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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Probability and Statistics
- Sprache: Englisch
Praise for the Third Edition
“. . . an easy-to read introduction to survival analysis which covers the major concepts and techniques of the subject.” —Statistics in Medical Research
Updated and expanded to reflect the latest developments, Statistical Methods for Survival Data Analysis, Fourth Edition continues to deliver a comprehensive introduction to the most commonly-used methods for analyzing survival data. Authored by a uniquely well-qualified author team, the Fourth Edition is a critically acclaimed guide to statistical methods with applications in clinical trials, epidemiology, areas of business, and the social sciences. The book features many real-world examples to illustrate applications within these various fields, although special consideration is given to the study of survival data in biomedical sciences.
Emphasizing the latest research and providing the most up-to-date information regarding software applications in the field, Statistical Methods for Survival Data Analysis, Fourth Edition also includes:
- Marginal and random effect models for analyzing correlated censored or uncensored data
- Multiple types of two-sample and K-sample comparison analysis
- Updated treatment of parametric methods for regression model fitting with a new focus on accelerated failure time models
- Expanded coverage of the Cox proportional hazards model
- Exercises at the end of each chapter to deepen knowledge of the presented material
Statistical Methods for Survival Data Analysis is an ideal text for upper-undergraduate and graduate-level courses on survival data analysis. The book is also an excellent resource for biomedical investigators, statisticians, and epidemiologists, as well as researchers in every field in which the analysis of survival data plays a role.
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Contents
Preface
CHAPTER 1: Introduction
1.1 PRELIMINARIES
1.2 CENSORED DATA
1.3 SCOPE OF THE BOOK
CHAPTER 2: Functions of Survival Time
2.1 DEFINITIONS
2.2 RELATIONSHIPS OF THE SURVIVAL FUNCTIONS
EXERCISES
CHAPTER 3: Examples of Survival Data Analysis
3.1 EXAMPLE 3.1: COMPARISON OF TWO TREATMENTS AND THREE DIETS
3.2 EXAMPLE 3.2: COMPARISON OF TWO SURVIVAL PATTERNS USING LIFE TABLES
3.3 EXAMPLE 3.3: FITTING SURVIVAL DISTRIBUTIONS TO TUMOR-FREE TIMES
3.4 EXAMPLE 3.4: COMPARING SURVIVAL OF A COHORT WITH THAT OF A GENERAL POPULATION — RELATIVE SURVIVAL
3.5 EXAMPLE 3.5: IDENTIFICATION OF RISK FACTORS FOR INCIDENT EVENTS
3.6 EXAMPLE 3.6: IDENTIFICATION OF RISK FACTORS FOR THE PREVALENCE OF AGE-RELATED MACULAR DEGENERATION
3.7 EXAMPLE 3.7: IDENTIFICATION OF SIGNIFICANT RISK FACTORS FOR INCIDENT HYPERTENSION USING RELATED DATA (REPEATED MEASUREMENTS) IN A LONGITUDINAL STUDY
EXERCISES
CHAPTER 4: Nonparametric Methods of Estimating Survival Functions
4.1 PRODUCT-LIMIT ESTIMATES OF SURVIVORSHIP FUNCTION
4.2 NELSON–AALEN ESTIMATES OF SURVIVORSHIP FUNCTION
4.3 LIFE-TABLE ANALYSIS
4.4 RELATIVE SURVIVAL RATES
4.5 STANDARDIZED RATES AND RATIOS
EXERCISES
CHAPTER 5: Nonparametric Methods for Comparing Survival Distributions
5.1 COMPARISON OF TWO SURVIVAL DISTRIBUTIONS
5.2 THE MANTEL AND HAENSZEL TEST
5.3 COMPARISON OF K (K > 2) SAMPLES
EXERCISES
CHAPTER 6: Some Well-Known Parametric Survival Distributions and Their Applications
6.1 EXPONENTIAL DISTRIBUTION
6.2 WEIBULL DISTRIBUTION
6.3 LOGNORMAL DISTRIBUTION
6.4 GAMMA, GENERALIZED GAMMA, AND EXTENDED GENERALIZED GAMMA DISTRIBUTIONS
6.5 LOG-LOGISTIC DISTRIBUTION
6.6 OTHER SURVIVAL DISTRIBUTIONS
EXERCISES
CHAPTER 7: Estimation Procedures for Parametric Survival Distributions without Covariates
7.1 GENERAL MAXIMUM LIKELIHOOD ESTIMATION PROCEDURE
7.2 EXPONENTIAL DISTRIBUTION
7.3 WEIBULL DISTRIBUTION
7.4 LOGNORMAL DISTRIBUTION
7.5 THE EXTENDED GENERALIZED GAMMA DISTRIBUTION
7.6 THE LOG-LOGISTIC DISTRIBUTION
7.7 GOMPERTZ DISTRIBUTION
7.8 GRAPHICAL METHODS
EXERCISES
CHAPTER 8: Tests of Goodness-of-Fit and Distribution Selection
8.1 GOODNESS-OF-FIT TEST STATISTICS BASED ON ASYMPTOTIC LIKELIHOOD INFERENCES
8.2 TESTS FOR APPROPRIATENESS OF A FAMILY OF DISTRIBUTIONS
8.3 SELECTION OF A DISTRIBUTION BY USING BIC OR AIC PROCEDURE
8.4 TESTS FOR A SPECIFIC DISTRIBUTION WITH KNOWN PARAMETERS
8.5 HOLLANDER AND PROSCHAN’S TEST FOR APPROPRIATENESS OF A GIVEN DISTRIBUTION WITH KNOWN PARAMETERS
EXERCISES
CHAPTER 9: Parametric Methods for Comparing Two Survival Distributions
9.1 LOG-LIKELIHOOD RATIO TEST FOR COMPARING TWO SURVIVAL DISTRIBUTIONS
9.2 COMPARISON OF TWO EXPONENTIAL DISTRIBUTIONS
9.3 COMPARISON OF TWO WEIBULL DISTRIBUTIONS
9.4 COMPARISON OF TWO GAMMA DISTRIBUTIONS
EXERCISES
CHAPTER 10: Parametric Methods for Regression Model Fitting and Identification of Prognostic Factors
10.1 PRELIMINARY EXAMINATION OF DATA
10.2 GENERAL STRUCTURE OF PARAMETRIC REGRESSION MODELS AND THEIR ASYMPTOTIC LIKELIHOOD INFERENCE
10.3 EXPONENTIAL AFT MODEL
10.4 WEIBULL AFT MODEL
10.5 LOGNORMAL AFT MODEL
10.6 THE EXTENDED GENERALIZED GAMMA AFT MODEL
10.7 LOG-LOGISTIC AFT MODEL
10.8 OTHER PARAMETRIC REGRESSION MODELS
10.9 MODEL SELECTION METHODS
EXERCISES
CHAPTER 11: Identification of Risk Factors Related to Survival Time: Cox Proportional Hazards Model
11.1 THE PROPORTIONAL HAZARDS MODEL
