1. Univariate Survival Analysis Prof. L. Duchateau Ghent University

2. Model specification  Most survival models are defined in terms of the hazard with the hazard at time t for subject i the baseline hazard at time t the incidence vector for subject i the parameter vector

3. Hazard function  Density function  Cumulative distribution function  Survival function  Hazard function

5. Alternative models  Hazard model: Baseline hazard function parametric Baseline hazard function unspecified Summary measure: hazard ratio  Accelerated failure time (AFT) model: Typically parametric Summary measure: accelerator factor

6. Parametric hazard model: Analytical solution  Assume constant baseline hazard (exponential lifetimes) with only control and treated group with = 0 for control and =1 for treated  Likelihood for exponential:

7. Likelihood specification constant hazard  Define as number of events in treated (control) group and  Define as at risk time in treated (control) group and

8. Solution from likelihood specification  Maximise the log likelihood function leading to

9. Analytical solution reconstitution data #The analytical solution DT<-sum(stat[trt==1]); DC<-sum(stat[trt==0]); yT<-sum(timerec[trt==1]);yC<-sum(timerec[trt==0]); lambda<-DC/yC;HR<-(DT/yT)/(DC/yC) lambda;HR

10. Exercise Obtain the analytical solution for the diagnosis data set First rework the data: #Read the data diag<-read.table("timetodiag.csv",header=T,sep=";") #Create 5 column vectors, five different variables timetodiag<-c(diag$t1,diag$t2) stat<-c(diag$c1,diag$c2) technique<-c(rep(0,106),rep(1,106)) dogid<-c(diag$dogid,diag$dogid) diagnosis<- data.frame(dogid=dogid,technique=technique,timetodiag=t imetodiag,stat=stat)

11. Variance of the estimates?  Obtain the Hessian, i.e., the matrix of the second derivatives of the log likelihood which is  The information matrix is then

12. Inverse of observed information matrix  The observed information matrix is thus and the asymptotic variance-covariance matrix is

13. Variance estimators #The observed information matrix I<-matrix(data=c((DT+DC)/(lambda^2), yT*HR,yT*HR,lambda*yT*HR), nrow = 2, ncol = 2) V<-solve(I) V;sqrt(V) [,1] [,2] [1,] 0.0006494837 -0.003003433 [2,] -0.0030034326 0.026234568 [,1] [,2] [1,] 0.02548497 NaN [2,] NaN 0.1619709

14. Exercise Obtain the asymptotic variance estimates for the parameters of the diagnosis data set

15. Maximizer solution reconstitution data #(negative) loglikelihood exponential with l=exp(p[1]), beta=p[2] loglikelihood.exponential<-function(p){ cumhaz<- exp(p[1])*timerec*(exp(p[2]*trt)) hazard<-stat*log(exp(p[1])*exp(p[2]*trt)) loglik<-sum(hazard)-sum(cumhaz) -loglik} #Apply minimizer to minus loglikelihood function res<-nlm(loglikelihood.exponential,c(-1,0)) res;lambda<-exp(res$estimate[1]);HR<- exp(res$estimate[2]) lambda;HR

16. Variances from maximizer solution #Apply minimizer to obtain Hessian matrix res<-nlm(loglikelihood.exponential,c(-1,0),hessian=T) solve(res$hessian) [,1] [,2] [1,] 0.01388753 -0.01388753 [2,] -0.01388753 0.02623197

17. Use parameters of interest as input #(negative) loglikelihood exponential with l=p[1], HR=p[2] loglikelihood.exponentialHR<-function(p){ cumhaz<- p[1]*timerec*(exp(log( p[2])*trt)) hazard<-stat*log(p[1]*exp(log(p[2])*trt)) loglik<-sum(hazard)-sum(cumhaz) -loglik} #Apply minimizer to obtain Hessian matrix res<- nlm(loglikelihood.exponentialHR,c(lambda,HR),hessian=T,iterl im=1) solve(res$hessian) [,1] [,2] [1,] 0.0006509066 -0.003590494 [2,] -0.0035904937 0.037368602

18. Exercise Obtain the parameter estimates and their variance for the diagnosis data set using the maximizer

19. Standard software? #Univariate model-exponential library(survival) res.unadjust<- survreg(Surv(timerec,stat)~trt,dist="exponential",data=reconstituti on) res.unadjust summary(res.unadjust) lambda<-res.unadjust$coef[1]; beta<-res.unadjust$coef[2];HR<-exp(beta) lambda;beta;HR

20. Loglinear model representation  Hazard model with parametric baseline hazard can be rewritten in a loglinear model representation  Most often used:

21. Examples Weibull distributions – varying 

22. Survival function for Weibull hazard model  Assume

23. Survival function for Weibull loglinear model  Assume with  From this follows that and thus  Based on the Gumbel assumption, the survival function becomes

24. Two presentations for Weibull event times and thus:

25. Two presentations for exponential event times and thus:

26. survreg function #unadjusted model-exponential res.unadjust<- survreg(Surv(timerec,stat)~trt,dist="exponential",data=reconstituti on) res.unadjust;summary(res.unadjust) mu<-res.unadjust$coef[1];alpha<-res.unadjust$coef[2]; lambda<- exp(-mu);beta<- -alpha;HR<-exp(beta) lambda;beta;HR

27. Exercise Obtain the parameter estimates of the diagnosis data set using the survreg function in R

28. survreg function variances #unadjusted model-exponential res.unadjust<- survreg(Surv(timerec,stat)~trt,dist="exponential",data=reconstituti on) res.unadjust;summary(res.unadjust) mu<-res.unadjust$coef[1];alpha<-res.unadjust$coef[2]; lambda<- exp(-mu);beta<- -alpha;HR<-exp(beta) lambda;beta;HR res.unadjust$var (Intercept) trt (Intercept) 0.01388889 -0.01388889 trt -0.01388889 0.02623457

29. The delta method on variance  Obtaining the variance of using the variance of

30. The delta method - general  Original parameters  Interest in univariate cont. function  Use one term Taylor expansion of with

31. The delta method - specific  Interest in univariate cont. function  The one term Taylor expansion of With

32. survreg function variances #unadjusted model-exponential-variances of transformed variables lambda<- exp(-mu);beta<- -alpha;HR<-exp(beta) Vlambda<- res.unadjust$var[1,1]*(lambda^2) Vbeta<- res.unadjust$var[2,2]

33. Exercise Obtain the variances parameter estimates of the diagnosis data set using the survreg function in R, and applying the delta method