High-Dimensional Variable Selection for Survival Data

The minimal depth of a maximal subtree is a dimensionless order statistic measuring the predictiveness of a variable in a survival tree. We derive the distribution of the minimal depth and use it for high-dimensional variable selection using random survival forests. In big p and small n problems (where p is the dimension and n is the sample size), the distribution of the minimal depth reveals a “ceiling effect” in which a tree simply cannot be grown deep enough to properly identify predictive variables. Motivated by this limitation, we develop a new regularized algorithm, termed RSF-Variable Hunting. This algorithm exploits maximal subtrees for effective variable selection under such scenarios. Several applications are presented demonstrating the methodology, including the problem of gene selection using microarray data. In this work we focus only on survival settings, although our methodology also applies to other random forests applications, including regression and classification settings. All examples presented here use the R-software package randomSurvivalForest.

[1]  F. Harrell,et al.  Evaluating the yield of medical tests. , 1982, JAMA.

[2]  W. Loh,et al.  Tree-Structured Classification via Generalized Discriminant Analysis. , 1988 .

[3]  D. Harrington,et al.  Counting Processes and Survival Analysis , 1991 .

[4]  W. Loh,et al.  SPLIT SELECTION METHODS FOR CLASSIFICATION TREES , 1997 .

[5]  G. Ridgeway.  The State of Boosting ∗ , 1999 .

[6]  E Graf,et al.  Assessment and comparison of prognostic classification schemes for survival data. , 1999, Statistics in medicine.

[7]  Leo Breiman,et al.  Statistical Modeling: The Two Cultures (with comments and a rejoinder by the author) , 2001 .

[8]  Leo Breiman,et al.  Statistical Modeling: The Two Cultures (with comments and a rejoinder by the author) , 2001, Statistical Science.

[9]  David E. Misek,et al.  Gene-expression profiles predict survival of patients with lung adenocarcinoma , 2002, Nature Medicine.

[10]  Yudong D. He,et al.  Gene expression profiling predicts clinical outcome of breast cancer , 2002, Nature.

[11]  L. Staudt,et al.  The use of molecular profiling to predict survival after chemotherapy for diffuse large-B-cell lymphoma. , 2002, The New England journal of medicine.

[12]  Danh V. Nguyen,et al.  Partial least squares proportional hazard regression for application to DNA microarray survival data , 2002, Bioinform..

[13]  R. Tibshirani,et al.  Diagnosis of multiple cancer types by shrunken centroids of gene expression , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[14]  L. Staudt,et al.  The proliferation gene expression signature is a quantitative integrator of oncogenic events that predicts survival in mantle cell lymphoma. , 2003, Cancer cell.

[15]  Leo Breiman,et al.  Random Forests , 2001, Machine Learning.

[16]  R. Tibshirani,et al.  Semi-Supervised Methods to Predict Patient Survival from Gene Expression Data , 2004, PLoS biology.

[17]  K. Lunetta,et al.  Screening large-scale association study data: exploiting interactions using random forests , 2004, BMC Genetics.

[18]  R. Tibshirani,et al.  Use of gene-expression profiling to identify prognostic subclasses in adult acute myeloid leukemia. , 2004, The New England journal of medicine.

[19]  Jiang Gui,et al.  Partial Cox regression analysis for high-dimensional microarray gene expression data , 2004, ISMB/ECCB.

[20]  Ramón Díaz-Uriarte,et al.  Gene selection and classification of microarray data using random forest , 2006, BMC Bioinformatics.

[21]  K. Lunetta,et al.  Identifying SNPs predictive of phenotype using random forests , 2005, Genetic epidemiology.

[22]  Hongzhe Li,et al.  Boosting proportional hazards models using smoothing splines, with applications to high-dimensional microarray data , 2005, Bioinform..

[23]  H. Zou,et al.  Regularization and variable selection via the elastic net , 2005 .

[24]  Mee Young Park,et al.  L 1-regularization path algorithm for generalized linear models , 2006 .

[25]  K. Hornik,et al.  Unbiased Recursive Partitioning: A Conditional Inference Framework , 2006 .

[26]  M. Schumacher,et al.  Consistent Estimation of the Expected Brier Score in General Survival Models with Right‐Censored Event Times , 2006, Biometrical journal. Biometrische Zeitschrift.

[27]  Shuangge Ma,et al.  Additive Risk Models for Survival Data with High‐Dimensional Covariates , 2006, Biometrics.

[28]  Torsten Hothorn,et al.  Model-based boosting in high dimensions , 2006, Bioinform..

[29]  Jian Huang,et al.  Regularized Estimation in the Accelerated Failure Time Model with High‐Dimensional Covariates , 2006, Biometrics.

[30]  Mee Young Park,et al.  L1‐regularization path algorithm for generalized linear models , 2007 .

[31]  Udaya B. Kogalur,et al.  Random Survival Forests for R , 2007 .

[32]  Hao Helen Zhang,et al.  Adaptive Lasso for Cox's proportional hazards model , 2007 .

[33]  Jian Huang,et al.  Clustering threshold gradient descent regularization: with applications to microarray studies , 2007, Bioinform..

[34]  H. Ishwaran.  Variable importance in binary regression trees and forests , 2007, 0711.2434.

[35]  C. Sotiriou,et al.  Meta-analysis of gene expression profiles in breast cancer: toward a unified understanding of breast cancer subtyping and prognosis signatures , 2007, Breast Cancer Research.

[36]  K. Doksum,et al.  Nonparametric Variable Selection: The EARTH Algorithm , 2008 .

[37]  Hemant Ishwaran,et al.  An interferon-related gene signature for DNA damage resistance is a predictive marker for chemotherapy and radiation for breast cancer , 2008, Proceedings of the National Academy of Sciences.

[38]  Hemant Ishwaran,et al.  Random Survival Forests , 2008, Wiley StatsRef: Statistics Reference Online.

[39]  M. West,et al.  Bayesian Weibull tree models for survival analysis of clinico-genomic data. , 2008, Statistical methodology.

[40]  H. Ishwaran,et al.  A novel approach to cancer staging: application to esophageal cancer. , 2009, Biostatistics.

[41]  Udaya B. Kogalur,et al.  Consistency of Random Survival Forests. , 2008, Statistics & probability letters.

[42]  Harald Binder,et al.  The benefit of data-based model complexity selection via prediction error curves in time-to-event data , 2011, Comput. Stat..