1. The parametric frailty model workshop
Luc Duchateau
Ghent University, Belgium

2. Event times in clusters of varying size Many large clusters (1)
One level of clustering, but varying cluster size
Example: Time to culling of heifer cows as a function of early somatic cell count

3. Parametric gamma frailty model Extension: Piecewise constant h(t)
Time to culling as function of SCC and herd as frailty term
Take first constant baseline hazard

4. Time to culling: data
setwd("c://docs//onderwijs//survival//Flames//");culling<-read.table('culling.csv',header=T,sep=";") culltime<-culling$timetocul*12/ logSCC<-culling$logSCC;herd<-as.factor(culling$herd);status<-culling$status n<-length(levels(as.factor(herd)));di<-aggregate(status,by=list(herd),FUN=sum)[,2];r<-sum(di)

5. Time to culling: constant baseline function
#parametric exponential covariate lnscc1 with exp(theta) likelihood.exp<-function(p){ cumhaz<-exp(logSCC*p[1])*exp(p[3])*culltime cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2] lnhaz<-status* (logSCC*p[1]+log(exp(p[3]))) lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2] lik<-r*log(exp(p[2]))-sum((di+1/exp(p[2]))*log(1+cumhaz*exp(p[2])))+sum(lnhaz)-n*log(gamma(1/exp(p[2])))+sum(log(gamma(di+1/exp(p[2])))) -lik }

6. Time to culling: constant baseline maximise function
initial<-c(0,log(0.2),log(0.01)); t<-nlm(likelihood.exp,initial,print.level=2) beta<-t$estimate[1];theta<-exp(t$estimate[2]);lambda<-exp(t$estimate[3]) beta;theta;lambda

7. Time to culling: constant baseline function for standard errors
likelihood.exp<-function(p){ cumhaz<-exp(logSCC*p[1])*p[3]*culltime cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2] lnhaz<-status* (logSCC*p[1]+log(p[3])) lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2] lik<-r*log(p[2])-sum((di+1/p[2])*log(1+cumhaz*p[2]))+sum(lnhaz)-n*log(gamma(1/p[2]))+sum(log(gamma(di+1/p[2]))) -lik } initial<-c(beta,theta,lambda);t<-nlm(likelihood.exp,initial,iterlim=1,hessian=T) beta<-t$estimate[1];theta<-t$estimate[2];lambda<-t$estimate[3] beta;theta;lambda stderr<-sqrt(diag(solve(t$hessian)));stderr

8. Parametric gamma frailty model Extension: Piecewise constant h(t)
Time to culling as function of SCC and herd as frailty term
Take first constant baseline hazard

9. Parametric gamma frailty model Extension: Piecewise constant h(t)
Constant baseline hazard incorrect
Use piecewise constant baseline hazard

10. Time to culling: PW constant baseline: function
endp1<-70*12/365.25;endp2<-300*12/ likelihood.piecexp<-function(p){ cumhaz<-exp(logSCC*p[1])*exp(p[3])*pmin(culltime,endp1) cumhaz<-cumhaz+exp(logSCC*p[1])*exp(p[4])*pmax(0,pmin(endp2-endp1,culltime-endp1)) cumhaz<-cumhaz+exp(logSCC*p[1])*exp(p[5])*pmax(0,culltime-endp2) cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2] lnhaz<-status*(logSCC*p[1]+log(as.numeric(culltime<endp1)*exp(p[3])+as.numeric(endp1<=culltime)*as.numeric(culltime<endp2)*exp(p[4])+as.numeric(culltime>=endp2)*exp(p[5]))) lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2] lik<-r*log(exp(p[2]))-sum((di+1/exp(p[2]))*log(1+cumhaz*exp(p[2])))+sum(lnhaz)-n*log(gamma(1/exp(p[2])))+sum(log(gamma(di+1/exp(p[2])))) -lik } initial<-c(0,log(0.2),log(0.01),log(0.01),log(0.01)) t<-nlm(likelihood.piecexp,initial,print.level=2) beta<-t$estimate[1] theta<-exp(t$estimate[2]) lambda1<-exp(t$estimate[3]) lambda2<-exp(t$estimate[4]) lambda3<-exp(t$estimate[5]) beta theta lambda1 lambda2 lambda3

