Parametric Survival Analysis in Health Economics

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

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.

Parametric Survival Analysis

• Time to event is model using a parametric (mathematical) model. For example, a exponential model.

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Progression Free Survival

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.

Objective

To present a systematic approach to parametric survival

and how it can be performed using the software STATA.

Parametric Models

• Exponential

• Weibull

• Log-Normal

• Gompertz

The Mathematical

Functions are Different

How to choose a Model?

The Data Choose

Gompertz

Exponential Weibull

Log-Normal

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

Weibull Model

Hazard Function

Survival Function

λ=1 λ=2 λ=5

p=0.2

p=1.0

p=1.3

LogNormal Modelσ=0.5

μ=0

Hazard Function

Survival Functionμ=0.5

μ=20

σ=1.0 σ=1.5

Gompertz Modelθ=0.2

α=-0.01

Hazard Function

Survival Functionα= 0

α=0.006

θ=0.5 θ=1.2

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 !!!!!!

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

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Survival

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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

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.

Graphic: Survival Functions

• Compare Exponential Survival with KM Survival

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Survivor function Exponencial

Survival

KM Survival

Exponential Survival

Graphics: Cumulative Hazard0

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Cumulative Hazard Kaplan-Meier

Cumulative Hazard

Exponential Cum Hazard

KM Cum Hazard

Survival LinearizationExponential Model

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Graphic: Survival Functions

• Compare Weibull Survival with KM Survival

KM Survival

Weibull Survival

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Survivor function Weibull

Survival

Graphics: Cumulative Hazard

Weibull Cum Hazard

KM Cum Hazard

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Cumulative Hazard Kaplan-Meier

Cumulative Hazard

Survival LinearizationWeibull Model

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Exponential Model

Weibull Model

Log-Normal Model

Gompertz Model

Graphical Results

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

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

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

Step 6:Survival Extrapolation

Observed Data Extrapolation

Is the extrapolated portion

Clinically and Biologically

Suitable?

External Data

Weibull Survival

Expert Opinion

Survival Extrapolation

Observed Data Extrapolation

Log-Normal Survival

Weibull Survival

Exponential Survival

Gompertz Survival

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.

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.

Thanks

patricia.ziegelmann@ufrgs.br

“All the Models are WrongBut Some are Useful”