Modelling Survival Data in Medical Research

✍ Scribed by David Collett

• Publisher: CRC Press/Chapman & Hall
• Year: 2023
• Tongue: English
• Leaves: 557
• Series: Chapman & Hall/CRC Texts in Statistical Science
• Edition: 4
• Category: Library

✦ Synopsis

Modelling Survival Data in Medical Research, Fourth Edition, describes the analysis of survival data, illustrated using a wide range of examples from biomedical research. Written in a non-technical style, it concentrates on how the techniques are used in practice. Starting with standard methods for summarising survival data, Cox regression and parametric modelling, the book covers many more advanced techniques, including interval-censoring, frailty modelling, competing risks, analysis of multiple events, and dependent censoring.

This new edition contains chapters on Bayesian survival analysis and use of the R software. Earlier chapters have been extensively revised and expanded to add new material on several topics. These include methods for assessing the predictive ability of a model, joint models for longitudinal and survival data, and modern methods for the analysis of interval-censored survival data.

Features:

• Presents an accessible account of a wide range of statistical methods for analysing survival data
• Contains practical guidance on modelling survival data from the author’s many years of experience in teaching and consultancy
• Shows how Bayesian methods can be used to analyse survival data
• Includes details on how R can be used to carry out all the methods described, with guidance on the interpretation of the resulting output
• Contains many real data examples and additional data sets that can be used for coursework
• All data sets used are available in electronic format from the publisher’s website

Modelling Survival Data in Medical Research, Fourth Edition, is an invaluable resource for statisticians in the pharmaceutical industry and biomedical research centres, research scientists and clinicians who are analysing their own data, and students following undergraduate or postgraduate courses in survival analysis.

