Pro gradu –thesis Tuija Hevonkorpi. Basic of survival analysis Weibull model Frailty models ...

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 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).

)(1)( tFtS )()( tTPtS

t

duuftTPtF0

)()()(

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

t

tTtTtPth

t

)|(lim)(

0

)()( tHetS duuhtHt

0

)()(, where

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

)()()(

))(log()()(

))(exp()(

0

tSthtf

tSduuhtH

tHtSt

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

1)( tth

)exp()();( 0 Txthxth

hazard decreases monotonically

hazard increases monotonicallyreduces to constant exponential hazard1

1

1

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 %

First Placebo 30 25.21

Active 89 74.79

Second Placebo 29 25.66

Active 84 74.34

Pain episode Treatment group

Time to pain relief

N %

First Placebo 5 min 12 54.44

10 min 5 22.73

15 min 4 18.18

20 min 1 4.55

Active 5 min 36 56.25

10 min 17 26.56

15 min 6 9.38

20 min 3 4.69

30 min 2 3.13

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

2

1

)))(log(()))()(exp(

11log()()log())(log())(log(

jjiji thTRTtHTRT

ddd

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 ui is the predicted frailty term and H0(t) the Weibull baseline hazard,

))()exp(1

1( 0 tHTRT

))(exp( 0 tHui

t

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

tuTRT

tuTRT

i

ii

)exp(

))log()1()log()log()((

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.

))(log())(log( tSthij

)log(1)(

utS

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-likelihood AIC

No frailty 1408.7 1416.7

Gamma frailty 1199.6 1207.6

Log-normal frailty 1251.0 1259.0

AFT model with frailty

1250.5 1258.5

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?