Time to event analysis for repeated pain episodes using frailty models

1. Time to eventanalysisfor repeated pain episodesusingfrailtymodels Pro gradu –thesis Tuija Hevonkorpi

2. Contents • Basic of survival analysis • Weibull model • Frailty models • Accelerated failure time model • Case study

3. Basics of survival analysis • Analysis of data from a given time origin until occurence of a specific point in time • Two main difficulties: Observed survival time are often incomplete Specifying the true survival time

4. Basics of survivalanalysisCensoring • Occurs when the favoured endpoint is not observed • Complicates the exact distribution theory and the estimation of quantiles • Special statistical models and methods for analysing data arises

5. Basics of survivalanalysisSurvivaltime • Moment in time when the patient was recruited until endpoint occurs • Only calculated for those who encounter the endpoint • Survivor function summarises the distribution of the survival times • Censoring time and survival time are statistically independent random variables

6. Basics of survivalanalysisSurvivaltime

7. Basics of survivalanalysisSurvivorfunction • Describes patient´s probability to survive from the time origin t0 over a specific time t. • The probability that survival time is less than t is described with the distribution function of T, F(t).

8. Basics of survivalanalysisHazardfunction • The approximate probability for a patient encountering the endpoint in the next point in time ti+1, on condition that the endpoint has not been encountered at time ti • Connection b/w the hazard and the survivor function can be easily made , where

9. Basics of survivalanalysisSummary • Useful connections between the functions used in the analysis of survival data

10. Weibull model • Survivor function of the Weibull distribution is at the same time a proportional hazards model and an accelerated failure time (AFT) model • Mathematically easy to handle • Characterised by the scale, , and the shape, , parameter • The hazard function: , for 0≤ t < ∞ • The proportional hazards model for a patient i is

11. Weibullmodel hazard decreases monotonically hazard increases monotonically reduces to constant exponential hazard

12. Frailty models • Often survival times are not independent • More than one endpoint occuring for one patient – repeated event times within a patient • The random effect is refered to as frailty • Frailty is unobserved variation between patients - the most frail encounter the endpoint earlier than those not so frail

13. Accelerated failure time model –AFT model • An alternative way to model failure time data • Hazard function does not have to follow a specific distribution • Regression parameters are robust towards the neglected covariates • Best described by the survivor function • The per cent of patients in the group A that live longer than t, is equal to the per cent of patients in the group B that live longer than t • The survival time is speeded up or slowed down by the effect of the explanatory variable

14. Case study • Main objective is to evaluate the time to significant pain relief with the active medication group compared to placebo group • Three models: • A proportional hazards model with Weibull distributed event times and gamma frailty term • A proportional hazards model with Weibull distributed event times and log-normal frailty term • An AFT model with log-normal distributed event times and log-normal frailty term

15. Case study • Patients were randomised in 3:1 ratio in the two treatment groups • 113 patients experienced two pain episodes, 6 patients only one.

16. Case study

17. Case studyModelwith gamma frailty • The model for the log-likelihood function with gamma frailty effect which in the NLMIXED-procedure can be written for patient i as

18. Case studyModelwith gamma frailty • Kaplan-Meier estimate and the population survivor function for the two treatment groups separately

19. Case studyModelwith gamma frailty • The population survivor function is calculated as • The subject specific estimated survivor functions are obtained from where ui is the predicted frailty term and H0(t) the Weibull baseline hazard,

20. Case studyModelwith gamma frailty • Individual and population survivor function estimate for the active treatment group

21. Case studyModelwithlog-normalfrailty • There is no explicit form for the the marginal likelihood • Instead of integrating out the frailty, numerical integration is done using the NLMIXED-procedure in SAS software • The log-likelihood function is of from

22. Case studyAFT modelwithfrailty • AFT model with log-normally distributed event times and log-normal frailty term • The log-likelihood function is where , in where is the cumulative distribution function of the standard normal distribution.

23. Case studyAFT modelwithfrailty • The functionscrossbecausedifferenttreatment is given to differentpatientswhen the usualre-parametrisation of the survivorfunction of the AFT modeldoesnotoccurnecessary

24. Case studyDiscussion • All but log-normal frailty and AFT with frailty differ from each other statistically significantly • In all analyses, the hazard ratio, or the accelerator factor in AFT model, was calculated, and the difference between the two models was not statistically significant

25. Questions?