Survival Models
and Data Analysis
REGINA C. ELANDT-JOHNSON
Department of Biostatistics
School of Public Health
University of North Carolina at Chapel Hill
NORMAN L. JOHNSON
Department of Statistics.
University of North Carolina at Chapel Hill
JOHN WILEY AND SONS,
New York • Chichester • Brisbane • Toronto • Singapore
Contents
PART 1. SURVIVAL MEASUREMENTS AND CONCEPTS
1.
SURVIVAL DATA
1.1
1.2
1.3
1.4
1.5
1.6
2.
Scope of the Book
Sources of Data
Types of Variables
Exposure to Risk
Use of Probability Theory
The Collection of Survival Data
MEASURES OF MORTALITY AND MORBIDITY, RATIOS,
PROPORTIONS, AND MEANS
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Introduction
Ratios and Proportions
2.2.1
Ratios
2.2.2
Proportion
Rates of Continuous Processes
2.3.1
Absolute Rate
2.3.2
Relative Rate
2.3.3
Average (Central) Rate
Rates for Repetitive Events
Crude Birth Rate
Mortality Measures Used in Vital Statistics
2.6.1
The Concept of Population Exposed to Risk
2.6.2
Crude Death Rate
2.6.3
Age Specific Death Rates
2.6.4
Cause Specific Mortality Used in Vital Statistics
Relationships Between Crude and Age Specific Rates
Standardized Mortality Ratio (SMR): Indirect
Standardization
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2.9
Direct Standardization
2.10 Evaluation of Person-Years of Exposed to Risk in
Long-Term Studies
2.10.1 'Exact'Dates for Each Individual Available
2.10.2 Only Years of Birth, Entry, and Departure
Available
2.11 Prevalence and Incidence of a Disease
2.11.1 Prevalence
2.11.2 Incidence
2.12 Association Between Disease and Risk Factor.
Relative Risk and Odds Ratio
2.12.1 Relative Risk
2.12.2 Odds Ratio
3.
SURVIVAL DISTRIBUTIONS
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
Introduction
Survival Distribution Functions
(
Hazard Function (Force of Mortality)
Conditional Probabilities of Death (Failure) and
Central Rate
Truncated Distributions
Expectation and Variance of Future Lifetime
Median of Future Lifetime
Transformations of Random Variables
Location-Scale Families of Distributions
Some Survival Distributions
Some Models of Failure
3.11.1 Series System
3.11.2 Parallel System
Probability Integral Transformation
Compound Distributions
Miscellanea
3.14.1 Interpolation
3.14.2 Method of Statistical Differentials
Maximum Likelihood Estimation and Likelihood
Ratio Tests
3.15.1 Construction of Likelihood Functions
3.15.2 Maximum Likelihood Estimation
3.15.3 Expected Values, Variances and Covariances
-of the MLE's
3.15.4 Assessing Goodness of Fit
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Contents
™
PART 2. MORTALITY EXPERIENCES AND LIFE TABLES
4.
LIFE TABLES: FUNDAMENTALS AND CONSTRUCTION
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
5.
Introduction
•
Life Table: Basic Definition and Notation
Force of Mortality. Mathematical Relationships
Among Basic Life Table Functions
Central Death Rate
Interpolation for Life Table Functions
Some Approximate Relationships Between nqx and
„**
4.6.1
Expected Fraction of the Last n Years of Life
4.6.2
Special Cases
4.6.3
Exponential Approximation
Some Approximations to ju.x
Concepts of Stationary and Stable Populations
4.8.1
Stationary Population
4.8.2
Stable Population
Construction of an Abridged Life Table from
Mortality Experience of a Current Population
4.9.1
Evaluation of nMx
4.9.2
Estimation of Jx
4.9.3
Estimation of nqx
4.9.4
Evaluation of the Life Table Functions
Some Other Approximations Used in Construction of
Abridged Life Tables
Construction of a Complete Life Table
From an Abridged Life Table
Selection
Select Life Tables
Some Examples
Construction of Select Tables
COMPLETE MORTALITY DATA, ESTIMATION OF SURVIVAL
FUNCTION
5.1
5.2
5.3
Introduction. Cohort Mortality Data
Empirical Survival Function
Estimation of Survival Function
From Grouped Mortality Data
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Contents
5.3.1
5.3.2
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
6.
Grouping into Fixed Intervals
Grouping Based on Fixed Numbers of
Deaths
Joint Distribution of the Numbers of Deaths
Distribution of Pt
Covariance of Pt and Py (i <j)
Conditional Distribution of q{
Greenwood's Formula for the (Conditional) Variance
of P,
Estimation of Curve of Deaths
Estimation of Central Death Rate and Force of
Mortality in [/,,/,+ 1)
Summary of Results
INCOMPLETE MORTALITY DATA: FOLLOW-UP STUDIES
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Basic Concepts and Terminology
6.1.1
Incomplete Data
6.1.2
Truncation and Censoring
6.1.3
Follow-up Studies
6.1.4
Other Kinds of Follow-up
6.1.5
Topics of the Chapter
Actuarial Estimator of q{ From Grouped Data
6.2.1
Amount of Person-Time Units. Central
Death Rates
6.2.2
Effective Number of Initial Exposed to Risk.
