Parametric Survival Analysis in Health Economics
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Patrícia Ziegelmann, Letícia Hermann
UFRGS – Federal University of Rio Grande do Sul – BrazilIATS – Health Technology Assessment Institute – Brazil
June 2012
Parametric Survival Analysis
in Health Economics
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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.
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Parametric Survival Analysis
• Time to event is model using a parametric (mathematical) model. For example, a exponential model.
.6.7
.8.9
1E
xpon
enci
al2
0 1000 2000 3000 4000Time in Days
Progression Free Survival
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Motivation
Observed Data Extrapolation
RCTs follow-up lengths are usually shorter than time horizon of economic evaluations. Parametric Survival analysis can be used to predict full survival.
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Objective
To present a systematic approach to parametric survival
and how it can be performed using the software STATA.
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Parametric Models
• Exponential
• Weibull
• Log-Normal
• Gompertz
The Mathematical
Functions are Different
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How to choose a Model?
The Data Choose
Gompertz
Exponential Weibull
Log-Normal
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Exponential Model
λ=0.2 λ=0.5
λ=1.0 λ=2.0
Hazard Function
Survival Function
• Constant Hazard• λ is the decreasing survival rate
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Weibull Model
Hazard Function
Survival Function
λ=1 λ=2 λ=5
p=0.2
p=1.0
p=1.3
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LogNormal Modelσ=0.5
μ=0
Hazard Function
Survival Functionμ=0.5
μ=20
σ=1.0 σ=1.5
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Gompertz Modelθ=0.2
α=-0.01
Hazard Function
Survival Functionα= 0
α=0.006
θ=0.5 θ=1.2
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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 !!!!!!
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Step 1: Kaplan Meyer
• Fit a survival curve using KM (Kaplan Meyer): it is a nonparametric estimator and a descritive analysis.
Stata Comand: sts graph
0.0
0.2
0.4
0.6
0.8
1.0
0 1000 2000 3000 4000Time in Days
Survival
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0.0
0.2
0.4
0.6
0.8
1.0
0 1000 2000 3000 4000Time in Years
Survivor function Exponencial2
Survival
Step 2: Parametric Fit
• Fit a model: for each parametric function fit the best curve.
Stata Comand: streg, dist(exp) nohr
λ=0.0016 λ=0.00016 λ=0.00013
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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.
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Graphic: Survival Functions
• Compare Exponential Survival with KM Survival
0.0
0.2
0.4
0.6
0.8
1.0
0 1000 2000 3000 4000Time in Years
Survivor function Exponencial
Survival
KM Survival
Exponential Survival
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Graphics: Cumulative Hazard0
.00
.51
.01
.5C
umu
lativ
e H
aza
rd
0 1000 2000 3000 4000analysis time
Cumulative Hazard Kaplan-Meier
Cumulative Hazard
Exponential Cum Hazard
KM Cum Hazard
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Survival LinearizationExponential Model
0.0
0.5
1.0
1.5
-log
(S(t
))
0 1000 2000 3000 4000t
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Graphic: Survival Functions
• Compare Weibull Survival with KM Survival
KM Survival
Weibull Survival
0.0
0.2
0.4
0.6
0.8
1.0
0 1000 2000 3000 4000Time in Years
Survivor function Weibull
Survival
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Graphics: Cumulative Hazard
Weibull Cum Hazard
KM Cum Hazard
0.0
0.5
1.0
1.5
Cum
ula
tive
Ha
zard
0 1000 2000 3000 4000analysis time
Cumulative Hazard Kaplan-Meier
Cumulative Hazard
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Survival LinearizationWeibull Model
0.0
0.5
1.0
1.5
-log
(S(t
))
0 1000 2000 3000 4000t
-5-4
-3-2
-10
log
(-lo
g(S
(t))
)
3 4 5 6 7 8ln(t)
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Exponential Model
Weibull Model
Log-Normal Model
Gompertz Model
Graphical Results
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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
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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
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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
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Step 6:Survival Extrapolation
Observed Data Extrapolation
Is the extrapolated portion
Clinically and Biologically
Suitable?
External Data
Weibull Survival
Expert Opinion
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Survival Extrapolation
Observed Data Extrapolation
Log-Normal Survival
Weibull Survival
Exponential Survival
Gompertz Survival
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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.
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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.