This document presents an overview of parametric survival analysis and how it can be used to extrapolate survival data from clinical trials for economic evaluations. It discusses several parametric models including exponential, Weibull, log-normal and Gompertz. It outlines a systematic approach to choosing the best fitting parametric model, including using graphical methods, formal statistical tests of nested models, and information criteria to compare non-nested models. The document applies this process to cardiac patient survival data as a case study and discusses challenges in extrapolating survival curves beyond the observed data.

1. Parametric Survival Analysis
in Health Economics
Patrícia Ziegelmann, Letícia Hermann
UFRGS – Federal University of Rio Grande do Sul – Brazil
IATS – Health Technology Assessment Institute – Brazil
June 2012

2. Survival Analysis
• Statistical models suitable to analyse time to event data with censure.
• Censure: when the event of interest is not observed (because, for example,
lost to follow-up or the end of study follow-up).
• Right Censure: event time > censure time.
• No informative Censure: the censure is independent of the end point event.

3. Parametric
Survival Analysis
• Time to event is model using a parametric (mathematical)
model. For example, a exponential model.
Progression Free Survival
1
.9
Exponencial2
.8
.7
.6
0 1000 2000 3000 4000
Time in Days

4. Motivation
RCTs follow-up lengths are usually shorter than time horizon of economic
evaluations. Parametric Survival analysis can be used to predict full survival.
Observed Data Extrapolation

5. Objective
To present a systematic approach to parametric survival
and how it can be performed using the software STATA.

6. Parametric Models
• Exponential
The Mathematical
• Weibull
Functions are Different
• Log-Normal
• Gompertz

7. How to choose a Model?
Exponential Weibull
Gompertz
Log-
Normal
The Data Choose

8. Exponential Model
λ=0.2 λ=0.5
Hazard Function
Survival Function
• Constant Hazard
λ=1.0 λ=2.0 • λ is the decreasing survival rate

9. Weibull Model
λ=1 λ=2 λ=5
p=0.2
Hazard Function
Survival Function
p=1.0
p=1.3

10. LogNormal Model
σ=0.5 σ=1.0 σ=1.5
μ=0
Hazard Function
Survival Function
μ=0.5
μ=20

11. Gompertz Model
θ=0.2 θ=0.5 θ=1.2
α=-0.01
Hazard Function
Survival Function
α= 0
α=0.006

12. Case Study
• Data from cardiac patients (Hospital in Porto Alegre, Brazil).
• Primary Outcome: all cause mortality.
• Follow-up Time: 4,000 days.
•n = 165 (only 31 all cause death). Lots of Censure !!!!!!

13. Step 1: Kaplan Meyer
• Fit a survival curve using KM (Kaplan Meyer): it is a
nonparametric estimator and a descritive analysis.
Survival
1.0
0.8
0.6
0.4
0.2
0.0
0 1000 2000 3000 4000
Time in Days
Stata Comand: sts graph

14. Step 2: Parametric Fit
• Fit a model: for each parametric function fit the best curve.
λ=0.0016 λ=0.00016 λ=0.00013
Survival
1.0
0.8
0.6
0.4
0.2
0.0
0 1000 2000 3000 4000
Time in Years
Survivor function Exponencial2
Stata Comand: streg, dist(exp) nohr

15. Step 3: Model Fitting
• Graphical Methods: for each parametric curve
Simple method to choose a model.
Has uncertainty and may be inaccurate.
In practice: can be used to check a “bad” fit.

