Competing Risks. Competing Risks: A Practical Perspective M. Pintilie 2006 John Wiley &Sons, Ltd. ISBN:

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1 Competing Risks Competing Risks: A Practical Perspective M. Pintilie 2006 John Wiley &Sons, Ltd. ISBN:

2 STATISTICS IN PRACTICE Advisory Editors Stephen Senn University of Glasgow, UK Marian Scott University of Glasgow, UK Peter Bloomfield North Carolina State University, USA Founding Editor Vic Barnett Nottingham Trent University, UK Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study. With sound motivation and many worked practical examples, the books show in down-to-earth terms how to select and use an appropriate range of statistical techniques in a particular practical field within each title s special topic area. The books provide statistical support for professionals and research workers across a range of employment fields and research environments. Subject areas covered include medicine and pharmaceutics; industry, finance and commerce; public services; the earth and environmental sciences, and so on. The books also provide support to students studying statistical courses applied to the above areas. The demand for graduates to be equipped for the work environment has led to such courses becoming increasingly prevalent at universities and colleges. It is our aim to present judiciously chosen and well-written workbooks to meet everyday practical needs. Feedback of views from readers will be most valuable to monitor the success of this aim. A complete list of titles in this series appears at the end of the volume.

3 Competing Risks A Practical Perspective Melania Pintilie Ontario Cancer Institute, Canada

4 Copyright 2006 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) (for orders and customer service enquiries): Visit our Home Page on All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or ed to permreq@wiley.co.uk, or faxed to (+44) Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA , USA Wiley-VCH Verlag GmbH, Boschstr. 12, D Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore John Wiley & Sons Canada Ltd, Freemont Blvd, Mississauga, ONT, L5R 4J3 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging in Publication Data Pintilie, Melania. Competing risks : a practical perspective / Melania Pintilie. p. cm. Includes bibliographical references and index. ISBN-13: (cloth : alk. paper) ISBN-10: (cloth : alk. paper) 1. Competing risks. I. Title. QA273.P dc British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN (HB) ISBN (HB) Typeset in 11/13pt Photina by Integra Software Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

5 What we have to learn to do, we learn by doing. Aristotle

6 Contents Preface Acknowledgements xi xiii 1 Introduction Historical notes Defining competing risks Use of the Kaplan Meier method in the presence of competing risks Testing in the competing risk framework Sample size calculation Examples Tamoxifen trial Hypoxia study Follicular cell lymphoma study Bone marrow transplant study Hodgkin s disease study 18 2 Survival basic concepts Introduction Definitions and background formulae Introduction Basic mathematical formulae Common parametric distributions Censoring and assumptions Estimation and hypothesis testing Estimating the hazard and survivor functions Nonparametric testing: log-rank and Wilcoxon tests Proportional hazards model Software for survival analysis Closing remarks 38 3 Competing risks definitions Recognizing competing risks Practical approaches Common endpoints in medical research 40 vii

7 viii Contents 3.2 Two mathematical definitions Competing risks as bivariate random variable Competing risks as latent failure times Fundamental concepts Competing risks as bivariate random variable Competing risks as latent failure times Discussion of the two approaches Closing remarks 51 4 Descriptive methods for competing risks data Product-limit estimator and competing risks Cumulative incidence function Heuristic estimation of the CIF Nonparametric maximum likelihood estimation of the CIF Calculating the CIF estimator Variance and confidence interval for the CIF estimator Software and examples Using R Using SAS Closing remarks 70 5 Testing a covariate Introduction Testing a covariate Gray s method Pepe and Mori s method Software and examples Using R Using SAS Closing remarks 84 6 Modelling in the presence of competing risks Introduction Modelling the hazard of the cumulative incidence function Theoretical details Model-based estimation of the CIF Using R Cox model and competing risks Checking the model assumptions Proportionality of the cause-specific hazards Proportionality of the hazards of the CIF Linearity assumption Closing remarks 113

8 Contents ix 7 Calculating the power in the presence of competing risks Introduction Sample size calculation when competing risks are not present Calculating power in the presence of competing risks General formulae Comparing cause-specific hazards Comparing hazards of the subdistributions Probability of event when the exponential distribution is not a valid assumption Examples Introduction Comparing the cause-specific hazard Comparing the hazard of the subdistribution Closing remarks Other issues in competing risks Conditional probability function Introduction Nonparametric estimation of the CP function Variance of the CP function estimator Testing a covariate Using R Using SAS Comparing two types of risk in the same population Theoretical background Using R Discussion Identifiability and testing independence Parametric modelling Introduction Modelling the marginal distribution Modelling the Weibull distribution Food for thought 149 Problem 1: Estimation of the probability of the event of interest 149 Problem 2: Testing a covariate 152 Problem 3: Comparing the event of interest between two groups when the competing risks are different for each group 154 Problem 4: Information needed for sample size calculations 158 Problem 5: The effect of the size of the incidence of competing risks on the coefficient obtained in the model 159 Problem 6: The KLY test and the non-proportionality of hazards 161 Problem 7: The KLY and Wilcoxon tests 162