11.2 THE PARTIAL LIKELIHOOD FUNCTION
11.3 IDENTIFICATION OF SIGNIFICANT COVARIATES
11.4 ESTIMATION OF THE SURVIVORSHIP FUNCTION WITH COVARIATES
11.5 ADEQUACY ASSESSMENT OF THE PROPORTIONAL HAZARDS MODEL
EXERCISES
CHAPTER 12: Identification of Prognostic Factors Related to Survival Time: Non-Proportional Hazards Models
12.1 MODELS WITH TIME-DEPENDENT COVARIATES
12.2 STRATIFIED PROPORTIONAL HAZARDS MODEL
12.3 COMPETING RISKS MODEL
12.4 RECURRENT EVENT MODELS
12.5 MODELS FOR RELATED OBSERVATIONS
EXERCISES
CHAPTER 13: Identification of Risk Factors Related to Dichotomous and Polychotomous Outcomes
13.1 UNIVARIATE ANALYSIS
13.2 LOGISTIC AND CONDITIONAL LOGISTIC REGRESSION MODELS FOR DICHOTOMOUS OUTCOMES
13.3 MODELS FOR POLYCHOTOMOUS OUTCOMES
13.4 MODELS FOR RELATED OBSERVATIONS
EXERCISES
APPENDIX: Statistical Tables
References
Index
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Library of Congress Cataloging-in-Publication Data:
Lee, Elisa T.Statistical methods for survival data analysis / Elisa T. Lee, University of Oklahoma, College of Public Health, Oklahoma City, OK, John Wenyu Wang, University of Oklahoma, College of Public Health, Oklahoma City, OK. – Fourth edition. pages cm Includes bibliographical references and index.
ISBN 978-1-118-09502-7 (cloth)1. Medicine–Research–Statistical methods. 2. Failure time data analysis. 3. Prognosis–Statistical methods. I. Wang, John Wenyu. II. Title. R853.S7L43 2013 610.72′7–dc23
2013009388
Printed in the United States of America
ISBN: 9781118095027
10 9 8 7 6 5 4 3 2 1
To the memory of our parents
Mr. Chi-Lan Tan and Mrs. Hwei-Chi Lee Tan (E.T.L.)Mr. Beijun Zhang and Mrs. Xiangyi Wang (J.W.W.)
Preface
The purpose of the Fourth Edition of Statistical Methods for Survival Data Analysis, similar to that of the previous editions, is to provide a comprehensive introduction of the most commonly used methods for analyzing survival data. The term “survival data” includes not only the time to a certain event (survival time) but also the occurrence (e.g., yes or no) of an event because, in many cases, the exact time of the event is unknown and the only available information is that the event has occurred. This group of methods has been applied to many areas, such as medicine, epidemiology, social sciences, criminology, and marketing.
The book is intended for students, applied statisticians, epidemiologists, medical researchers, and others who are interested in survival data analysis. It covers the basic survival functions, and parametric and nonparametric methods for estimating and comparing these functions. Parametric methods that can be used to search for a suitable mathematical model for the survival time are also introduced. In addition, parametric and/or nonparametric methods for the identification of significant risk factors for the time to an event or the occurrence of an event are discussed. The Fourth Edition continues to be application-oriented with many examples taken from real research situations.
The mathematical level of the Fourth Edition is kept at a minimum. Readers with only college algebra should find the book readable and understandable. Complicated mathematical derivations are excluded and only in a few sections, some knowledge of calculus and matrix algebra is needed. These sections can be skipped without loss of continuity.
Most of the chapters in the Fourth Edition have been revised with additional discussions and/or examples. Examples in Chapter 3 have been updated. In particular, an example to identify significant risk factors for incident hypertension with repeated measurements is included. In Chapter 4, the method for estimating the survivorship function proposed by Nelson and Aalen has been added. We have shortened the discussion on graphical methods for fitting a parametric model to survival times and incorporated it to Chapter 7.
The chapter on the proportional hazards model (Chapter 11) has been expanded to cover more discussion on the assessment of the model by using several residual-based methods including the martingale-based residuals proposed by Lin, Wei, and Ying. The application of the proportional hazards model to the development of risk prediction equations is illustrated by an example. In the chapter on non-proportional hazards model (Chapter 12), we have expanded the section on models for related observations. This section introduces the frailty model based on the proportional hazards model and the accelerated failure time model. Chapter 13 is devoted to the identification of risk factors that are related to a dichotomous outcome (e.g., the occurrence of an event) or polychotomous outcome (e.g., the occurrence of an event with more than two subcategories). A new section has been added on models for related observations that can be used in family studies or longitudinal studies with repeated measures.
The computer software package R has become popular in statistical analysis. In this new edition, we have included the use of R in various chapters in addition to SAS and SPSS, when applicable. Computer codes for these software packages are given for many of the examples. In addition, the number of exercise problems has been increased in most chapters. Supplementary material for this book can be found by entering ISBN 9781118095027 at booksupport.wiley.com.
