Survival Models and Data Analysis

478 Pages·1980·14.16 MB·English

by  Regina C. Elandt-Johnson, Norman L. Johnson

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
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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