Survival Models and Data Analysis Survival Models and Data Analysis REGINA C.ELANDT-JOHNSON DepartmentofBiostatistics School ofPublic Health UniversityofNorth Carolina at Chapel Hill NORMANL. JOHNSON DepartmentofStatistics UniversityofNorth Carolina at Chapel Hill WILEYCLASSICSLIBRARY EDITIONPUBLISHED 1999 AWiley-IntersciencePublication JOHNWILEY& SONS, INC. NewYork • Chichester • Weinheim • Brisbane • Singapore • Toronto This textisprinted onacid-free paper.@ Copyright©1980byJohnWiley& Sons,lnc.Allrightsreserved. PublishedsimultaneouslyinCanada. WileyClassicsLibrary EditionPublished 1999. Nopartofthispublicationmaybereproduced, stored inaretrievalsystemortransmitted inany formorbyanymeans,electronic, mechanical, photocopying,recording,scanningorotherwise, exceptaspermittedunderSection 107or 108ofthe1976UnitedStatesCopyright Act,without eitherthepriorwrittenpermission ofthePublisher,orauthorizationthroughpaymentofthe appropriateper-copy feetotheCopyrightClearanceCenter,222Rosewood Drive,Danvers, MA 01923, (978) 750-8400, fax(978)750-4744. RequeststothePublisherforpermission shouldbe addressedtothePermissionsDepartment,JohnWiley& Sons,Inc.,605ThirdAvenue,NewYork, NY 10158-0012, (212)850-6011, fax(212)850-6008, E-Mail:[email protected]. LibraryofCongress CataloginginPublication Data: Elandt-Johnson,ReginaC.1918- Survival modelsanddataanalysis. (Wiley series inprobability andmathematical statistics,appliedsection) Includesindex. I.Failuretimedataanalysis. 2.Mortality. 3.Medical statistics. 4.Competingrisks. I.Johnson,Norman Lloyd,joint author. II.Title. QA276.E39 312'.01'51 79-22836 ISBN0-471-03174-7 ISBN0-471-34992-5(WileyClassics Paperback Edition) 10 9 8 7 6 5 4 3 2 I Preface This book contains material and techniques developed in several different disciplines: vital statistics, epidemiology, demography, actuarial science, reliability theory, statistical methods, among others. Despite this diversity of origin, these techniques are all relevant to aspects of the analysis of survival data. Survivaldata can take so many different forms, spanning from results of small-scale laboratory tests to massive records from long-term clinical trials. Therefore it is impossible to lay down universal rules of procedure. Attempts to do so, even if apparently successful, are likely to lead to an uncritical, authoritarian approach, following whatever is currently re- garded as the "correct approach." We have tried to set out general principles to be used in each particular case. A number of the exercises require considerable independent thought and cannot be said to have a unique "correct" answer. They call rather for sound appraisal of a situa- tion. We have also endeavored to avoid the useof hidden assumptions. Much of our analysis is, indeed, based on assumptions (of independence, stabil- ity, etc.), but wehave always sought to make it clear what assumptions are being made, and we encourage the reader to consider what might be the effects of departures from these assumptions. The remarkable increase of activity in the statistical analysis of survival data over the last two decades, largely stimulated by problems arising in the analysis of clinical trials, has resulted in a considerable volume of writing on the topic. A major purpose of this book is to act as a guide for using this literature, assist in the choice of appropriate methods, and warn against uncritical use. The content of the book might be subclassified according to several different criteria-statistical approach, relevant scientific disciplines, types of applications, and so on. We have, in fact, divided the book into four broad parts. v vi Part I introduces the type of data to be analyzed and