Springer Series in Statistics

Transcription

1 Springer Series in Statistics Series editors Peter Bickel, CA, USA Peter Diggle, Lancaster, UK Stephen E. Fienberg, Pittsburgh, PA, USA Ursula Gather, Dortmund, Germany Ingram Olkin, Stanford, CA, USA Scott Zeger, Baltimore, MD, USA

2 More information about this series at

3 Gerhard Tutz Matthias Schmid Modeling Discrete Time-to-Event Data 123

4 Gerhard Tutz LMU Munich Munich, Germany Matthias Schmid University of Bonn Bonn, Germany ISSN ISSN X (electronic) Springer Series in Statistics ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

5 Preface In recent years, a large variety of textbooks dealing with time-to-event analysis has been published. Most of these books focus on the statistical analysis of observations in continuous time. In practice, however, one often observes discrete event times either because of grouping effects or because event times are intrinsically measured on a discrete scale. Statistical methodology for discrete event times has been mainly presented in journal articles and a few book chapters. In this book we introduce basic concepts and give several extensions that allow to model discrete time data adequately. In particular, modeling discrete time-to-event data strongly profits from the smoothing and regularization methods that have been developed in recent decades. The presented approaches include methods that allow to find much more flexible models than in the early times of survival modeling. The book is aimed at applied statisticians, students of statistics and researchers from areas like biometrics, social sciences and econometrics. The mathematical level is moderate, instead we focus on basic concepts and data analysis. Objectives The main aims of the book are to provide a thorough introduction to basic and advanced concepts of discrete hazard modelling, to exploit the relationship between hazard models and generalized linear models, to demonstrate how existing statistical software can be used to fit discrete timeto-event models, and to illustrate the statistical methodology for discrete time-to-event models by considering applications from the social sciences, economics and biomedical sciences. v

6 vi Preface Special Topics This book provides a comprehensive treatment of statistical methodology for discrete time-to-event models. Special topics include non-parametric modeling of survival (e.g., by using smooth baseline hazards and/or smooth predictor effects), methods for the evaluation of model fit and prediction accuracy of discrete timeto-event models, regularized estimation techniques for predictor selection in high-dimensional covariate spaces, and tree-based methods for discrete time-to-event analysis. In addition, each section of the book contains a set of exercises on the respective topics. Implementation and Software All numerical results presented in this book were obtained by using the R System for Statistical Computing (R Core Team 2015). Hence readers are able to reproduce all the results by using freely available software. Various functions and tools for the analysis of discrete time-to-event data are collected in the R package discsurv (Welchowski and Schmid 2015). We are grateful to many colleagues for valuable discussions and suggestions, in particular to Kaveh Bashiri, Moritz Berger, Jutta Gampe, Andreas Groll, Wolfgang Hess, Stephanie Möst, Vito M. R. Mugeo, Margret Oelker, Hein Putter, Micha Schneider and Steffen Unkel. Silke Janitza carefully read preliminary versions of the book and helped to reduce the number of mistakes. We also thank Helmut Küchenhoff for late but substantial suggestions. Special thanks go to Thomas Welchowski for his excellent programming work and to Pia Oberschmidt for assisting us in compiling the subject index. München, Germany Bonn, Germany April 2015 Gerhard Tutz Matthias Schmid

7 Contents 1 Introduction Survivaland Time-to-EventData Continuous Versus Discrete Survival Overview Examples The Life Table Life Table Estimates DistributionalAspects SmoothLife Table Estimators HeterogeneousIntervals Kaplan MeierEstimator Life Tables in Demography Literatureand FurtherReading Software Exercises Basic Regression Models The Discrete Hazard Function ParametricRegression Models Logistic Discrete Hazards: The Proportional ContinuationRatio Model AlternativeModels Discrete and Continuous Hazards Concepts for Continuous Time The Proportional Hazards Model Estimation StandardErrors Time-VaryingCovariates Continuous Versus Discrete Proportional Hazards Subject-SpecificIntervalCensoring Literatureand FurtherReading vii

8 viii Contents 3.9 Software Exercises Evaluation and Model Choice Relevance ofpredictors: Tests Residuals and Goodness-of-Fit No Censoring Deviancein the Case of Censoring MartingaleResiduals MeasuringPredictivePerformance Predictive Deviance and R 2 Coefficients PredictionErrorCurves DiscriminationMeasures Choice of Link FunctionandFlexible Links Families of Response Functions Nonparametric Estimation of Link Functions Literatureand FurtherReading Software Exercises Nonparametric Modeling and Smooth Effects Smooth Baseline Hazard Estimation SmoothLife Table Estimates AdditiveModels Time-VaryingCoefficients Penalty for Smooth Time-Varying Effects andselection Time-VaryingEffects and AdditiveModels Inclusionof CalendarTime Literatureand FurtherReading Software Exercises Tree-Based Approaches Recursive Partitioning Recursive Partitioning Based on Covariate-Free Discrete Hazard Models Recursive Partitioning with Binary Outcome Ensemble Methods Bagging Random Forests Literatureand FurtherReading Software Exercises

9 Contents ix 7 High-Dimensional Models: Structuring and Selection of Predictors Penalized Likelihood Approaches Boosting GenericBoosting AlgorithmforArbitraryOutcomes Application to Discrete Hazard Models Extensionto AdditivePredictors Literatureand FurtherReading Software Exercises Competing Risks Models ParametricModels MultinomialLogitModel OrderedTarget Events GeneralForm SeparateModelingof Single Targets Maximum Likelihood Estimation Variable Selection Literatureand FurtherReading Software Exercises Frailty Models and Heterogeneity Discrete Hazard Frailty Model Individual and Population Level Hazard Basic Frailty Model Including Covariates Modeling with Frailties Estimation of Frailty Models Extensions to Additive Models Including Frailty Variable Selection in Frailty Models Fixed-EffectsModel Finite MixtureModels Extensions to Covariate-Dependent Mixture Probabilities The Cure Model Estimationfor Finite Mixtures Sequential Models in Item Response Theory Literatureand FurtherReading Software Exercises Multiple-Spell Analysis Multiple Spells Estimation Multiple Spells as Repeated Measurements

10 x Contents 10.3 GeneralizedEstimation Approachto Repeated Measurements Literatureand FurtherReading Software Exercises References List of Examples Subject Index Author Index