Survival Analysis

Statistics for Biology and Health

Survival Analysis

David G. Kleinbaum

Mitchel Klein

A Self-Learning Text

Third Edition

Statistics for Biology and Health

Series Editors

M. Gail, K. Krickeberg, J.M. Samet, A. Tsiatis, W. Wong

For further volumes:

http://www.springer.com/series/2848

David G. Kleinbaum

Mitchel Klein

Survival Analysis

A Self‐Learning Text

Third Edition

Series Editors

M. Gail

National Cancer Institute

Rockville, MD 20892

USA

K. Krickeberg

Le Chatelet €

F-63270 Manglieu

France

J.M. Samet

Department of Epidemiology

School of Public Health

Johns Hopkins University

615 Wolfe Street

Baltimore, MD 21205

USA

SAS1 and all other SAS Institute Inc. product or service names are registered trademarks or

trademarks of SAS Institute Inc. in the USA and other countries. 1 indicates USA registration.

SPSS1 is a registered trademark of SPSS Inc.

STATA1 and the STATA1 logo are registered trademarks of StataCorp LP.

ISSN 1431-8776

ISBN 978-1-4419-6645-2 e-ISBN 978-1-4419-6646-9

DOI 10.1007/978-1-4419-6646-9

Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011938018

# Springer Science+Business Media, LLC 1996, 2005, 2012

All rights reserved. This work may not be translated or copied in whole or in part without the written

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David G. Kleinbaum

Department of Epidemiology

Rollins School of Public Health at

Emory University

1518 Clifton Road NE

Atlanta, GA 30322

USA

[email protected]

A. Tsiatis W. Wong

Department of Statistics Department of Statistics

North Carolina State University Stanford University

Raleigh, NC 27695 Stanford, CA 94305

USA USA

Mitchel Klein

Department of Epidemiology

Rollins School of Public Health at

Emory University

1518 Clifton Road NE

Atlanta, GA 30322

USA

[email protected]

To Rachel Robinson

Morris Dees

Aung San Suu Kyi

John Lewis

And

countless other persons, well-known or unknown,

who have had the courage to stand up for their beliefs for the

benefit of humanity.

Preface

This is the third edition of this text on survival analysis,

originally published in 1996. As in the first and second

editions, each chapter contains a presentation of its topic

in “lecture-book” format together with objectives, an out￾line, key formulae, practice exercises, and a test. The “lec￾ture-book” format has a sequence of illustrations and

formulae in the left column of each page and a script in

the right column. This format allows you to read the script

in conjunction with the illustrations and formulae that

highlight the main points, formulae, or examples being

presented.

This third edition has expanded the second edition by

adding one new chapter, additional sections and clarifica￾tions to several chapters, and a revised computer appendix.

The new chapter is Chapter 10, “Design Issues for

Randomized Trials,” which considers how to compute

sample size when designing a randomized trial involving

time-to-event data.

We have expanded Chapter 1 to clarify the distinction

between random, independent, and noninformative cen￾soring assumptions often made about survival data. We

also added a section in Chapter 1 that introduces the

Counting Process data layout that is discussed in later

chapters (3, 6, and 8).

We added sections in Chapter 2 to describe how to obtain

confidence intervals for the Kaplan–Meier (KM) curve and

the median survival time obtained from a KM curve.

We have expanded Chapter 3 on the Cox Proportional

Hazards (PH) Model by describing the use of age as the

time scale instead of time-on-follow-up as the outcome

variable. We also added a section that clarifies how to

obtain confidence intervals for PH models that contain

product terms that reflect effect modification of exposure

variables of interest.

vii

We have added sections that describe the derivation of the

(partial)likelihood functions for the stratified Cox (SC)model

in Chapter 5 and the extended Cox model in Chapter 6.

We have expanded Chapter 9 on competing risks to

describe the Fine and Gray model for a subdistribution

hazard that allows for a multivariable analysis involving a

cumulative incidence curve (CIC). We also added a numer￾ical example to illustrate the calculation of a conditional

probability curve (CPC) defined from a CIC.

The Computer Appendix in the second edition of this text

provided step-by-step instructions for using the computer

packages STATA, SAS, and SPSS to carry out the survival

analyses presented in the main text. We expanded this

Appendix to include the free internet-based computer soft￾ware package call R. We have also updated our description

of STATA (version 10.0), SAS (version 9.2), and SPSS

(version PASW 18). The application of these computer

packages to survival data is described in separate self￾contained sections of the Computer Appendix, with the

analysis of the same datasets illustrated in each section.

In addition to the above new material, the original nine

chapters have been modified slightly to correct for errata

in the second edition and to add or modify exercises

provided at the end of some chapters.