9 x Contents A Theoretical background 165 A.1 Nonparametric maximum likelihood estimation for the survivor function in the discrete case 165 A.2 Confidence interval for survivor function 167 A.3 The Variance for Gray s test 168 A.4 Derivation of the parameters for the exponential latent failure time model 169 A.5 Likelihood of a mixture of exponentials in the bivariate approach 171 B Analysing competing risks data using R and SAS 173 B.1 The R software and cmprsk package 173 B.1.1 Downloading and installation 173 B.1.2 Getting help 174 B.1.3 Operators in R 175 B.1.4 Objects in R 175 B.1.5 The cmprsk package and datasets 178 B.2 Importing datasets in SAS 181 B.3 Other programs written for R 182 B.3.1 CIF variance based on the delta method 182 B.3.2 Pepe Mori test for the difference between two CIFs 183 B.3.3 Conditional probability and its variance 185 B.3.4 Plotting the conditional probability 187 B.3.5 Testing the conditional probability 187 B.3.6 Calculating the conditional probability of observing the event of interest in a time period knowing that the patient was free of any event at the beginning of the period 189 B.3.7 Comparing two types of risk in the same population 191 B.3.8 Calculating the power 192 B.4 SAS macros for competing risk analysis 195 B.4.1 Cumulative incidence and conditional probability 195 B.4.2 Pepe Mori test for the comparison of two CIFs 199 B.4.3 Pepe Mori test for the comparison of two CPs 203 References 209 Index 215

10 Preface This book describes the methodology for the analysis of competing risks from the applied perspective. Although most of the examples are drawn from medical research, situations involving competing risks occur in many other areas, such as sociology, economics and engineering. Chapter 1 is a non-mathematical introduction to the topic of competing risks. The problems involved in the analysis of competing risks need to be understood by both the statistician and the scientist. Thus, this chapter gives enough detail to enable a meaningful discussion to take place between the scientist and the applied statistician. Some general techniques for survival analysis are included to facilitate the understanding of the theory of competing risks (Chapter 2). Chapters 3 6 cover the analysis of competing risks using nonparametric or semiparametric techniques. The sample size calculations under different scenarios are presented in Chapter 7. A brief overview of other aspects of the analysis of competing risks is included in Chapter 8, and examples are provided in Chapter 9 to complement the material in Chapter 1. Although Chapters 2 8 contain a certain amount of mathematical detail, an understanding of the concepts can be achieved through studying the examples given. For ease of exposition, subscripts are omitted whenever possible. Appendix A contains the mathematical proofs which were considered too elaborate to be included in the respective chapters. Details on how to perform the analyses are included in each chapter and are supplemented with R functions and SAS macros incorporated in Appendix B. For those readers who wish to extend their knowledge of the topic of competing risks, references to current theoretical developments are provided at the end of each chapter. xi

11 xii Preface The datasets, the supplementary R functions and the SAS macros used in this book can be downloaded from I would be grateful to be informed of any errors, ambiguities or oversights found in this book. Melania Pintilie pintilie@uhnres.utoronto.ca

12 Acknowledgements I would first like to express my warmest gratitude to Ellen Maki whose participation in shaping this book was crucial. Ellen contributed substantially to Chapters 2 and 4 and had many suggestions throughout. Richard Hill s help was invaluable. He thoroughly reviewed the material and generously shared his experience and expertise. I thank Janet Raboud for her careful reading and the highly competent advice she gave me. Richard and Janet s helpful comments contributed considerably to enhancing the clarity of the presentation of the material. I would also like to acknowledge the input given by Gina Lockwood and Ruth Croxford who kindly reviewed parts of this manuscript. Finally, I express my appreciation to the anonymous reviewers who read the manuscript and made many helpful comments and suggestions. Special thanks go to Tony Panzarella who, as the manager of the Biostatistics Department of the Princess Margaret Hospital/Ontario Cancer Institute, reviewed the work and encouraged this endeavour from its earliest stages. The datasets used were from the Princess Margaret Hospital, a leading cancer centre in North America. I appreciate the generosity of the researchers, Mary Gospodarowicz, Anthony Fyles and Hans Messner, who graciously agreed to let me use their datasets as examples. It was a pleasure to work with the team at Wiley, notably Rob Calver, who sowed the seeds for this book, and Kathryn Sharples, Simon Lightfoot and Wendy Hunter, who diligently helped me accomplish this project. Last, but not least, I wish to thank my husband, Alex, whose help and support gave me the necessary confidence to continue. xiii

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