Estimation of qt
6.2.3
Estimation of Survival Function
Some Maximum Likelihood Estimators of qt
6.3.1
Failure Time Alone Regarded as a Random
Variable
6.3.2
Failure Time and Censoring Time Regarded
as Random Variables
Some Other Estimators of qt
6.4.1
Moment Estimator of qt
6.4.2
Estimator of qi Based on Reduced Sample
Comparison of Various Estimators of qt
Estimation of Curve of Deaths
Product-Limit Method of Estimating the Survival
Function From Individual Times at Death
Estimation of Survival Function Using the
Cumulative Hazard Function
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Contents
7.
xl
FITTING PARAMETRIC SURVIVAL DISTRIBUTIONS
7.1
7.2
Introduction
Some Methods of Fitting Parametric Distribution
Functions
7.3
Exploitation of Special Forms of Survival Function
7.4
Fitting Different Distribution Functions over
Successive Periods of Time
7.5
Fitting a "Piece-Wise" Parametric Model to a Life
Table: An Example
7.6
Mixture Distributions
7.7
Cumulative Hazard Function Plots—Nelson's
Method for Ungrouped Data
•
7.7.1
Complete Data
7.7.2
Incomplete Data
7.8
Construction of the Likelihood Function for Survival
Data: Some Examples
7.9
Minimum Chi-Square and Minimum Modified ChiSquare
7.10 Least Squares Fitting
7.11 Fitting a Gompertz Distribution to Grouped Data:
An Example
7.12 Some Tests of Goodness of Fit
7.12.1 Graphical'Test"
7.12.2 Kolmogorov-Smirnov Statistics. Limiting
Distribution
7.12.3 Anderson-Darling ^-Statistic
7.12.4 Chi-Square Test for Grouped Data
8.
COMPARISON OF MORTALITY EXPERIENCES
8.1
8.2
8.3
8.4
Introduction
Comparison of Two Life Tables
8.2.1
Graphical Displays
8.2.2
Conditional Probabilities qx
8.2.3
Conditional Expectations and Median of
Future Lifetime
Comparison of Mortality Experience with a
Population Life Table
8.3.1
Test Based on Median of Future Lifetime
8.3.2
Test Based on Expected Future Lifetime
Some Distribution-Free Methods for Ungrouped
Data
_
8.4.1
Two Sample Kolmogorov-Smirnov Test
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Contents
8.4.2
Two Sample Wilcoxon Test for Complete
Data
8.4.3
Modified Wilcoxon Tests for Incomplete
Data
8.5
Special Problems Arising in Clinical Trials and
Progressive Life Testing
8.5.1
Early Decision
8.5.2
Multistage Testing
8.5.3
Testing for Trends in Mortality Patterns
8.5.4
Staggered Entries and Withdrawals
8.6
Censored Kolmogorov-Smirnov (or Tsao-Conover)
Test
8.7
Truncated Data. Pearson's Conditional X2 Test
8.7.1
Two Sample Problem .
8.7.2
Extension to k Treatments
8.8
Testing for Consistent Differences in Mortality.
Mantel-Haenszel and Logrank Tests
8.8.1
Mantel-Haenszel Test
8.8.2
Logrank Test
8.8.3
Extension to r Experiments (Classes)
8.9
Parametric Methods
8.10 Sequential Methods
PART 3. MULTIPLE TYPES OF FAILURE
9.
THEORY OF COMPETING CAUSES: PROBABILISTIC APPROACH
9.1
9.2
9.3
9.4
9.5
9.6
Causes of Death: Basic Assumptions
Some Basic Problems
"Times Due to Die"
The Overall and 'Crude' Survival Functions
9.4.1
The Overall Survival Function
9.4.2
The Crude and Net Hazard Rates
9.4.3
The Crude Probability Distribution for Cause
ca
Case when Xl,...,Xk are Independent
Equivalence and Nonidentifiability Theorems in
Competing Risks
9.6.1
Equivalent Models of Survival Distribution
"Functions
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Contents
9.6.2
9.7
9.8
9.9
10.
M U L T I P L E DECREMENT LIFE TABLES
10.1
10.2
10.3
10.4
10.5
10.6
11.
Nonidentifiability of the Member of a
Parametric Family of Distributions
Proportional Hazard Rates
Examples
Heterogeneous Populations: Mixture of Survival
Functions
Multiple Decrement Life Tables: Notation
Definitions of the M D L T Functions
Relationships A m o n g Functions of Multiple
Decrement Life Tables
Crude Forces of Mortality
Construction of Multiple Decrement Life Tables from
Population (Cross-Sectional) Mortality Data
10.5.1 Mortality D a t a : Evaluation of naqx a n d ^q^
10.5.2 Construction of M D L T
Some Major Causes of Death: An Example of
Constructing the M D L T
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SINGLE DECREMENT LIFE TABLES ASSOCIATED W I T H MULTIPLE
DECREMENT LIFE TABLES: THEIR INTERPRETATION AND
MEANING
309
11.1
11.2
309
11.3
12.