The authors’ Web site for this textbook has the following

Web-link: http://www.sph.emory.edu/dklein/surv3.htm.

This Web site includes information on how to order this

second edition from the publisher and a freely download￾able zip-file containing data-files for examples used in the

textbook.

Suggestions

for Use

This text was originally intended for self-study, but in the

15 years since the first edition was published, it has also

been effectively used as a text in a standard lecture-type

classroom format. The text may also be used to supplement

material covered in a course or to review previously

learned material in a self-instructional course or self￾planned learning activity. A more individualized learning

program may be particularly suitable to a working profes￾sional who does not have the time to participate in a regu￾larly scheduled course.

viii Preface

In working with any chapter, the learner is encouraged

first to read the abbreviated outline and the objectives

and then work through the presentation. The reader is

then encouraged to read the detailed outline for a summary

of the presentation, work through the practice exercises,

and, finally, complete the test to check what has been

learned.

Recommended

Preparation

The ideal preparation for this text on survival analysis is a

course on quantitative methods in epidemiology and a

course in applied multiple regression. Also, knowledge of

logistic regression, modeling strategies, and maximum￾likelihood techniques is crucial for the material on the

Cox and parametric models described in Chapters 3–9.

Recommended references on these subjects, with sug￾gested chapter readings are:

Kleinbaum D, Kupper L, Nizam A, and Muller K, Applied

Regression Analysis and Other Multivariable Methods,

Fourth Edition, Cengage Publishers, 2007, Chapters 1–16,

22–23.

Kleinbaum D, Kupper L and Morgenstern H, Epidemio￾logic Research: Principles and Quantitative Methods, John

Wiley and Sons, Publishers, New York, 1982, Chapters

20–24.

Kleinbaum D and Klein M, Logistic Regression: A Self￾Learning Text, Third Edition, Springer Publishers,

New York, 2010, Chapters 4–7, 11.

Kleinbaum D, ActivEpi-A CD Rom Electronic Textbook on

Fundamentals of Epidemiology, Springer Publishers,

New York, 2002, Chapters 13–15.

A first course on the principles of epidemiologic research

would be helpful, since all chapters in this text are written

from the perspective of epidemiologic research. In parti￾cular, the reader should be familiar with the basic charac￾teristics of epidemiologic study designs, and should have

some idea of the frequently encountered problem of

controlling for confounding and assessing interaction/

effect modification. The above reference, ActivEpi, pro￾vides a convenient and hopefully enjoyable way to review

epidemiology.

Preface ix

Acknowledgments

We thank Dr. Val Gebski of the NHMRC Clinical Trials

Centre, Sydney, Australia for providing continued insight

on current methods of survival analysis and review of new

additions to the manuscript for this edition.

Finally, David Kleinbaum and Mitch Klein thank Edna

Kleinbaum and Becky Klein for their love, support, com￾panionship, and sense of humor during the writing of this

third edition.

xi

Contents

Preface vii

Acknowledgments xi

Chapter 1 Introduction to Survival Analysis 1

Introduction 2

Abbreviated Outline 2

Objectives 3

Presentation 4

Detailed Outline 44

Practice Exercises 50

Test 52

Answers to Practice Exercises 54

Chapter 2 Kaplan-Meier Survival Curves

and the Log-Rank Test 55

Introduction 56

Abbreviated Outline 56

Objectives 57

Presentation 58

Detailed Outline 83

Practice Exercises 87

Test 91

Answers to Practice Exercises 93

Chapter 3 The Cox Proportional Hazards

Model and Its Characteristics 97

Introduction 98

Abbreviated Outline 98

Objectives 99

Presentation 100

Detailed Outline 145

Practice Exercises 149

Test 153

Answers to Practice Exercises 157

xiii

Chapter 4 Evaluating the Proportional Hazards

Assumption 161

Introduction 162

Abbreviated Outline 162

Objectives 163

Presentation 164

Detailed Outline 188

Practice Exercises 191

Test 194

Answers to Practice Exercises 197

Chapter 5 The Stratified Cox Procedure 201

Introduction 202

Abbreviated Outline 202

Objectives 203

Presentation 204

Detailed Outline 228

Practice Exercises 231

Test 234

Answers to Practice Exercises 237

Chapter 6 Extension of the Cox Proportional

Hazards Model for Time-Dependent

Variables 241

Introduction 242

Abbreviated Outline 242

Objectives 243

Presentation 244

Detailed Outline 278

Practice Exercises 281

Test 285

Answers to Practice Exercises 287

Chapter 7 Parametric Survival Models 289

Introduction 290

Abbreviated Outline 290

Objectives 291

Presentation 292

Detailed Outline 345

Practice Exercises 351

Test 356

Answers to Practice Exercises 359

xiv Contents