Flexible Parametric Survival Analysis Using Stata: Beyond the Cox Model
Authors: 
Patrick Royston and Paul C. Lambert 
Publisher: 
Stata Press 
Copyright: 
2011 
ISBN13: 
9781597180795 
Pages: 
347; paperback 
Price: 
$54.00 



Comment from the Stata technical group
Researchers wishing to fit regression models to survival data have long
faced the difficult task of choosing between the Cox model and a parametric
survival model, such as Weibull. Cox models are fit using Stata’s
stcox command, and parametric models are fit using streg,
which offers five parametric forms in addition to Weibull. While the Cox
model makes minimal assumptions about the form of the baseline hazard
function, prediction of hazards and other related functions for a given set
of covariates is hindered by this lack of assumptions; the resulting
estimated curves are not smooth and do not possess information about what
occurs between the observed failure times. Parametric models offer nice,
smooth predictions by assuming a functional form of the hazard, but often
the assumed form is too structured for use with real data, especially if
there exist significant changes in the shape of the hazard over time.
Flexible Parametric Survival Analysis Using Stata: Beyond the Cox Model
is concerned with obtaining a compromise between Cox and
parametric models that retains the desired features of both types of models.
The book is aimed at researchers who are familiar with the basic concepts of
survival analysis and with the stcox and streg commands in
Stata. As such, it is an excellent complement to
An Introduction to Survival Analysis
Using Stata by Cleves, Gould, Gutierrez, and Marchenko.
This book is written for
Stata 12 but is fully compatible with Stata 11 as well.
Much of the text is dedicated to estimation with Royston–Parmar models
using the stpm2 command, which is maintained by the authors and
available from the Statistical Software Components (SSC) archive at
http://www.repec.org.
Royston–Parmar models are highly flexible alternatives to the
exponential, Weibull, loglogistic, and lognormal models (fit using
streg) that allow extension from proportional hazards to proportional
odds and to scaled probit models. Additional flexibility is obtained by the
use of restricted cubic spline functions as alternatives to the linear
functions of log time used in standard models. The authors demonstrate
fitting these models and graphing predicted hazards, cumulative hazards, and
survival functions with real data from breast cancer and prostate cancer
studies.
After some introductory material on the motivation behind flexible
parametric models and on working with survival data in Stata, the authors
proceed by demonstrating that Cox models may instead be expressed as Poisson
models by splitting the time scale at the observed failures. The
Poissonmodel expression allows for extension by changing how the time scale is
split and by introducing restricted cubic splines and fractional
polynomials.
Royston–Parmar models are then introduced, followed by
material on model building and diagnostics for these models. Considerable
attention is then given to timedependent effects, how these may be modeled,
and how to interpret the graphs of the predicted functions that the models
produce. This material is followed by a chapter on relative survival models,
such as those used for populationbased cancer studies. This chapter is
very thorough, relates well to the previous material, and is an ideal
introduction for those new to the concepts of relative survival and excess
mortality. The final chapter is devoted to advanced topics, such as
determining the number needed to treat (NNT), handling multipleevent data,
and analyzing competing risks.
Table of contents
List of tables
List of figures
Preface
1 Introduction
1.1 Goals
1.2 A brief review of the Cox proportional hazards model
1.3 Beyond the Cox model
1.3.1 Estimating the baseline hazard
1.3.2 The baseline hazard contains useful information
1.3.3 Advantages of smooth survival functions
1.3.4 Some requirements of a practical survival analysis
1.3.5 When the proportionalhazards assumption is breached
1.4 Why parametric models?
1.4.1 Smooth baseline hazard and survival functions
1.4.2 Timedependent HRs
1.4.3 Modeling on different scales
1.4.4 Relative survival
1.4.5 Prediction out of sample
1.4.6 Multiple time scales
1.5 Why not standard parametric models?
1.6 A brief introduction to
stpm2
1.6.1 Estimation (model fitting)
1.6.2 Postestimation facilities (prediction)
1.7 Basic relationships in survival analysis
1.8 Comparing models
1.9 The delta method
1.10 Adofile resources
1.11 How our book is organized
2 Using stset and stsplit
2.1 What is the
stset command?