basic concepts used in their analysis. Part 2 deals with problems related to univariate survival functions. These include construction of life tables from population (cross-sectional) data and from experimental-type follow-up data. Considerable space is devoted to fitting parametric distributions and comparisons of two or more mortality experiences. Part 3 is concerned with multiple-failure data. Time as well as cause of death are identified. Parametric and nonparametric theories of competing causes and estimation of different kinds of failure distributions are pre- sented in some detail. Part 4 presents some more advanced topics, including speculative mathematical models of biological processes of disease progression and aging. These are not intended to be definitive. Rather, they give the reader some ideas of ways in which models may be constructed. Some readers may find the mathematical level uneven. This is because mathematical techniques are used as they are needed, and never for their own sake. We take this opportunity to acknowledge the help we have received while working on this book. The typing was done by Joyce Hill (in major part), June Maxwell, and Mary Riddick. We are especially grateful to Anna Colosi, who did all the calculations and obtained graphical presenta- tions using an electronic computer. REGINA C. ELANDT-JOHNSON NORMAN L. JOHNSON ChapelHill, North Carolina October 1979 Contents PART 1. SURVIVAL MEASUREMENTS ANDCONCEPTS 1. SURVIVAL DATA 3 1.1 Scope of the Book 3 1.2 Sources of Data 4 1.3 Types ofVariables 5 1.4 Exposure to Risk 6 1.5 Use of ProbabilityTheory 6 1.6 The Collection of Survival Data 7 2. MEASURES OF MORTALITY AND MORBIDITY. RATIOS, PROPORTIONS, AND MEANS 9 2.1 Introduction 9 2.2 Ratiosand Proportions to 2.2.1 Ratios to 2.2.2 Proportion 11 2.3 Rates ofContinuous Processes 12 2.3.1 Absolute Rate 12 2.3.2 Relative Rate 13 2.3.3 Average (Central) Rate 14 2.4 Rates for Repetitive Events 16 2.5 Crude Birth Rate 17 2.6 Mortality Measures Used in Vital Statistics 18 2.6.1 The Conceptof Population Exposed to Risk 18 2.6.2 Crude Death Rate 20 2.6.3 Age Specific Death Rates 21 2.6.4 Cause Specific Mortality Used in Vital Statistics 21 2.7 Relationships Between Crude and Age Specific Rates 22 2.8 Standardized Mortality Ratio (SMR): Indirect Standardization 22 vii Cooteobi 2.9 Direct Standardization 25 2.10 Evaluation of Person-Years of Exposed to Risk in Long-Term Studies 25 2.10.1 'Exact' Dates for Each Individual Available 26 2.10.2 Only Years of Birth, Entry, and Departure Available 29 2.11 Prevalence and Incidence of a Disease 31 2.11.1 Prevalence 31 2.11.2 Incidence 32 2.12 Association Between Disease and Risk Factor. Relative Risk and Odds Ratio 35 2.12.1 Relative Risk 36 2.12.2 Odds Ratio 37 3. SURVIVAL DISTRIBUTIONS 50 3.1 Introduction 50 3.2 Survival Distribution Functions 50 3.3 Hazard Function (Force of Mortality) 51 3.4 Conditional Probabilities of Death (Failure) and Central Rate 52 3.5 Truncated Distributions 53 3.6 Expectation and Variance of Future Lifetime 55 3.7 Median of Future Lifetime 56 3.8 Transformations of Random Variables 57 3.9 Location-Scale Families of Distributions 58 3.10 Some Survival Distributions 59 3.11 Some Modelsof Failure 64 3.11.1 SeriesSystem 64 3.11.2 Parallel System 65 3.12 Probability Integral Transformation 66 3.13 Compound Distributions 67 3.14 Miscellanea 67 3.14.1 Interpolation 67 3.14.2 Method of Statistical Differentials 69 3.15 Maximum Likelihood Estimation and Likelihood Ratio Tests 72 3.15.1 Construction of Likelihood Functions 73 3.15.2 Maximum Likelihood Estimation 73 3.15.3 Expected Values, Variances and Covariances of the MLE's 74 3.15.4 Assessing Goodness of Fit 74