We thank the many researchers, teachers, and students who have used the Third Edition of the book for their valuable comments and suggestions. Steve Quigley of John Wiley and Sons invited us to work on the Fourth Edition. We thank him for his enthusiasm and encouragement.
Finally, we are most grateful to have so much love and support from our spouses, Samuel (E.T.L.) and Alice (J.W.W.). We also want to thank our children, their spouses, and grandchildren — Vivian, Benedict, Jennifer, Jan, Annelisa, Mira-Rose, Sophie-An, Kiri-Anna, Samuel, and Benjamin (E.T.L.); and Xing, Liya, and Cynthia (J.W.W.) — for the constant love and joy they have given us.
Elisa T. LeeJohn Wenyu Wang
CHAPTER 1
Introduction
1.1 PRELIMINARIES
This book is written for biomedical researchers, epidemiologists, consulting statisticians, students taking a first course on survival data analysis, and others interested in survival time study. It deals with the statistical methods for analyzing survival data derived from laboratory studies of animals, clinical and epidemiologic studies of humans, and other appropriate applications.
Survival time can be broadly defined as the time to the occurrence of a given event. This event can be the development of a disease, response to a treatment, relapse, or death. Therefore, survival time can be tumor-free time, the time from the start of treatment to response, length of remission, and time to death. Survival data can include survival time, response to a given treatment, and patient characteristics related to response, survival, and the development of a disease. The study of survival data has focused on predicting the probability of response, survival, or mean lifetime, comparing the survival distributions of experimental animals or of human patients and the identification of risk and/or prognostic factors related to response, survival, and the development of a disease. In this book, special consideration is given to the study of survival data in biomedical sciences, though all the methods are suitable for applications in industrial reliability, social sciences and business. Examples of survival data in these fields are lifetime of electronic devices, components or systems (reliability engineering), felons’ time to parole (criminology), duration of first marriage (sociology), length of newspaper or magazine subscription (marketing) and workmen’s compensation claims (insurance), and their various influencing risk or prognostic factors.
1.2 CENSORED DATA
Many researchers consider survival data analysis to be merely the application of two conventional statistical methods to a special type of problem: parametric if the distribution of survival times is known to be normal and nonparametric if the distribution is unknown. This assumption would be true if the survival times of all the subjects were exact and known. However, some survival times are not. Further, the survival distribution is often skewed or far from being normal. Thus, there is a need for new statistical techniques. One of the most important developments is due to a special feature of survival data in the life sciences that occurs when some subjects in the study have not experienced the event of interest at the end of the study or time of analysis. For example, some patients may still be alive or disease-free at the end of the study period. The exact survival times of these subjects are unknown. These are called censored observations or censored times and can also occur when individuals are lost to follow-up after a period of study. When there are no censored observations, the set of survival times is complete. There are three types of censoring.
1.2.1 Type I Censoring
Animal studies usually start with a fixed number of animals, to which treatment or treatments are given. Because of time and/or cost limitations, the researcher often cannot wait for the event of interest to occur to all the animals. One option is to observe for a fixed period of time, say 6 months, after which the surviving animals are sacrificed. Survival times recorded for the animals that had the event during the study period are the times from the start of the experiment to the time the event occurred. These are called exact or uncensored observations. The survival times of the sacrificed animals are not exactly known, but are recorded as at least the length of the study period. These are called censored observations. Some animals could be lost or die accidentally. Their survival times, from the start of experiment to loss or death, are also censored observations. In type I censoring, if there are no accidental losses, all censored observations equal the length of the study period.
For example, suppose that six rats have been exposed to carcinogens by injecting tumor cells into their foot-pads. The times to develop a tumor of a given size are observed. The investigator decides to terminate the experiment after 30 weeks. Figure 1.1 plots the development times of the tumors. Rats A, B, and D develop tumors after 10, 15, and 25 weeks, respectively. Rats C and E do not develop tumors by the end of the study; their tumor-free times are thus 30+ weeks. Rat F died accidentally without any tumors after 19 weeks of observation. The survival data (tumor-free times) are 10, 15, 30+, 25, 30+, and 19+ weeks. (The plus indicates a censored observation.)
Figure 1.1 Example of type I censored data.
Figure 1.2 Example of type II censored data.
1.2.2 Type II Censoring
Another option in animal studies is to wait until a fixed portion of the animals have had the event of interest, say 80 of 100, after which the surviving animals are sacrificed. In this case, if there are no accidental losses, the censored observations equal the largest uncensored observation. For example, in an experiment of six rats (Figure 1.2), the investigator may decide to terminate the study after four of the six rats have developed tumors. The survival or tumor-free times are then 10, 15, 35+, 25, 35, and 19+ weeks.
Figure 1.3 Example of type III censored data.
1.2.3 Type III Censoring
In most clinical and epidemiologic studies, the period of study is fixed and patients enter the study at different times during that period. Some may have the event before the end of the study; their exact survival times are known. Others may withdraw before the end of the study and are lost to follow-up. Still others may be alive at the end of the study. For “lost” persons, survival times are at least from their entrance to the last contact. For persons who still have no event, survival times are at least from entry to the end of the study. The latter two kinds of observations are censored observations. Since the entry times are not simultaneous, the censored times are also different. This is type III censoring. For example, suppose that six patients with acute leukemia enter a clinical study during a total study period of 1 year. Suppose also that all six respond to treatment and achieve remission. The remission times are plotted in Figure 1.3. Patients A, C, and E achieve remission at the beginning of the second, fourth, and ninth months, and relapse after four, six and three months, respectively. Patient B achieves remission at the beginning of the third month but is lost to follow-up four months later; the remission duration is thus at least four months. Patients D and F achieve remission at the beginning of the fifth and tenth months, respectively, and are still in remission at the end of the study; their remission times are thus at least eight and three months. The respective remission times of the six patients are 4, 4+, 6, 8+, 3, and 3+ months.
