Pro gradu –thesis Tuija Hevonkorpi

 Basic of survival analysis  Weibull model  Frailty models  Accelerated failure time model  Case study

 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

 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

 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

 Describes patient´s probability to survive from the time origin t 0 over a specific time t.  The probability that survival time is less than t is described with the distribution function of T, F(t).

 The approximate probability for a patient encountering the endpoint in the next point in time t i+1, on condition that the endpoint has not been encountered at time t i  Connection b/w the hazard and the survivor function can be easily made, where

 Useful connections between the functions used in the analysis of survival data

 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

hazard decreases monotonically hazard increases monotonically reduces to constant exponential hazard

 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

 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

 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

 Patients were randomised in 3:1 ratio in the two treatment groups  113 patients experienced two pain episodes, 6 patients only one. Pain episode Treatment group N% FirstPlacebo Active SecondPlacebo Active

Pain episodeTreatment group Time to pain relief N% FirstPlacebo5 min min min min14.55 Active5 min min min min min23.13

 The model for the log-likelihood function with gamma frailty effect which in the NLMIXED- procedure can be written for patient i as

 Kaplan-Meier estimate and the population survivor function for the two treatment groups separately

 The population survivor function is calculated as  The subject specific estimated survivor functions are obtained from where u i is the predicted frailty term and H 0 (t) the Weibull baseline hazard,

 Individual and population survivor function estimate for the active treatment group

 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

 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.

 The functions cross because different treatment is given to different patients when the usual re- parametrisation of the survivor function of the AFT model does not occur necessary

Model- 2 log-likelihoodAIC No frailty Gamma frailty Log-normal frailty AFT model with frailty  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

Questions?