Elimination, Prevention, a n d Control of a Disease
Mortality Pattern from Cause Ca Alone: 'Private'
Probabilities of Death
11.2.1 How Might the S D F from Cause Ca Alone be
Interpreted
11.2.2 Estimable, Although not Observable, Waiting
Time Distributions
Estimation of Waiting Time Distribution for Cause
Ca: Single Decrement Life Table
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ESTIMATION AND TESTING HYPOTHESES IN COMPETING RISK
ANALYSIS
12.1
Introduction. Experimental D a t a
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Contents
12.2
12.3
12.4
12.5
PART 4.
13.
Grouped Data. Nonparametric Estimation
12.2.1 Complete Data
12.2.2 Follow-up Data
12.2.3 Truncated Samples
Grouped Data. Fitting Parametric Models
12.3.1 Choice of a Joint Survival Function
12.3.2 Fitting Crude Parametric Distribution to
Mortality Data from Each Cause Separately
Cohort Mortality Data with Recorded Times at
Death or Censoring. Nonparametric Estimation
12.4.1 Complete Data
12.4.2 Incomplete Data
Cohort Mortality Data with Recorded Times at
Death or Censoring. Parametric Estimation
13.3
13.4
13.5
13.6
13.7
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SOME MORE ADVANCED TOPICS
CONCOMITANT VARIABLES IN LIFETIME DISTRIBUTIONS MODELS
13.1
13.2
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Concomitant Variables
The Role of Concomitant Variables in Planning
Clinical Trials
General Parametric Model of Hazard Function with
Observed Concomitant Variables
13.3.1 Types of Concomitant Variables
13.3.2 General Model
13.3.3 Some Other Expressions for the General
Model
Additive Models of Hazard Rate Functions
Multiplicative Models
13.5.1 Exponential-Type Hazard Functions
13.5.2 Gompertz and Weibull Models with
Covariates
Estimation in Multiplicative Models
13.6.1 Construction of the Likelihood Function
13.6.2 Multiple Failures
Assessment of the Adequacy of a Model: Tests of
Goodness of Fit
13.7.1 Cumulative Hazard Plottings
13.7.2 ^.Method of Half-Replicates
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Contents
13.8
13.9
13.10
13.11
13.12
13.13
13.14
14.
Selection of Concomitant Variables13.8.1 Likelihood Ratio Tests of Composite
Hypotheses
12.8.2 Step-Down Procedure
13.8.3 Step-Up Procedure
Treatment-Covariate Interaction
Logistic Linear Model
Time Dependent Concomitant Variables
Concomitant Variables Regarded as Random
Variables
Posterior Distribution of Concomitant Variables
Concomitant Variables in Competing Risk Models
AGE OF ONSET DISTRIBUTIONS
14.1
14.2
14.3
14.4
14.5
14.6
15.
XT
Introduction
Models of Onset Distributions
14.2.1 Incidence Onset Distribution
14.2.2 Waiting Time Onset Distribution
14.2.3 Life Tables and Onset Distributions
Estimation of Incidence Onset Distribution from
Cross-Sectional Incidence Data
Estimation of Incidence Onset Distribution from
Prevalence Data
14.4.1 Estimation of Age Specific Incidence from
Prevalence Data: No Differential Mortality
14.4.2 Affected Individuals Are Subject to
Differential Mortality
Estimation of Waiting Time Onset Distribution from
Population Data
Estimation of Waiting Time Onset Distribution from
Retrospective Data
14.6.1 No Differential Mortality Between Affected
and Unaffected
14.6.2 Effects of Differential Mortality
MODELS OF AGING AND CHRONIC DISEASES
15.1
15.2
Introduction
Aging and Chronic Diseases
15.2.1 Biological Aging
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Contents
15.3
15.4
15.5
15.6
15.7
15.2.2 Hazard Rate: A Measure of Aging
15.2.3 Models of Chronic Diseases
Some Models of Carcinogenesis
15.3.1 A One-Hit Model of Carcinogenesis:
Gompertz Distribution
15.3.2 Multi-Hit Models of Carcinogenesis: Parallel
Systems and Weibull Distribution
Some "Mosaic" Models of a Chronic Disease
"Fatal Shock" Models of Failure
15.5.1 A Two Component Series System
15.5.2 Generalization of the "Fatal Shock" Model
to n Components
Irreversible Markov Processes in Illness-Death
Modeling
15.6.1 Basic Concepts
15.6.2 A Two Component Parallel System: Two
Distinct States Before Failure
15.6.3 Extension to /--Component Parallel System
with r States
15.6.4 A Model of Disease Progression
Reversible Models: The Fix-Neyman Model
Author index
Subject index
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