Type I and Type II censored observations are also called singly censored data and Type III progressively censored data. Another commonly used name for Type III censoring is random censoring. All of these types of censoring are right censoring or censoring to the right. There are also left censoring and interval censoring cases. Left censoring occurs when it is known that the event of interest occurred prior to a certain time, t, but the exact time of occurrence is unknown. For example, an epidemiologist wishes to know the age at diagnosis in a follow-up study of diabetic retinopathy. At the time of the examination, a 50-year-old participant was found to have already developed retinopathy but there is no record of the exact time at which initial evidence was found. Thus, the age at the examination, that is, 50, is a left censored observation. It means that the age of diagnosis for this patient is at most 50 years.
Interval censoring occurs when the event of interest is known to have occurred between times a and b. For example, if medical records indicate that at age 45, the patient in the aforementioned example did not have retinopathy, his age at diagnosis is between 45 and 50 years.
We will study descriptive and analytic methods for complete, singly censored, and progressively censored survival data using numerical and graphical techniques. Analytic methods discussed include parametric and nonparametric. Parametric approaches are used either when a suitable model or distribution is fitted to the data or when a distribution can be assumed for the population from which the sample is drawn. Commonly used survival distributions are the exponential, Weibull, lognormal, gamma, and log-logistic. If a survival distribution is found to fit the data properly, the survival pattern can then be described by the parameters in a compact way. Statistical inference can be based on the distribution chosen. If the search for an appropriate model or distribution is too time consuming or not economical or no theoretical distribution adequately fits the data, nonparametric methods, which are generally easy to apply, should be considered.
1.3 SCOPE OF THE BOOK
Chapters 1–3 define survival functions and gives examples of survival data analysis. Survival distribution is most commonly described by three functions: survivorship function (also called cumulative survival rate or survival function), probability density function, and hazard function (hazard rate or age-specific rate). Chapter 2 defines these three functions and their equivalence relationships. Chapter 3 illustrates survival data analysis with seven examples taken from actual research situations. Clinical and laboratory data are systematically analyzed in progressive steps and the results are interpreted. Section and chapter numbers are given for quick reference. The actual calculations are given as example or left as exercises in the chapters where the methods are discussed. Four sets of data are provided in the exercise section for the reader to analyze. These data are referred to subsequent chapters.
In Chapters 4 and 5 we introduce some of the most widely used nonparametric methods for estimating and comparing survival distributions. Chapter 4 deals with the nonparametric methods for estimating the three survival functions: the Kaplan and Meier product-limit (PL) estimate, the Nelson–Aalen estimate, and the life-table technique (population life tables and clinical life tables). Also covered is standardization of rates by direct and indirect methods including the standardized mortality or morbidity ratio (SMR). Chapter 5 is devoted to nonparametric techniques for comparing survival distributions. A common practice is to compare the survival experiences of two or more groups differing in their treatment or in a given characteristic. Several nonparametric tests are described.
Chapters 6–9 introduce the parametric approach to survival data analysis. Although nonparametric methods play an important role in survival studies, parametric techniques cannot be ignored. Chapter 6 introduces and discusses the exponential, Weibull, lognormal, gamma, and log-logistic survival distributions. Practical applications of these distributions taken from the literature are included.
An important part of survival data analysis is model or distribution fitting. Once an appropriate statistical model for survival time has been constructed and its parameters estimated, the information can help predict survival, develop optimal treatment regimens, plan future clinical or laboratory studies, and so on. Chapter 7 discusses the analytical estimation procedures for survival distributions. Most of the estimation procedures are based on the maximum likelihood method. Mathematical derivations are omitted; only formulas for the estimates and examples are given. The graphical technique is a simple informal way to select a statistical model and estimate its parameters. When a statistical distribution is found to fit the data well, the parameters can be estimated by analytical methods. Chapter 7 also introduces graphical methods, probability plotting, hazard plotting, and the Cox–Snell residual method for survival distribution fitting. Chapter 8 discusses several tests of goodness-of-fit and distribution selection. Chapter 9 describes several parametric methods for comparing survival distributions.
An important topic that has received much attention is the identification of prognostic factors related to survival time. For example, who is likely to survive longest after mastectomy and what are the most important factors that influence that survival? Another subject important to biomedical researchers and epidemiologists alike is the identification of the risk factors related to the development of a given disease or the response to a given treatment. What are the factors most closely related to the development of a given disease? Who is more likely to develop lung cancer, diabetes, or coronary disease? In many diseases, such as cancer, patients who respond to treatment have a better prognosis than patients who do not. The question, then, relates to which factors would influence response significantly. Who is more likely to respond to treatment and thus perhaps survive longer?
Chapters 10–13 deal with prognostic/risk factors and survival times. Chapters 10 and 11 introduce, respectively, a parametric model called the accelerated failure time model and the Cox proportional hazards model for identifying important prognostic factors. Chapter 12 discusses several nonproportional hazards models and a model to handle related survival times, namely, the frailty model. The last chapter, Chapter 13 introduces the linear logistic, conditional logistic and other regression models for the identification of risk factors that are related to dichotomous, polychotomous, or related outcomes.
The appendix gives several statistical tables for convenience.
Most nonparametric techniques discussed here are easy to understand and simple to apply. Parametric methods require an understanding of the survival distributions. Unfortunately, most of the survival distributions are not simple. Readers without calculus may find it difficult to apply them on their own. However, if the main purpose is not model fitting, most parametric techniques can be substituted for by their nonparametric competitors. In fact, a large percentage of the survival studies in clinical or epidemiological journals are analyzed by nonparametric methods. Researchers not interested in survival model fitting should read the chapters and sections on nonparametric methods. Computer programs for survival data analysis are available in several available software packages, for example, SAS (SAS, Version 9.3, 2011, SAS Institute Inc., Cary, NC), SPSS (SPSS, Version 19, 2010, IBM Corporation, Armonk, NY) and R (R, Version 2.15.2, 2012, R Foundation for Statistical Computing). These computer programs are referred in various chapters when applicable. Computer programming codes are given for most of the examples.