2.2 Some key concepts
2.3 Syntax of the
stset command
2.4 Variables created by the
stset command
2.5 Examples of using
stset
2.5.1 Standard survival data
2.5.2 Using the scale( ) option
2.5.3 Date of diagnosis and date of exit
2.5.4 Date of diagnosis and date of exit with the scale( ) option
2.5.5 Restricting the followup time
2.5.6 Lefttruncation
2.5.7 Age as the time scale
2.6 The
stsplit command
2.6.1 Timedependent effects
2.6.2 Timevarying covariates
2.7 Conclusion
3 Graphical introduction to the principal datasets
3.1 Introduction
3.2 Rotterdam breast cancer data
3.3 England and Wales breast cancer data
3.4 Orchiectomy data
3.5 Conclusion
4 Poisson models
4.1 Introduction
4.2 Modeling rates with the Poisson distribution
4.3 Splitting the time scale
4.3.1 The piecewise exponential model
4.3.2 Time as just another covariate
4.4 Collapsing the data to speed up computation
4.5 Splitting at unique failure times
4.5.1 Technical note: Why the Cox and Poisson approaches are equivalent
4.6 Comparing a different number of intervals
4.7 Fine splitting of the time scale
4.8 Splines: Motivation and definition
4.8.1 Calculating splines
4.8.2 Restricted cubic splines
4.8.3 Splines: Application to the Rotterdam data
4.8.4 Varying the number of knots
4.8.5 Varying the location of the knots
4.8.6 Estimating the survival function
4.9 FPs: Motivation and definition
4.9.1 Application to Rotterdam data
4.9.2 Higher order FP models
4.9.3 FP function selection procedure
4.10 Discussion
5 Royston–Parmar models
5.1 Motivation and introduction
5.1.1 The exponential distribution
5.1.2 The Weibull distribution
5.1.3 Generalizing the Weibull
5.1.4 Estimating the hazard function
5.2 Proportional hazards models
5.2.1 Generalizing the Weibull
5.2.2 Example
5.2.3 Comparing parameters of PH(1) and Weibull models
5.3 Selecting a spline function
5.3.1 Knot positions
Example
5.3.2 How many knots?
5.4 PO models
5.4.1 Introduction
5.4.2 The loglogistic model
5.4.3 Generalizing the loglogistic model
5.4.4 Comparing parameters of PO(1) and loglogistic models
Example
5.5 Probit models
5.5.1 Motivation
5.5.2 Generalizing the probit model
5.5.3 Comparing parameters of probit(1) and lognormal models
5.5.4 Comments on probit and POs models
5.6 Royston–Parmar (RP) models
5.6.1 Models with θ not equal to 0 or 1
5.6.2 Example
5.6.3 Likelihood function and parameter estimation
5.6.4 Comparing regression coefficients
5.6.5 Model selection
5.6.6 Sensitivity to number of knots
5.6.7 Sensitivity to location of knots
5.7 Concluding remarks
6 Prognostic models
6.1 Introduction
6.2 Developing and reporting a prognostic model
6.3 What does the baseline hazard function mean?
6.3.1 Example
6.4 Model selection
6.4.1 Choice of scale and baseline complexity
Example
6.4.2 Selection of variables and functional forms
Example
6.5 Quantitative outputs from the model
6.5.1 Survival probabilities for individuals
6.5.2 Survival probabilities across the risk spectrum
6.5.3 Survival probabilities at given covariate values
6.5.4 Survival probabilities in groups
6.5.5 Plotting adjusted survival curves
6.5.6 Plotting differences between survival curves
6.5.7 Centiles of the survival distribution
6.6 Goodness of fit
6.6.1 Example
6.7 Discrimination and explained variation
6.7.1 Example
6.7.2 Harrell’s C index of concordance
6.8 Outofsample prediction: Concept and applications
6.8.1 Extrapolation of survival functions: Basic technique
6.8.2 Extrapolation of survival functions: Further investigations
6.8.3 Validation of prognostic models: Basics
6.8.4 Validation of prognostic models: Further comments
6.9 Visualization of survival times
6.9.1 Example
6.10 Discussion
7 Timedependent effects
7.1 Introduction
7.2 Definitions
7.3 What do we mean by a TD effect?
7.4 Proportional on which scale?
7.5 Poisson models with TD effects
7.5.1 Piecewise models
7.5.2 Using restricted cubic splines
7.6 RP models with TD effects
7.6.1 Piecewise HRs
7.6.2 Continuous TD effects
7.6.3 More than one TD effect
7.6.4 Stratification is the same as including TD effects
7.7 TD effects for continuous variables
7.8 Attained age as the time scale
7.8.1 The orchiectomy data
7.8.2 Proportional hazards model
7.8.3 TD model
7.9 Multiple time scales
7.10 Prognostic models with TD effects
7.10.1 Example
7.11 Discussion
8 Relative survival
8.1 Introduction
8.2 What is relative survival?
8.3 Excess mortality and relative survival
8.3.1 Excess mortality
8.3.2 Relative survival is a ratio
8.4 Motivating example
8.5 Lifetable estimation of relative survival
8.5.1 Using strs
8.6 Poisson models for relative survival
8.6.1 Piecewise models
8.6.2 Restricted cubic splines
8.7 RP models for relative survival
8.7.1 Likelihood for relative survival models
8.7.2 Proportional cumulative excess hazards
8.7.3 RP models on other scales
8.7.4 Application to England and Wales breast cancer data
8.7.5 Relative survival models on other scales
8.7.6 Timedependent effects
8.8 Some comments on model selection
8.9 Age as a continuous variable
8.10 Concluding remarks
9 Further topics
9.1 Introduction
9.2 Number needed to treat
9.2.1 Example
9.3 Average and adjusted survival curves
9.3.1 Renal data
9.4 Modeling distributions with RP models
9.4.1 Example 1: Rotterdam breast cancer data
9.4.2 Example 2: CD4 lymphocyte data
9.4.3 Example 3: Prostate cancer data
9.5 Multiple events
9.5.1 Introduction
9.5.2 The AG model
9.5.3 The WLW model
9.5.4 The PWP model
9.5.5 Multiple events in RP models
9.5.6 Summary
9.6 Bayesian RP models
9.6.1 Introduction
9.6.2 The “zeros trick” in WinBUGS
9.6.3 Fitting a RP model
9.6.4 Summary
9.7 Competing risks
9.7.1 Summary
9.8 Period analysis
9.8.1 Introduction
9.8.2 What is period analysis?
9.8.3 Application to England and Wales breast cancer data
9.9 Crude probability of death from relative survival models
9.9.1 Introduction
9.9.2 Application to England and Wales breast cancer data
9.9.3 Conclusion
9.10 Final remarks
References
Author index
Subject index