Univariate Survival Analysis Prof. L. Duchateau Ghent University

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

Hazard function  Density function  Cumulative distribution function  Survival function  Hazard function

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

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:

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

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

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

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)

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

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

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,] [2,] [,1] [,2] [1,] NaN [2,] NaN

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

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

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,] [2,]

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,] [2,]

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

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

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

Examples Weibull distributions – varying 

Survival function for Weibull hazard model  Assume

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

Two presentations for Weibull event times and thus:

Two presentations for exponential event times and thus:

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

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

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) trt

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

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

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

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]

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