CHAPTER 2
Functions of Survival Time
Survival times are data that measure the time to a certain event such as failure, death, response, relapse, the development of a given disease, parole, or divorce. These times are subject to random variations, and like any random variables, form a distribution. The distribution of survival times is usually described or characterized by three functions: (1) the survivorship function, (2) the probability density functions, and (3) the hazard function. These three functions are mathematically equivalent — if one of them is given, the other two can be derived.
In practice, the three functions can be used to illustrate different aspects of the data. A basic problem in survival data analysis is to estimate from the sampled data one or more of these three functions and to draw inferences about the survival pattern in the population.
In Section 2.1, we define the three functions and in Section 2.2, discuss the equivalence relationship among the three functions.
2.1 DEFINITIONS
Let T denote the survival time. The distribution of T can be characterized by the following three equivalent functions.
2.1.1 Survivorship Function (or Survival Function)
This function, denoted by S(t), is defined as the probability that an individual survives longer than t:
(2.1.1)
Figure 2.1 (a) Low survival rate or short survival time and (b) high survival rate or longer survival.
From the definition of the cumulative distribution function F(t) of T,
(2.1.2)
Here S(t) is a nonincreasing function of time t with the properties,
That is, the probability of surviving at least at the time 0 is 1 and that of surviving an infinite time is 0.
The function S(t) is also known as the cumulative survival rate. To depict the course of survival, Berkson (1942) recommended a graphic presentation of S(t). The graph of S(t) is called the survival curve. A steep survival curve, such as the one in Figure 2.1a, represents low survival rate or short survival time. A gradual or flat survival curve such as in Figure 2.1b represents high survival rate or longer survival.
The survivorship function or the survival curve is used to find the 50th percentile (the median) and other percentiles (e.g., 25th and 75th) of survival time and to compare survival distributions of two or more groups. The median survival times in Figures 2.1a and b are approximately 5 and 36 units of time, respectively. The mean is generally used to describe the central tendency of a distribution, but in survival distributions the median is often better because a small number of individuals with exceptionally long or short lifetimes will cause the mean survival time to be disproportionately large or small.
In practice, if there are no censored observations, the survivorship function is estimated as the proportion of patients surviving longer than t:
(2.1.3)
where the circumflex denotes an estimate of the function. When censored observations are present, the numerator of (2.1.3) cannot always be determined. For example, consider the following set of survival data, 4, 6, 6+, 10+, 15, 20. Using (2.1.3), we can compute . However, we cannot obtain since the exact number of patients surviving longer than 11 is unknown. Either the third or the fourth patient (6+ and 10+) could survive longer than or less than 11. Thus, when censored observations are present, (2.1.3) is no longer appropriate for estimating S(t). Nonparametric methods of estimating S(t) for censored data will be discussed in Chapter 4.
2.1.2 Probability Density Function (or Density Function)
Like any other continuous random variable, the survival time T has a probability density function defined as the limit of the probability that an individual fails in the short interval t to t + Δt per unit width Δt, or simply the probability of failure in a small interval per unit time. It can be expressed as
(2.1.4)
The graph of f (t) is called the density curve. Figures 2.2a and b give two examples of the density curve. The density function has the following two properties:
In practice, if there are no censored observations, the probability density function f(t) is estimated as the proportion of patients dying in an interval per unit width:
(2.1.5)
Similar to the estimation of S(t), when censored observations are present, (2.1.5) is not applicable. We will discuss an appropriate method in Chapter 4.
The proportion of individuals that fail in any time interval and the peaks of high frequency of failure can be found from the density function. The density curve in Figure 2.2a gives a pattern of high failure rate at the beginning of the study and decreasing failure rate as time increases. In Figure 2.2b, the peak of high failure frequency occurs at approximately 1.7 units of time. The proportion of individuals that fail between 1 and 2 units of time is equal to the shaded area between the density curve and the axis. The density function is also known as the unconditional failure rate.
Figure 2.2 (a) Monotonically decreasing failure rate and (b) increasing and decreasing failure rate.
2.1.3 Hazard Function
The hazard function h(t) of survival time T gives the conditional failure rate. This is defined as the probability of failure during a very small time interval, assuming that the individual has survived to the beginning of the interval, or as the limit of the probability that an individual fails in a very short interval, t + Δt, given that the individual has survived to time t:
(2.1.6)
The hazard function can also be defined in terms of the cumulative distribution function F(t) and the probability density function f(t):
(2.1.7)
The hazard function is also known as the instantaneous failure rate, force of mortality, conditional mortality rate, and age-specific failure rate. If t in (2.1.6) is age, it is a measure of the proneness to failure as a function of the age of the individual in the sense that the quantity h(t)Δt is the expected proportion of age t individuals who will fail in the short time interval t + Δt. The hazard function thus gives the risk of failure per unit time during the aging process. It plays an important role in survival data analysis.
Figure 2.3 Examples of the hazard function.
In practice, when there are no censored observations, the hazard function is estimated as the proportion of individuals failing in an interval per unit time, given that they have survived to the beginning of the interval:
(2.1.8)
Actuaries usually use the average hazard rate of the interval in which the number of individuals failing per unit time in the interval is divided by the average number of survivors at the midpoint of the interval:
(2.1.9)
The actuarial estimate in (2.1.9) gives a higher hazard rate than (2.1.8) and thus a more conservative estimate.
The hazard function may increase, decrease, remain constant, or indicate a more complicated process. Figure 2.3 plots several kinds of hazard function. For example, patients with acute leukemia who do not respond to treatment have an increasing hazard rate, h1(t), h2(t) is a decreasing hazard function that, for example, indicates the risk of soldiers wounded by bullets who undergo surgery. The main danger is the operation itself and this danger decreases if the surgery is successful. An example of a constant hazard function, h3(t), is the risk of healthy individuals between 18 and 40 years of age whose main risks of death are accidents. The bathtub curve, h4(t), describes the process of human life. During an initial period, the risk is high (high infant mortality). Subsequently, h(t) stays approximately constant until a certain time, after which it increases because of wear-out failures. Finally, patients with tuberculosis have risks that increase initially, then decrease after treatment. Such an increasing then decreasing hazard function is described by h5(t).
