Modelling Survival Data in Medical Research

Description

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 dataContains practical guidance on modelling survival data from the author’s many years of experience in teaching and consultancyShows how Bayesian methods can be used to analyse survival dataIncludes details on how R can be used to carry out all the methods described, with guidance on the interpretation of the resulting outputContains many real data examples and additional data sets that can be used for courseworkAll 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. 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