16. Graphic: Survival Functions
• Compare Exponential Survival with KM Survival
Survival
1.0
KM Survival
0.8
Exponential Survival
0.6
0.4
0.2
0.0
0 1000 2000 3000 4000
Time in Years
Survivor function Exponencial

17. Graphics: Cumulative Hazard
Cumulative Hazard
1.5
Exponential Cum Hazard
Cumulative Hazard
1.0
KM Cum Hazard
0.5 0.0
0 1000 2000 3000 4000
analysis time
Cumulative Hazard Kaplan-Meier

18. Survival Linearization
Exponential Model
1.5
1.0
-log(S(t))
0.5
0.0
0 1000 2000 3000 4000
t

19. Graphic: Survival Functions
• Compare Weibull Survival with KM Survival
Survival
1.0
0.8
KM Survival
0.6
Weibull Survival
0.4
0.2
0.0
0 1000 2000 3000 4000
Time in Years
Survivor function Weibull

20. Graphics: Cumulative Hazard
Cumulative Hazard
1.5
Weibull Cum Hazard
KM Cum Hazard
Cumulative Hazard
0.5 0.0 1.0
0 1000 2000 3000 4000
analysis time
Cumulative Hazard Kaplan-Meier

21. Survival Linearization
Weibull Model
0
1.5
-1
1.0
log(-log(S(t)))
-2
-log(S(t))
-3
0.5
-4
-5
0.0
3 4 5 6 7 8
0 1000 ln(t)
2000 3000 4000
t

22. Graphical Results
Exponential Model
Weibull Model
Log-Normal Model
Gompertz Model

23. Step 4: Nested Model Test
Exponential, Weibull and Log-Normal are particular cases of Gamma Model
Nule Hypoteses: The Model is Suitable
A formal statistical test that compare Likelihoods
Exponential Don not need
Gompertz It is not gamma nested
Weibull P-value = 0.9999 Do not reject
Log-Normal P-value = 0.2379 Do not reject

24. Step 5: Model Comparison (AIC e BIC)
• AIC (Akaike´s Information Criterion) anBIC (Bayesian Information Criterion)
are formal Statistical tests to compare model fitting.
• The models compared do not need to be nested.
• Smaller values means better fittings.
Model AIC BIC
Weibull 208.5774 214.7893
LogNormal 209.9704 216.1822
Gompertz 208.6454 214.8573

25. AIC (Akaike´s Information Criterion)
BIC (Bayesian Information Criterion)
• Statistical Tests to compare model fitting.
• The models compared do not need to be nested.
• Smaller values means better fittings.
Model AIC BIC
Exponencial 206.7846 209.8906
Weibull 208.5774 214.7893
LogNormal 209.9704 216.1822
Gompertz 208.6454 214.8573

26. Step 6:Survival Extrapolation
Weibull Survival
Is the extrapolated portion
Clinically and Biologically
Suitable?
External Data
Observed Data Extrapolation Expert Opinion

27. Survival Extrapolation
Log-Normal Survival
Exponential Survival
Weibull Survival
Observed Data Extrapolation Gompertz Survival

28. Discussion
• A large number of economic evaluations need extrapolation to estimate
full survival.
• Parametric Survival Analysis is a helpfull tool for extrapolation. But...
• Alternative Models should be considered.
• The models should be formally compared .
• Reviews should report the methodological process conducted in order to be
transparent and justify their results.
• A good model should provide a good fit to the observed data and the
extrapolated portion should be clinically and biologically plausible.

29. Main References
• COLLETT, D. Modelling Survival Data in Medical Research. 2ª edition. Chapman & Hall, 2003.
• HOSMER, D. W. JR.; LEMESHOW, S. Applied Survival Analysis: regression modeling of time
to event data. John Wiley & Sons, 1999.
• LATIMER, N., Survival Analysis for Economic Evaluations Alongside Clinical Trials –
Extrapolation with Patient-Level Data, Technical Report by NICE
(http://www.nicedsu.org.uk/NICE DSU TSD Survival analysis_finalv2.pdf).
• LEE, E. T.; WANG, J. W. Statistical Methods for Survival Data Analysis.3ª edition. New Jersey:
John Wiley & Sons,2003.

30. Thanks
patricia.ziegelmann@ufrgs.br
“All the Models are Wrong
But Some are Useful”