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
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. Progression Free Survival 1 .9 Exponencial2 .8 .7 .6 0 1000 2000 3000 4000 Time in Days
Motivation RCTs follow-up lengths are usually shorter than time horizon of economicevaluations. Parametric Survival analysis can be used to predict full survival. Observed Data Extrapolation
ObjectiveTo present a systematic approach to parametric survivaland how it can be performed using the software STATA.
Parametric Models• Exponential The Mathematical• Weibull Functions are Different• Log-Normal• Gompertz
How to choose a Model? Exponential Weibull Gompertz Log- Normal The Data Choose
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
Weibull Model λ=1 λ=2 λ=5p=0.2 Hazard Function Survival Functionp=1.0p=1.3
LogNormal Model σ=0.5 σ=1.0 σ=1.5μ=0 Hazard Function Survival Functionμ=0.5μ=20
Gompertz Model θ=0.2 θ=0.5 θ=1.2α=-0.01 Hazard Function Survival Functionα= 0α=0.006
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. Survival 1.0 0.8 0.6 0.4 0.2 0.0 0 1000 2000 3000 4000 Time in DaysStata Comand: sts graph
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 Exponencial2Stata Comand: streg, dist(exp) nohr
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 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
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
Survival LinearizationExponential Model 1.5 1.0-log(S(t)) 0.5 0.0 0 1000 2000 3000 4000 t
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
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
Graphical ResultsExponential ModelWeibull ModelLog-Normal ModelGompertz Model
Step 4: Nested Model TestExponential, Weibull and Log-Normal are particular cases of Gamma ModelNule Hypoteses: The Model is SuitableA formal statistical test that compare LikelihoodsExponential Don not needGompertz It is not gamma nestedWeibull P-value = 0.9999 Do not rejectLog-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 BICExponencial 206.7846 209.8906Weibull 208.5774 214.7893LogNormal 209.9704 216.1822Gompertz 208.6454 214.8573
Step 6:Survival ExtrapolationWeibull Survival Is the extrapolated portion Clinically and Biologically Suitable? External Data Observed Data Extrapolation Expert Opinion
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.
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