An Introduction to Survival Analysis Using Stata, Second Edition

Mario Cleves, William W. Gould, Roberto G. Gutierrez, and Yulia Marchenko

An Introduction to Survival Analysis Using Stata, Second Edition is the ideal tutorial for professional data analysts who want to learn survival analysis for the first time or who are well versed in survival analysis but not as dexterous in using Stata to analyze survival data. This text also serves as a valuable reference to those who already have experience using Stata’s survival analysis routines.

The second edition has been updated for Stata 10, containing a new chapter on power and sample-size calculations for survival studies and sections that describe how to fit regression models (stcox and streg) in the presence of complex survey data. Other enhancements include discussions about nonparametric estimation of mean/median survival, survival graphs with embedded at-risk tables, better hazard graphs through the use of boundary kernels, and concordance measures for assessing the predictive accuracy of the Cox model, as well as an expanded discussion of model building strategies including the use of fractional polynomials.

Survival analysis is a field of its own requiring specialized data management and analysis procedures. Toward this end, Stata provides the st family of commands for organizing and summarizing survival data. The authors of this text are also the authors of Stata’s st commands.

This book provides statistical theory, step-by-step procedures for analyzing survival data, an in-depth usage guide for Stata’s most widely used st commands, and a collection of tips for using Stata to analyze survival data and present the results. This book develops from first principles the statistical concepts unique to survival data and assumes only a knowledge of basic probability and statistics and a working knowledge of Stata.

The first three chapters of the text cover basic theoretical concepts: hazard functions, cumulative hazard functions, and their interpretations; survivor functions; hazard models; and a comparison of nonparametric, semiparametric, and parametric methodologies. Chapter 4 deals with censoring and truncation. The next three chapters cover the formatting, manipulation, stsetting, and error checking involved in preparing survival data for analysis using Stata’s st analysis commands. Chapter 8 covers nonparametric methods, including the Kaplan–Meier and Nelson–Aalen estimators, and the various nonparametric tests for the equality of survival experience.

Chapters 9–11 discuss Cox regression and include various examples of fitting a Cox model, obtaining predictions, interpreting results, building models, and model diagnostics. The next four chapters cover parametric models, which are fit using Stata’s streg command. These chapters include detailed derivations of all six parametric models currently supported in Stata and methods for determining which model is appropriate, as well as information on obtaining predictions, stratification, and advanced topics such as frailty models. The final chapter is devoted to power and sample-size calculations for survival studies.