Table 2.1 Survival Data and Estimated Survival Functions of 40 Myeloma Patients
The cumulative hazard function is defined as
(2.1.10)
It will be shown in Section 2.2 that
(2.1.11)
The following example illustrates how these functions can be estimated from a complete sample of grouped survival times without censored observations.
Example 2.1 The first three columns of Table 2.1 give the survival data of 40 patients with myeloma. The survival times are grouped into intervals of 5 months. The estimated survivorship function, density function, hazard function, and cumulative hazard function are also given, with the corresponding graphs plotted in Figure 2.4a–d.
Figure 2.4 Estimated survival functions of myeloma patients: (a) survivorship function, (b) density function, (c) hazard function, and (d) cumulative hazard function.
From Table 2.1 or Figure 2.4a, the median survival time of myeloma patients is approximately 17.5 months and the peak of high frequency of death occurs in 5–10 months. In addition, the hazard function shows an increasing trend and reaches its peak at approximately 32.5 months and then fluctuates.
2.2 RELATIONSHIPS OF THE SURVIVAL FUNCTIONS
The three functions defined in Section 2.1 are mathematically equivalent. Given any one of them, the other two can be derived. Readers not interested in the mathematical relationship among the three survival functions can skip this section without loss of continuity.
Hence, if f(t) is known, the survivorship function can be obtained from the basic relationship between f(t), F(t), and (2.1.2). The hazard function can then be determined from (2.2.1). If S(t) is known, f(t) and h(t) can be determined from (2.2.2) and (2.2.1), respectively, or h(t) can be derived first from (2.2.3) and then f(t) from (2.2.1). If h(t) is given, S(t) and f(t) can be obtained, respectively, from (2.2.4) and (2.2.5). Thus, given any one of the three survival functions, the other two can easily be derived. The following example illustrates these equivalence relationships.
Example 2.2 Suppose that the survival time of a population has the following density function:
Using the definition of the cumulative distribution function,
From (2.1.2), we obtain the survivorship function
The hazard function can then be obtained from (2.2.1):
and the cumulative hazard function is
A complete treatment of this distribution is given in Section 6.1.
EXERCISES
Exercise Table 2.1 Survival Data
Year of Follow-up
Number Alive at Beginning of Interval
Number Dying in Interval
0–1
1100
240
1–2
860
180
2–3
680
184
3–4
496
138
4–5
358
118
5–6
240
60
6–7
180
52
7–8
128
44
8–9
84
32
≥9
52
28
Exercise Table 2.2 Life Table for the Total Population: United States, 1959–1961
Source: U.S. National Center for Health Statistics. (1964). Life Tables 1959–1961, 1(1), 8–9.
Age Interval
Number Living at Beginning of Age Interval
Number Dying in Age Interval
0–1
100,000
2,593
1–5
97,407
409
5–10
96,998
233
10–15
96,765
214
15–20
96,551
440
20–25
96,111
594
25–30
95,517
612
30–35
94,905
761
35–40
94,144
1,080
40–45
93,064
1,686
45–50
91,378
2,622
50–55
88,756
4,045
55–60
84,711
5,644
60–65
79,067
7,920
65–70
71,147
10,290
70–75
60,857
12,687
75–80
48,170
14,594
80–85
33,576
15,034
85 and over
18,542
18,542
CHAPTER 3
Examples of Survival Data Analysis
The investigator who has assembled a large amount of data must decide what to do with it and what it indicates. In this chapter, we give seven examples of survival data analysis from actual research situations to illustrate the most commonly used methods discussed in this book. In Example 3.1, we analyze two sets of data obtained, respectively, from two and three treatment groups to compare the treatment’s abilities to prolong life. Example 3.2 is an example of the life-table technique for large samples. Example 3.3 gives tumor-free times of rats, and the investigator seeks a well-known distribution for the tumor-free time. Example 3.4 compares the survival of a cohort with that of a general population using the relative survival rate. In Example 3.5, we study the time to the development of coronary heart disease and several patient characteristics to identify important risk factors; the patient characteristics are analyzed individually and simultaneously for their predictive values. Example 3.6 introduces a case in which the interest is to identify risk factors for the prevalence of age-related macular degeneration (AMD). Example 3.7 is a longitudinal study of hypertension risk factors, where related data are observed. Several sets of real data are provided in this chapter for readers to use.
3.1 EXAMPLE 3.1: COMPARISON OF TWO TREATMENTS AND THREE DIETS
3.1.1 Comparison of Two Treatments
Thirty melanoma patients (Stages 2–4) were studied to compare the immunotherapies BCG (Bacillus Calmette–Guérin) and Corynebacterium parvum for their abilities to prolong remission and survival time. The age, sex, disease stage, treatment received, remission duration, and survival time are given in Table 3.1. All the patients were resected before treatment began and thus had no evidence of melanoma at the time of first treatment.
Table 3.1 Data for 30 Resected Melanoma Patients
Source: Data courtesy of Dr. Richard Ishmael.
The usual objective with this type of data is to determine the length of remission and survival and to compare the distributions of remission and survival time in each group. Before comparing the remission and survival distributions, we attempt to determine if the two treatment groups are comparable with respect to prognostic factors. Let us use the survival time to illustrate the steps. (The remission time could be similarly analyzed.)
Table 3.2 Kaplan–Meier Product-Limit Estimate of Survival Function S(t)
1. Estimate and plot the survival function of the two treatment groups. The resulting curves are called survival curves. Points on the curve estimate the proportion of patients who will survive at least a given period of time. Points on the curve estimate the proportion of patients who will survive at least a given period of time. For such small samples with progressively censored observations, the Kaplan–Meier product-limit (PL) method is appropriate for estimating the survival function. It does not require any assumptions about the form of the function that is being estimated. Section 4.1 will discuss this method in detail. Computer programs for the method can be found in SPSS, SAS, and R. Examples for computer codes will be given in Section 4.1.
