Flexible Parametric Survival Analysis Using Stata: Beyond the Cox Model

by Patrick Royston and Paul C. Lambert

 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 can be 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.

This text 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 et al. (2010).

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 Poisson-model expression allows 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 time-dependent effects, how these may be modeled, and how to interpret the graphs of the predicted functions the models produce. This material is followed by a chapter on relative survival models such as those used for population-based 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 multiple-event data, and analyzing competing risks.

Table of contents

List of Tables

List of Figures

Preface

1 Theory and Practice

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 proportional-hazards assumption is breached

1.4 Why parametric models?

                    1.4.1 Smooth baseline hazard and survival functions

          1.4.2 Time-dependent 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 Ado-file 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 follow-up time

         2.5.6 Left-truncation

                   2.5.7 Age as the time scale

2.6 The stsplit command

                   2.6.1 Time-dependent effects

         2.6.2 Time-varying 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

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 Out-of-sample 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 Time-dependent 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 Life-table 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 Time-dependent 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

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