Preface to the Second Edition

Preface to the Revised Edition

Preface to first edition

Notation and Typography

1 The problem of survival analysis

1.1 Parametric modeling

1.2 Semiparametric modeling

1.3 Nonparametric analysis

1.4 Linking the three approaches

2 Describing the distribution of failure times

2.1 The survivor and hazard functions

2.2 The quantile function

2.3 Interpreting the hazard and cumulative hazard

2.3.1 Interpreting the cumulative hazard

2.3.2 Interpreting the hazard rate

2.4 Means and medians

3 Hazard models

3.1 Parametric models

3.2 Semiparametric models

3.3 Analysis time (time at risk)

4 Censoring and truncation

4.1 Censoring

4.1.1 Right censoring

4.1.2 Interval censoring

4.1.3 Left censoring

4.2 Truncation

4.2.1 Left truncation (delayed entry)

4.2.2 Interval truncation (gaps)

4.2.3 Right truncation

5 Recording survival data

5.1 The desired format

5.2 Other formats

5.3 Example: Wide-form snapshot data

6 Using stset

6.1 A short lesson on dates

6.2 The purpose of the stset command

6.3 The syntax of the stset command

6.3.1 Specifying analysis time

6.3.2 Variables defined by stset

6.3.3 Specifying what constitutes failure

6.3.4 Specifying when subjects exit from the analysis

6.3.5 Specifying when subjects enter the analysis

6.3.6 Specifying the subject-id variable

6.3.7 Specifying the begin-of-span variable

6.3.8 Convenience options

7 After stset

7.1 Look at stset's output

7.2 List some of your data

7.3 Use stdescribe

7.4 Use stvary

7.5 Perhaps use stfill

7.6 Example: Hip fracture data

8 Nonparametric analysis

8.1 Inadequacies of standard univariate methods

8.2 The Kaplan–Meier estimator

8.2.1 Calculation

8.2.2 Censoring

8.2.3 Left truncation (delayed entry)

8.2.4 Interval truncation (gaps)

8.2.5 Relationship to the empirical distribution function

8.2.6 Other uses of sts list

8.2.7 Graphing the Kaplan–Meier estimate

8.3 The Nelson–Aalen estimator

8.4 Tests of hypothesis

8.4.1 The log-rank test

8.4.2 The Wilcoxon test

8.4.3 Other tests

8.4.4 Stratified tests

9 The Cox proportional hazards model

9.1 Using stcox

9.1.1 The Cox model has no intercept

9.1.2 Interpreting coefficients

9.1.3 The effect of units on coefficients

9.1.4 Estimating the baseline cumulative hazard and survivor functions

9.1.5 Estimating the baseline hazard function

9.1.6 The effect of units on the baseline functions

9.2 Likelihood calculations

9.2.1 No tied failures

9.2.2 Tied failures

The marginal calculation

The partial calculation

The Breslow approximation

The Efron approximation

9.2.3 Summary

9.3 Stratified analysis

9.3.1 Obtaining coefficient estimates

9.3.2 Obtaining estimates of baseline functions

9.4 Coxed Models with shared frailty

9.4.1 Parameter Estimation

9.4.2 Obtaining Estimates of Base Functions

9.5 Cox models with survey data

9.5.1 Declaring survey characteristics

9.5.2 Fitting a Cox model with survey data

9.5.3 Some caveats of analyzing survival data from complex survey designs

10 Model building using stcox

10.1 Indicator variables

10.2 Categorical variables

10.3 Continuous variables

10.3.1 Fractional polynomials

10.4 Interactions

10.5 Time-varying variables

10.5.1 Using stcox, tvc() texp()

10.5.2 Using stsplit

10.6 Modeling group effects: fixed-effects, random-effects, stratification, and clustering

11 The Cox model: Diagnostics

11.1 Testing the proportional hazards assumption

11.1.1 Tests based on re-estimation

11.1.2 Test based on Schoenfeld residuals

11.1.3 Graphical methods

11.2 Residuals

Reye's syndrome data

11.2.1 Determining functional form

11.2.2 Goodness of fit

11.2.3 Outliers and influential points

12 Parametric models

12.1 Motivation

12.2 Classes of parametric models

12.2.1 Parametric proportional hazards models

12.2.2 Accelerated failure time-models

12.2.3 Comparing the two parameterizations

13 A survey of parametric regression models in Stata

13.1 The exponential model

13.1.1 Exponential regression in the PH metric

13.1.2 Exponential regression in the AFT metric

13.2 Weibull regression

13.2.1 Weibull regression in the PH metric

Fitting null models

13.2.2 Weibull regression in the AFT metric

13.3 Gompertz regression (PH metric)

13.4 Log-normal regression (AFT metric)

13.5 Log-logistic regression (AFT metric)

13.6 Generalized gamma regression (AFT metric)

13.7 Choosing among parametric models

13.7.1 Nested models

13.7.2 Non-nested models

14 Post-estimation commands for parametric models

14.1 Use of predict after streg

14.1.1 Predicting the time of failure

14.1.2 Predicting the hazard and related functions

14.1.3 Calculating residuals

14.2 Using stcurve

15 Generalizing the parametric regression model

15.1 Using the ancillary() option

15.2 Stratified models

15.3 Frailty models

15.3.1 Unshared frailty models

15.3.2 Example: Kidney data

15.3.3 Testing for heterogeneity

15.3.4 Shared frailty models

16 Power and sample-size determination for survival analysis

16.1 Estimating sample size

16.1.1 Multiple-myeloma data

16.1.2 Comparing two survivor functions nonparametrically

16.1.3 Comparing two exponential survivor functions

16.1.4 Cox regression models

16.2 Accounting for withdrawal and accrual of subjects

16.2.1 The effect of withdrawal or loss to follow-up

16.2.2 The effect of accrual

16.2.3 Examples

16.3 Estimating power and effect size

16.4 Tabulating or graphing results

References

Author Index

Subject Index