Table 3.2 gives the PL estimate of the survival function, , for the two treatment groups. Note that S(t) is estimated only at death times; however, the censored observations are used to estimate S(t). The median survival time can be estimated by linear interpolation. For BCG patients, the median survival time was about 18.2 months. The median survival time for the C. parvum group cannot be calculated since 15 of the 19 patients were still alive. Most computer programs give not only but also the standard error of , and the 75-, 50-, and 25-percentile points.
Figure 3.1 plots the estimated survival function for patients receiving the two treatments: The median survival time (50-percentile point) for the BCG group can also be determined graphically. The survival curves clearly show that C. parvum patients had slightly better survival experience than BCG patients. For example, 50% of the BCG patients survived at least 18.2 months while about 61% of the C. parvum patients survived that long.
2. Examine the prognostic homogeneity of the two groups. The next question to ask is whether the difference in survival between the two treatment groups is statistically significant. Is the difference shown by the data significant or simply random variation in the sample? A statistical test of significance is needed. However, a statistical test without considering patient characteristics makes sense only if the two groups of patients are homogeneous with respect to prognostic factors. It has been assumed thus far that the patients in the two groups are comparable and that the only difference between them is treatment. Thus, before performing a statistical test it is necessary to examine the homogeneity between the two groups.
Figure 3.1 Survival curves of patients receiving BCG and C. parvum.
Although prognostic factors for melanoma patients are not well established, it has been reported that the female and the young have a better survival experience than the male and the old. Also, the disease stage plays an important role in survival. Let us check the homogeneity of the two treatment groups with respect to age, sex, and disease stage.
The age distributions are estimated and plotted in Figure 3.2. The median age is 39 for the BCG group and 43 for the C. parvum patients. To test the significance of the difference between the two age distributions, the two-sample t-test (Armitage et al., 2002; Daniel, 2009) or nonparametric tests such as the Mann–Whitney U-test or the Kolmogorov–Smirnov test (Marascuilo and McSweeney, 1985) are appropriate. However, the generalized Wilcoxon tests given in Section 5.1 can also be used, since they reduce to the Mann–Whitney U-test. Using Gehan’s generalized Wilcoxon test, the difference between the two age distributions is not found to be statistically significant. More about the test will be given in Section 5.1.
The number of male and female patients in the two treatment groups is given in Table 3.3. Sixty-four percent of the BCG patients and 42% of the C. parvum patients are women. A chi-square test can be used to compare the two proportions (see Section 13.1). It can be used only for r × c tables in which the entries are frequencies and not for tables in which the entries are mean values or medians of a certain variable. For a 2 × 2 table, the chi-square value can be computed by hand. Computer programs for the test can be found in many computer program packages, such as SAS, SPSS, and R.
The chi-square value for treatment by sex in Table 3.3 is 1.29 with 1 degree of freedom, which is not significant at the 0.05 or 0.10 level. Therefore, the difference between the two proportions is not statistically significant. The number of Stage 2 patients and the number of patients with more advanced disease in the two treatment groups are also given in Table 3.3. Eighteen percent of the BCG patients are at Stage 2 against 21% of the C. parvum patients. However, a chi-square test result shows that the difference is not significant.
Figure 3.2 Age distribution of two treatment groups.
Thus, we can say that the data do not show heterogeneity between the two treatment groups. If heterogeneity is found, the groups can be divided into subgroups of members who are similar in their prognoses.
3. Compare the two survival distributions. There are several parametric and nonparametric tests to compare two survival distributions. They are described in Chapters 5 and 9. Since we have no information on the survival distribution that the data follow, we would continue to use nonparametric methods to compare the two survival distributions. The four tests described in Sections 5.1.1–5.1.4 are suitable. The performance of these tests is discussed at the end of Section 5.1. We choose Gehan’s generalized Wilcoxon test here to demonstrate the analysis procedure only because of its simplicity of calculation.
In testing the significance of the difference between two survival distributions, the hypothesis is that the survival distribution of the BCG patients is the same as that of the C. parvum patients. Let S1(t) and S2(t) be the survival function of the BCG and C. parvum groups, respectively. The null hypothesis is
The alternative hypothesis chosen is two-sided:
since we have no prior information concerning the superiority of either of the two treatments. The slight difference between the two estimated survival curves could be due to random variation. The one-sided alternative H1 : S1(t) < S2(t) should be considered inappropriate.
Table 3.3 Treatment by Gender and Disease Stage
3.1.2 Comparison of Three Diets
A laboratory investigator interested in the relationship between diet and the development of tumors divided 90 rats into three groups and fed them with low, saturated, and unsaturated fat diets, respectively (King et al., 1979). The rats were of the same age and species and were in similar physical condition. An identical amount of tumor cells were injected into a foot pad of each rat. The rats were observed for 200 days. Many developed a recognizable tumor early in the study period. Some were tumor-free at the end of the 200 days. Rat 16 in the low-fat group and rat 24 in the saturated fat group died accidentally after 140 days and 170 days, respectively, with no evidence of tumor. Table 3.4 gives the tumor-free time, the time from injection to the time that a tumor develops or to the end of the study. Fifteen of the 30 rats on the low-fat diet developed a tumor before the experiment was terminated. Rat 16 that died had a tumor-free time of at least 140 days. The other 14 rats did not develop any tumor by the end of the experiment; their tumor-free times were at least 200 days. Among the 30 rats in the saturated fat diet group, 23 developed a tumor, 1 died tumor-free after 170 days, and 6 were tumor-free at the end of the experiment. All 30 rats in the unsaturated fat diet group developed tumors within 200 days. The two early deaths can be considered losses to follow-up. The data are singly censored if the two early deaths are excluded.
Table 3.4 Tumor-Free Time (Days) of 90 Rats on Three Different Diets
Source: King et al. (1979). Data are used by permission of the author.
