Parametric Survival Analysis in Health Economics

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”