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Author Biography
1. Survival analysis
1.1. Applications of survival analysis
1.2. Special features of survival data
1.2.1. Censoring
1.2.2. Independent censoring
1.2.3. Study time and patient time
1.3. Some examples
1.4. Survivor, hazard, and cumulative hazard functions
1.4.1. The survivor function
1.4.2. The hazard function
1.4.3. The cumulative hazard function
1.5. Computer software for survival analysis
1.6. Further reading
2. Some non-parametric procedures
2.1. Estimating the survivor function
2.1.1. Life-table estimate of the survivor function
2.1.2. Kaplan-Meier estimate of the survivor function
2.1.3. Nelson-Aalen estimate of the survivor function
2.2. Standard error of the estimated survivor function
2.2.1. Standard error of the Kaplan-Meier estimate
2.2.2. Standard error of other estimates
2.2.3. Confidence intervals for values of the survivor function
2.3. Estimating the hazard function
2.3.1. Life-table estimate of the hazard function
2.3.2. Kaplan-Meier type estimate
2.3.3. Estimating the cumulative hazard function
2.4. Estimating the median and percentiles of survival times
2.5. Confidence intervals for the median and percentiles
2.6. Comparison of two groups of survival data
2.6.1. Hypothesis testing
2.6.2. The log-rank test
2.6.3. The Wilcoxon test
2.6.4. The Peto-Peto test
2.6.5. Comparison of the log-rank, Wilcoxon, and Peto-Peto tests
2.7. Comparison of three or more groups of survival data
2.8. Stratified tests
2.9. Log-rank test for trend
2.10. Further reading
3. The Cox regression model
3.1. Modelling the hazard function
3.1.1. A model for the comparison of two groups
3.1.2. The general proportional hazards model
3.2. The linear component of the model
3.2.1. Including a variate
3.2.2. Including a factor
3.2.3. Including an interaction
3.2.4. Including a mixed term
3.3. Fitting the Cox regression model
3.3.1. Likelihood function for the model
3.3.2. Treatment of ties
3.3.3. The Newton-Raphson procedure
3.4. Confidence intervals and hypothesis tests
3.4.1. Confidence intervals for hazard ratios
3.4.2. Two examples
3.5. Comparing alternative models
3.5.1. The statistic −2 logˆL
3.5.2. Comparing nested models
3.6. Strategy for model selection
3.6.1. Variable selection procedures
3.7. Variable selection using the lasso
3.7.1. The lasso in Cox regression modelling
3.7.2. Data preparation
3.8. Non-linear terms
3.8.1. Testing for non-linearity
3.8.2. Modelling non-linearity
3.8.3. Fractional polynomials
3.9. Interpretation of parameter estimates
3.9.1. Models with a variate
3.9.2. Models with a factor
3.9.3. Models with combinations of terms
3.10. Estimating the hazard and survivor functions
3.10.1. The special case of no covariates
3.10.2. Some approximations to estimates of baseline functions
3.11. Risk-adjusted survivor function
3.11.1. Risk-adjusted survivor function for groups of individuals
3.12. Concordance, predictive ability, and explained variation
3.12.1. Measures of concordance
3.12.2. Predictive ability
3.12.3. Explained variation in the Cox regression model
3.12.4. Measures of explained variation
3.12.5. Model validation
3.13. Time-dependent ROC curves
3.13.1. Sensitivity and specificity
3.13.2. Modelling the probability of disease
3.13.3. ROC curves
3.13.4. Time-dependent ROC curves
3.14. Proportional hazards and the log-rank test
3.15. Further reading
4. Model checking in the Cox regression model
4.1. Residuals for the Cox regression model
4.1.1. Cox-Snell residuals
4.1.2. Modified Cox-Snell residuals
4.1.3. Martingale residuals
4.1.4. Deviance residuals
4.1.5. Schoenfeld residuals
4.1.6. Score residuals
4.2. Assessment of model fit
4.2.1. Plots based on the Cox-Snell residuals
4.2.2. Plots based on the martingale and deviance residuals
4.2.3. Checking the functional form of covariates
4.3. Identification of influential observations
4.3.1. Influence of observations on a parameter estimate
4.3.2. Influence of observations on the set of parameter estimates
4.3.3. Treatment of influential observations
4.4. Testing the assumption of proportional hazards
4.4.1. The log-cumulative hazard plot
4.4.2. Use of Schoenfeld residuals
4.4.3. Tests for non-proportional hazards
4.4.4. Adding a time-dependent variable
4.5. Recommendations
4.6. Further reading
5. Parametric regression models
5.1. Models for the hazard function
5.1.1. The exponential distribution
5.1.2. The Weibull distribution
5.1.3. The log-logistic distribution
5.1.4. The lognormal distribution
5.1.5. The Gompertz distribution
5.1.6. The gamma distribution
5.1.7. The inverse Gaussian distribution
5.1.8. Some other distributions
5.2. Assessing the suitability of a parametric model
5.3. Fitting a parametric model to a single sample
5.3.1. Likelihood function for randomly censored data
5.4. Fitting exponential and Weibull models
