités deconcentration particulières pour définir et calibrer les méthodes ainsi que pour comprendre les performances de l'inférence statistique. Cet article est une revue non exhaustive des inégalités de concentration qui sont utiles dans ce contexte.Mathematics Subject Classification: 62G05, 62G10, 62M07, 62M09, 60G55, 60E15.

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“…Lasso (and similar) methods for particular counting processes such as Cox model or multiplicative Aalen intensity have also been derived for instance in [8] or [19].…”

Section: Lasso Criterion and Other Counting Processesmentioning

confidence: 99%

Concentration inequalities, counting processes and adaptive statistics

Reynaud-Bouret

2014

ESAIM: Proc.

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“…However, when p ≫ n, computational issues inherent in these methods makes them nonapplicable to ultrahigh-dimensional statistical learning problems because of serious challenges in "computational expediency, statistical accuracy, and algorithmic stability" ( [6]). A recent work by [2] did establish the oracle properties of the regularized partial likelihood estimates under an ultrahigh dimensional setting. The results, however, required the optimizers to the penalized partial likelihood function to be unique and global, which is, in general, difficult to verify, especially when the dimension of covariates is exceedingly high.…”

Section: §1 Introductionmentioning

confidence: 99%

Feature selection of ultrahigh-dimensional covariates with survival outcomes: a selective review

Hong

2017

Appl. Math. J. Chin. Univ.

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Many modern biomedical studies have yielded survival data with high-throughput predictors. The goals of scientific research often lie in identifying predictive biomarkers, understanding biological mechanisms and making accurate and precise predictions. Variable screening is a crucial first step in achieving these goals. This work conducts a selective review of feature screening procedures for survival data with ultrahigh dimensional covariates. We present the main methodologies, along with the key conditions that ensure sure screening properties. The practical utility of these methods is examined via extensive simulations. We conclude the review with some future opportunities in this field. MR Subject Classification 97K80Keywords survival analysis; ultrahigh dimensional predictors; variable screening; sure screening property §1 Introduction Modern biomedical studies have generated abundant survival data with high dimensional biomarkers for various scientific purposes. For instance, identifying genomic profiles that are associated with cancer patients' survival may help with understanding disease progression processes and designing more effective gene therapies. With the advent of new biotechnologies, the emergence of high-throughput data, such as gene expressions, SNPs, methylation and next-generation RNA sequencing, has pushed the dimensionality of data to a larger scale. In these cases, the dimensionality of covariates may grow exponentially with the sample size and such data has been commonly referred to as ultrahigh dimensional data ([5]).When the number of covariates (p) is less than the sample size (n), the parametric regression, such as Weibull models, and the semiparametric regression, such as the Cox proportional hazards model and the Accelerated Failure Time (AFT) model, have been routinely used for modeling censored outcome data in many practical settings. When p > n, penalized ([20], [4], [25], [29]) and the oracle properties and statistical error bounds of estimation have been established ([13], [15]). However, when p ≫ n, computational issues inherent in these methods makes them nonapplicable to ultrahigh-dimensional statistical learning problems because of serious challenges in "computational expediency, statistical accuracy, and algorithmic stability" ([6]). A recent work by [2] did establish the oracle properties of the regularized partial likelihood estimates under an ultrahigh dimensional setting. The results, however, required the optimizers to the penalized partial likelihood function to be unique and global, which is, in general, difficult to verify, especially when the dimension of covariates is exceedingly high. HHS Public AccessA seminal paper by [5] has demonstrated a simple but useful way to deal with ultrahigh dimensional regression. First, a variable screening procedure is used as a fast and crude tool for reducing the dimensionality to a moderate size (usually below the sample size). In the second step, a more sophisticated technique, such as penalized likelihood methods, c...

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“…Belloni and Chernozhukov (2011a), Bradic et al (2011) and Wang et al (2012)), and to hazards models (e.g. Bradic et al (2012) and Lin and Lv (2013)). We contribute to this literature by considering a regression model with a possible change point and then deriving non-asymptotic oracle inequalities for both the prediction risk and the l 1 -estimation loss for regression coefficients under a sparsity scenario.…”

Section: Introductionmentioning

confidence: 99%

The lasso for high-dimensional regression with a possible change-point

Lee¹,

Seo²,

Shin³

2014

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Summary. We consider a high dimensional regression model with a possible change point due to a covariate threshold and develop the lasso estimator of regression coefficients as well as the threshold parameter. Our lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non-asymptotic oracle inequalities for both the prediction risk and the l 1 -estimation loss for regression coefficients. Since the lasso estimator selects variables simultaneously, we show that oracle inequalities can be established without pretesting the existence of the threshold effect. Furthermore, we establish conditions under which the estimation error of the unknown threshold parameter can be bounded by a factor that is nearly n 1 even when the number of regressors can be much larger than the sample size n. We illustrate the usefulness of our proposed estimation method via Monte Carlo simulations and an application to real data.

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Regularization for Cox’s proportional hazards model with NP-dimensionality

Cited by 121 publications

References 38 publications

Concentration inequalities, counting processes and adaptive statistics

Concentration inequalities, counting processes and adaptive statistics

Feature selection of ultrahigh-dimensional covariates with survival outcomes: a selective review

The lasso for high-dimensional regression with a possible change-point

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