11. Time to culling: PW constant baseline: function execution
initial<-c(0,log(0.2),log(0.01),log(0.01),log(0.01)) t<-nlm(likelihood.piecexp,initial,print.level=2) beta<-t$estimate[1] theta<-exp(t$estimate[2]) lambda1<-exp(t$estimate[3]) lambda2<-exp(t$estimate[4]) lambda3<-exp(t$estimate[5]) beta;theta;lambda1;lambda2;lambda3

12. Time to culling: PW constant baseline: function execution
#parametric piecewise exponential covariate lnscc1, natural likelihood.piecexp<-function(p){ cumhaz<-exp(logSCC*p[1])*p[3]*pmin(culltime,endp1) cumhaz<-cumhaz+exp(logSCC*p[1])*p[4]*pmax(0,pmin(endp2-endp1,culltime-endp1)) cumhaz<-cumhaz+exp(logSCC*p[1])*p[5]*pmax(0,culltime-endp2) cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2] lnhaz<-status*(logSCC*p[1]+log(as.numeric(culltime<endp1)*p[3]+as.numeric(endp1<=culltime)*as.numeric(culltime<endp2)*p[4]+as.numeric(culltime>=endp2)*p[5])) lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2] lik<-r*log(p[2])-sum((di+1/p[2])*log(1+cumhaz*p[2]))+sum(lnhaz)-n*log(gamma(1/p[2]))+sum(log(gamma(di+1/p[2]))) -lik } initial<-c(beta,theta,lambda1,lambda2,lambda3) t<-nlm(likelihood.piecexp,initial,iterlim=1,hessian=T) beta<-t$estimate[1];theta<-t$estimate[2];lambda1<-t$estimate[3] lambda2<-t$estimate[4];lambda3<-t$estimate[5] beta;theta;lambda1;lambda2;lambda3 stderr<-sqrt(diag(solve(t$hessian)));stderr

13. Parametric gamma frailty model Extension: Piecewise constant h(t)
Constant baseline hazard incorrect
Use piecewise constant baseline hazard

14. Parametric gamma frailty model Comparing PW vs constant
Constant vs piecewise constant model
P<

15. Parametric gamma frailty model Extension: Recurrent events (1)
Recurrent asthma data, drug vs placebo
Patient i has ni at risk periods
Different timescales

16. Parametric gamma frailty model Extension: Recurrent events (2)
Models expressed in terms of conditional hazard and fitted using marginal lilkehood
Calendar time model
Weibull baseline hazard

17. Calendar time: R program Reading the data
setwd("c://docs//onderwijs//survival//Flames//notas//") w <- read.table("asthma.dat",head=T) di<-aggregate(w$st.w,by=list(w$id.w),FUN=sum)[,2];r<-sum(di);n<-length(di) trt<-aggregate(w$trt.w,by=list(w$id.w),FUN=min)[,2] begin<-12*w$start.w/365.25;end<-12*w$stop.w/ ptno<-w$id.w;status<-w$st.w;gap<-end-begin;fevent<-w$fevent

18. Calendar time: R program function
#Weibull-calendar-nofirstevent likelihood.weibull.cp.nof<-function(p){ cumhaz<-p[3]*(end^p[4]-begin^p[4]) cumhaz<-aggregate(cumhaz,by=list(ptno),FUN=sum)[,2] lnhaz<-status*(log(p[3]*p[4]*end^(p[4]-1))) lnhaz<-aggregate(lnhaz,by=list(ptno),FUN=sum)[,2] lik<-r*log(p[2])-n*log(gamma(1/p[2]))+sum(log(gamma(di+1/p[2])))- sum((di+1/p[2])*log(1+p[2]*exp(p[1]*trt)*cumhaz))+ sum(di*p[1]*trt)+sum(lnhaz) -lik} r.weibull.cp.nof<-nlm(likelihood.weibull.cp.nof,c( ,0.5737, ,1.0293),hessian=T,print.level=2)

19. Parametric gamma frailty model Extension: Recurrent events (3)
Gap time model. Information used is
Weibull baseline hazard

20. Parametric gamma frailty model Extension: Recurrent events (4)
Gap time model with hazard for first event different and constant
Combining gap and calendar time

21. Parametric gamma frailty model Extension: Recurrent events (5)

22. Parametric gamma frailty model Extension: Recurrent events (6)