Evaluating Prediction Rules for t-Year Survivors With Censored Regression Models

Suppose that we are interested in establishing simple but reliable rules for predicting future t-year survivors through censored regression models. In this article we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate, sensitivity and specificity, and positive and negative predictive values. Specifically, under various working models, we derive consistent estimators for the above measures through substitution and cross-validation estimation procedures. Furthermore, we provide large-sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified. Confidence intervals, for example, for the difference of the precision measures between two competing rules can then be constructed. All of the proposals are illustrated with real examples, and their finite-sample properties are evaluated through a simulation study.

[1]  Jianming Ye.  On Measuring and Correcting the Effects of Data Mining and Model Selection , 1998 .

[2]  Laurence L. George,et al.  The Statistical Analysis of Failure Time Data , 2003, Technometrics.

[3]  B. Efron.  How Biased is the Apparent Error Rate of a Prediction Rule , 1986 .

[4]  R. Tibshirani,et al.  Improvements on Cross-Validation: The 632+ Bootstrap Method , 1997 .

[5]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[6]  Els Goetghebeur,et al.  Model evaluation based on the sampling distribution of estimated absolute prediction error , 2007 .

[7]  R Simon,et al.  Measures of explained variation for survival data. , 1990, Statistics in medicine.

[8]  Snigdhansu Chatterjee,et al.  © Institute of Mathematical Statistics, 2005 GENERALIZED BOOTSTRAP FOR ESTIMATING EQUATIONS , 2022 .

[9]  Lee-Jen Wei,et al.  The accelerated failure time model: a useful alternative to the Cox regression model in survival analysis. , 1992, Statistics in medicine.

[10]  Chaya S Moskowitz,et al.  Quantifying and comparing the accuracy of binary biomarkers when predicting a failure time outcome. , 2004, Statistics in medicine.

[11]  M. Pepe,et al.  Limitations of the odds ratio in gauging the performance of a diagnostic, prognostic, or screening marker. , 2004, American journal of epidemiology.

[12]  J. Kalbfleisch,et al.  The Statistical Analysis of Failure Time Data , 1980 .

[13]  D.,et al.  Regression Models and Life-Tables , 2022 .

[14]  J. Shao.  Bootstrap Model Selection , 1996 .

[15]  D J Spiegelhalter,et al.  Probabilistic prediction in patient management and clinical trials. , 1986, Statistics in medicine.

[16]  O. Borgan.  The Statistical Analysis of Failure Time Data (2nd Ed.). John D. Kalbfleisch and Ross L. Prentice , 2003 .

[17]  B. Efron.  The Estimation of Prediction Error , 2004 .

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

[19]  Shinichi Morishita,et al.  On Classification and Regression , 1998, Discovery Science.

[20]  Tianxi Cai,et al.  Semiparametric Box–Cox power transformation models for censored survival observations , 2005 .

[21]  Margaret Sullivan Pepe,et al.  Combining Several Screening Tests: Optimality of the Risk Score , 2002, Biometrics.

[22]  C. H. Oh,et al.  Some comments on , 1998 .

[23]  C. Mallows.  More comments on C p , 1995 .

[24]  D. Pollard.  Empirical Processes: Theory and Applications , 1990 .

[25]  Anthony C. Davison,et al.  Bootstrap Methods and Their Application , 1998 .

[26]  M. Pepe.  The Statistical Evaluation of Medical Tests for Classification and Prediction , 2003 .

[27]  J. Kalbfleisch,et al.  The Statistical Analysis of Failure Time Data , 1980 .

[28]  W. Newey,et al.  Large sample estimation and hypothesis testing , 1986 .

[29]  Zhiliang Ying,et al.  Towards a general asymptotic theory for Cox model with staggered entry , 1997 .

[30]  Z. Ying,et al.  Analysis of transformation models with censored data , 1995 .

[31]  R. Tibshirani,et al.  Model Search by Bootstrap “Bumping” , 1999 .

[32]  Michael Schemper,et al.  The explained variation in proportional hazards regression , 1990 .

[33]  Yuhyun Park,et al.  Estimating subject-specific survival functions under the accelerated failure time model , 2003 .

[34]  C. L. Mallows.  Some comments on C_p , 1973 .

[35]  T. Lumley,et al.  Time‐Dependent ROC Curves for Censored Survival Data and a Diagnostic Marker , 2000, Biometrics.

[36]  P. Heagerty,et al.  Survival Model Predictive Accuracy and ROC Curves , 2005, Biometrics.

[37]  Tianxi Cai,et al.  Application of the Time‐Dependent ROC Curves for Prognostic Accuracy with Multiple Biomarkers , 2006, Biometrics.

[38]  Colin L. Mallows,et al.  Some Comments on Cp , 2000, Technometrics.

[39]  G. Brier.  VERIFICATION OF FORECASTS EXPRESSED IN TERMS OF PROBABILITY , 1950 .

[40]  M Schemper,et al.  Explained variation for logistic regression. , 1996, Statistics in medicine.

[41]  Xiao-Hua Zhou,et al.  Statistical Methods in Diagnostic Medicine , 2002 .

[42]  B. Efron.  Estimating the Error Rate of a Prediction Rule: Improvement on Cross-Validation , 1983 .

[43]  Romain Neugebauer,et al.  Cross-Validated Bagged Prediction of Survival , 2006, Statistical applications in genetics and molecular biology.

[44]  Brian D. Ripley,et al.  Pattern Recognition and Neural Networks , 1996 .

[45]  Van,et al.  A gene-expression signature as a predictor of survival in breast cancer. , 2002, The New England journal of medicine.

[46]  G. McLachlan.  Discriminant Analysis and Statistical Pattern Recognition , 1992 .

[47]  B. Efron.  The jackknife, the bootstrap, and other resampling plans , 1987 .

[48]  P. Grambsch,et al.  Prognosis in primary biliary cirrhosis: Model for decision making , 1989, Hepatology.

[49]  John OQuigly and Ronghui Xu,et al.  Explained Variation in Proportional Hazards Regression , 2005 .

[50]  Christopher M. Bishop,et al.  Classification and regression , 1997 .