The investigator’s main interest here is to compare the three diets’ abilities to keep the rats tumor-free. To obtain information about the distribution of the tumor-free time, we can first estimate the survival (tumor-free) function of the three diet groups. The three survival functions are estimated using the Kaplan–Meier PL method and plotted in Figure 3.3. The median tumor-free times for the low, saturated, and unsaturated fat diet groups are 188, 107, and 91 days, respectively. Since the three groups are homogeneous, we can skip the step that checks for homogeneity and compare the three distributions of tumor-free time.
Figure 3.3 Survival curves of rats in three diet groups.
3.2 EXAMPLE 3.2: COMPARISON OF TWO SURVIVAL PATTERNS USING LIFE TABLES
When the sample of patients is so large that their groupings are meaningful, the life-table technique can be used to estimate the survival distribution. A method developed by Mantel and Haenszel (1959) and applied to life tables by Mantel (1966) can be used to compare two survival patterns in the life-table analysis.
Consider the data of male patients with localized cancer of the rectum diagnosed in Connecticut from 1935 to 1954 (Myers, 1969). A total of 388 patients were diagnosed between 1935 and 1944, and 749 patients were diagnosed between 1945 and 1954. For such large sample sizes, the data can be grouped and tabulated as shown in Table 3.5. The 10 intervals indicate the number of years after diagnosis. For the tabulated life tables, the survival function S(ti) can be estimated for each interval ti. In Section 4.2, we discuss the estimation procedures of S(ti) and density and hazard functions. The survival, density, and hazard functions are the three most important functions that characterize a survival distribution.
The column in Table 3.5 gives the estimated survival function for the two time periods; these are plotted in Figure 3.4. Patients diagnosed in the 1945–1954 period had considerably longer survival times (median 3.87 years) than patients diagnosed in the 1935–1944 period (median 1.58 years). Patients diagnosed in 1935–1944 had a 10-year survival rate of 0.1404, or 14%. The patients diagnosed in 1945–1954 had a 10-year survival rate of 0.2905, or 29%. In comparing two sets of survival data, one can compare the proportions of patients surviving some stated period such as 10 years, or the 10-year survival rates. However, one cannot anticipate that two survival patterns will always stand in a superior–inferior relationship. It is more desirable to make a whole-pattern comparison (see Section 5.2).
The Mantel–Haenszel method described in Section 5.2 is a whole-pattern comparison and can be used to compare two survival patterns in life tables. Application of this method to the data in Table 3.5 results in a chi-square value of 51.996 with one degree of freedom. We can conclude that the difference between the two survival patterns is highly significant (p < 0.001).
Figure 3.4 Survival curves for male patients with localized cancer of the rectum, diagnosed in Connecticut, 1935–1944 versus 1945–1954.
Table 3.5 Life Table for Male Patients with Localized Cancer of Rectum Diagnosed in Connecticut, 1935–1944 and 1945–1954a
Source: Myers (1969).
Estimates of the survival function or survival rate depend on the life-table interval used. If each interval is very short, resulting in a large number of intervals, the computation becomes very tedious and the advantage of the life table is not fully taken. One assumption underlying the life table is that the population has the same survival probability in each interval. If the interval length is long, this assumption may be violated and the estimates inaccurate; this should be avoided except for rough calculations. Although the length of each interval and the total number of intervals are important, they will not cause trouble in most clinical studies since the study periods normally cover a short period of time such as 1, 2, or 3 years. Life tables with about 10–20 intervals of several months to 1 year each are reasonable. The investigator should also consider the disease under study. If the variation in survival is large in a short period of time, the interval length should be short. However, in some demographic or other studies it is often of interest to cover a life span from birth to age 100 or more. The number of intervals would be very large if short intervals were used. In this case, 5-year intervals are sufficient to take into account the important variations in survival rate estimates (Shryock et al., 1971).
3.3 EXAMPLE 3.3: FITTING SURVIVAL DISTRIBUTIONS TO TUMOR-FREE TIMES
Consider the tumor-free times of the 30 rats that are fed a saturated fat diet in Table 3.4. Suppose that we are interested in a distribution to describe the tumor-free times of these rats but that no information is available as to which distribution will fit. We need to identify a distribution that fits the data well. If such a distribution can be found, the tumor-free time can then be described by the properties of the distribution, and the tumor-free time of other rats fed the same diet can be predicted. Parametric tests can be used to compare the diets. However, there are a large number of potential functions and distributions to choose from; the search could become an art as much as a scientific task.
In this book, we cover five commonly used survival distributions: exponential, Weibull, lognormal, gamma (and extended generalized gamma), and log-logistic. The first four distributions are special cases of the extended generalized gamma distribution and, therefore, these five distributions belong to a large family of distributions, namely the extended generalized gamma family. Also, the exponential distribution is a special case of the Weibull, and therefore, these two form a small family of two members, exponential and Weibull. The procedure is to compare the goodness of fit of a distribution with a higher level distribution. For example, we can test the null hypothesis that the distribution is exponential versus the alternative hypothesis that the distribution is Weibull, or test the null hypothesis that the distribution is Weibull versus the alternative that the distribution is the generalized gamma.
First, we use the maximum likelihood method to estimate the parameters of each of these five distributions and obtain the respective log-likelihood values. We use the log-likelihood values to perform the likelihood ratio tests to compare the goodness of fit of these distributions. Table 3.6 gives the log-likelihood (LL) values , the likelihood ratio test statistic (XL), and p-value corresponding to the test statistic. The last column of Table 3.6 gives the value of another simple criterion, called the Bayesian information criterion (BIC). This criterion is based on the log-likelihood value and the number of parameters of the distribution. The larger the BIC is, the better the fit. These methods and procedures are discussed in Chapter 8.
Table 3.6 Goodness-of-Fit Tests Based on Likelihood Inference for the Tumor-Free Times of Rats Fed with Saturated Fat Diet in Table 3.4