5.4.1. Fitting the exponential distribution
5.4.2. Fitting the Weibull distribution
5.4.3. Standard error of a percentile of the Weibull distribution
5.5. Comparison of two groups
5.5.1. Exploratory analysis
5.5.2. Fitting the model
5.6. The Weibull proportional hazards model
5.6.1. Fitting the model
5.6.2. Standard error of a percentile in the Weibull model
5.6.3. Log-linear form of the model
5.6.4. Exploratory analysis
5.7. Comparing alternative Weibull proportional hazards models
5.8. The Gompertz proportional hazards model
5.9. Model choice
5.10. Accelerated failure model for two groups
5.10.1. Comparison with the proportional hazards model
5.10.2. The percentile-percentile plot
5.11. The general accelerated failure time model
5.11.1. Log-linear form of the accelerated failure time model
5.12. Parametric accelerated failure time models
5.12.1. The Weibull accelerated failure time model
5.12.2. The log-logistic accelerated failure time model
5.12.3. The lognormal accelerated failure time model
5.13. Fitting and comparing accelerated failure time models
5.14. Explained variation in parametric models
5.14.1. Predictive ability of a parametric model
5.15. The proportional odds model
5.15.1. The log-logistic proportional odds model
5.16. Modelling cure rates
5.17. Effect of covariate adjustment
5.18. Further reading
6. Flexible parametric models
6.1. Piecewise exponential model
6.2. Modelling using spline functions
6.2.1. B-splines
6.2.2. Restricted cubic splines
6.2.3. Number and position of the knots
6.3. Flexible models for the hazard function
6.4. Flexible models for the log-cumulative hazard function
6.5. Flexible proportional odds models
6.6. Further reading
7. Model checking in parametric models
7.1. Residuals for parametric models
7.1.1. Standardised residuals
7.1.2. Cox-Snell residuals
7.1.3. Martingale residuals
7.1.4. Deviance residuals
7.1.5. Score residuals
7.2. Residuals for particular parametric models
7.2.1. Weibull distribution
7.2.2. Log-logistic distribution
7.2.3. Lognormal distribution
7.2.4. Analysis of residuals
7.3. Comparing observed and fitted survivor functions
7.4. Identification of influential observations
7.4.1. Influence of observations on a parameter estimate
7.4.2. Influence of observations on the set of parameter estimates
7.5. Testing proportional hazards in the Weibull model
7.6. Further reading
8. Time-dependent variables
8.1. Types of time-dependent variables
8.1.1. Time-dependent coefficients
8.2. Modelling with time-dependent variables
8.2.1. Fitting models with time-dependent variables
8.3. Coding of time-dependent variables
8.4. Estimation of the survivor function
8.5. Model comparison and validation
8.5.1. Comparison of treatments
8.5.2. Assessing model adequacy
8.6. Some applications of time-dependent variables
8.6.1. Some examples
8.7. Joint modelling of longitudinal and survival data
8.7.1. Longitudinal modelling
8.7.2. A joint model
8.7.3. Some extensions to the joint model
8.8. Further reading
9. Interval-censored survival data
9.1. Interval censoring
9.1.1. Current status data
9.2. Estimating the survivor function
9.2.1. Derivation of the estimated survivor function
9.3. Semi-parametric proportional hazards models
9.3.1. Semi-parametric Turnbull model
9.3.2. Piecewise exponential model for interval-censored data
9.4. Parametric models
9.5. Further reading
10. Frailty models
10.1. Introduction to frailty
10.1.1. Random effects
10.1.2. Individual frailty
10.1.3. Shared frailty
10.2. Modelling individual frailty
10.2.1. Frailty distributions
10.2.2. Observable survivor and hazard functions
10.3. The gamma frailty distribution
10.3.1. Impact of frailty on an observable hazard function
10.3.2. Impact of frailty on an observable hazard ratio
10.4. Fitting parametric frailty models
10.4.1. Gamma frailty
10.5. Fitting semi-parametric frailty models
10.5.1. Lognormal frailty effects
10.5.2. Gamma frailty effects
10.6. Comparing models with frailty
10.6.1. Testing for the presence of frailty
10.7. The shared frailty model
10.7.1. Fitting the shared frailty model
10.7.2. Comparing shared frailty models
10.8. Some other aspects of frailty modelling
10.8.1. Model checking
10.8.2. Correlated frailty models
10.8.3. Dependence measures
10.8.4. Numerical problems in model fitting
10.9. Further reading
11. Non-proportional hazards and institutional comparisons
11.1. Non-proportional hazards
11.1.1. Modelling the probability of an event at a given time
11.2. Stratified proportional hazards models
11.2.1. Non-proportional hazards between treatments
11.3. Restricted mean survival
11.3.1. Use of pseudo-values
11.4. Institutional comparisons
11.4.1. Interval estimate for the RAFR
11.4.2. Use of the Poisson regression model
11.4.3. Random institution effects
11.5. Further reading
12. Competing risks
12.1. Introduction to competing risks
12.2. Summarising competing risks data
12.2.1. Kaplan-Meier estimate of survivor function
12.3. Hazard and cumulative incidence functions
12.3.1. Cause-specific hazard function
12.3.2. Cause-specific cumulative incidence function
12.3.3. Some other functions of interest
12.4. Modelling cause-specific hazards
12.4.1. Likelihood functions for competing risks models
12.4.2. Parametric models for cumulative incidence functions
12.5. Modelling cause-specific incidence
12.5.1. The Fine and Gray competing risks model
12.6. Model checking
12.7. Further reading
13. Multiple events and event history modelling
13.1. Introduction to counting processes
13.1.1. Modelling the intensity function
13.1.2. Survival data as a counting process
13.1.3. Survival data in the counting process format
13.1.4. Robust estimation of the variance-covariance matrix
13.2. Modelling recurrent event data
13.2.1. The Anderson and Gill model
13.2.2. The Prentice, Williams, and Peterson model
13.3. Multiple events
13.3.1. The Wei, Lin, and Weissfeld model
13.4. Event history analysis
13.4.1. Models for event history analysis
13.5. Further reading
14. Dependent censoring
14.1. Identifying dependent censoring
14.2. Sensitivity to dependent censoring
14.2.1. A sensitivity analysis
14.2.2. Impact of dependent censoring
14.3. Modelling with dependent censoring
14.3.1. Cox regression model with dependent censoring
14.4. Further reading
15. Sample size requirements for a survival study
15.1. Distinguishing between two treatment groups
15.2. Calculating the required number of deaths
15.2.1. Derivation of the required number of deaths
15.3. Calculating the required number of patients
15.3.1. Derivation of the required number of patients
15.3.2. An approximate procedure
15.4. Further reading
16. Bayesian survival analysis
16.1. Bayes’ theorem
16.2. Bayesian inference
16.3. Bayesian models for survival data
16.3.1. Bayesian version of the simple exponential model
16.4. Incorporating prior knowledge
16.4.1. Non-informative prior information
16.4.2. Vague prior information
16.4.3. Substantial prior information
16.5. Summarising posterior information
16.5.1. Point estimates
16.5.2. Interval estimates
16.5.3. Bayesian hypothesis tests
16.6. Evaluating a posterior distribution
16.6.1. Rejection sampling
16.6.2. Sampling from a posterior distribution using MCMC
16.7. Predictive distributions
16.8. Bayesian model comparison
16.8.1. DIC statistic for comparing models
16.8.2. WAIC statistic for comparing models
16.9. Commentary
16.10. Further reading
17. Survival analysis with R
17.1. Introduction to R
17.2. Data input and editing
17.2.1. Reading and manipulating data from a file
17.2.2. R packages
17.3. Non-parametric procedures
17.4. The Cox regression model
17.4.1. Variable selection and the lasso
17.4.2. Measures of predictive ability and explained variation
17.4.3. Time-dependent ROC curves
17.5. Model checking in the Cox regression model
17.5.1. Analysis of residuals
17.5.2. Identification of influential observations
17.5.3. Testing the assumption of proportional hazards
17.6. Parametric survival models
17.7. Flexible parametric models
17.7.1. Piecewise exponential model
17.7.2. Models for the hazard function
17.7.3. Models for the log-cumulative hazard function
17.8. Model checking in parametric models
17.8.1. Influential values
17.8.2. Comparing observed and fitted survivor functions
17.9. Time-dependent variables
17.9.1. Time-varying coefficients
17.9.2. Joint modelling of longitudinal and survival data
17.10. Interval-censored data
17.10.1. NPMLE of the survivor function
17.10.2. Semi-parametric models for interval-censored data
17.10.3. Parametric models for interval-censored data
17.11. Frailty modelling
17.11.1. Fitting parametric frailty models with individual frailty
17.11.2. Fitting parametric frailty models with shared frailty
17.11.3. Fitting semi-parametric models with individual lognormal frailty
17.11.4. Fitting semi-parametric models with individual gamma frailty
17.11.5. Fitting semi-parametric models with shared frailty
17.12. Competing risks
17.12.1. Estimating and modelling cause-specific hazard functions
17.12.2. Estimating the cumulative incidence function
17.12.3. The Fine and Gray model for cumulative incidence
17.13. Multiple events and event history modelling
17.14. Dependent censoring
17.15. Bayesian survival analysis
17.15.1. Bayesian parametric modelling
17.15.2. Bayesian semi-parametric modelling
17.15.3. Flexible models for the hazard function
17.16. Further reading
A. Maximum likelihood estimation
A.1. Inference about a single unknown parameter
A.2. Inference about a vector of unknown parameters
B. Additional data sets
B.1. Chronic active hepatitis
B.2. Recurrence of bladder cancer
B.3. Survival of black ducks
B.4. Bone marrow transplantation
B.5. Chronic granulomatous disease
Bibliography
Index of examples
Index