1. Basics of the parametric frailty model
Luc Duchateau
Ghent University, Belgium

2. Overview Frailty distributions The parametric gamma frailty model
The parametric positive stable frailty model
The parametric lognormal frailty model

3. Frailty distributions
Power variance function family
Gamma
Inverse Gaussian
Positive stable
General PVF
Compound Poisson
Lognormal

4. Parametric gamma frailty model Frailty density function (1)
Two-parameter gamma density
One-parameter gamma density

5. Parametric gamma frailty model Frailty density function examples
One-parameter gamma density

6. Parametric gamma frailty model Laplace transform of frailty density
Characteristic function
Moment generating function
Laplace transform for positive r.v.

7. Parametric gamma frailty model Laplace transf. generates moments
Generate nth moment
Use nth derivative of Laplace transform
Evaluate at s=0

8. Parametric gamma frailty model Gamma Laplace transform

9. Parametric gamma frailty model Joint survival function (1)
Joint survival function in conditional model
Now use notation
For cluster with covariates

10. Parametric gamma frailty model Joint survival function (2)
Applied to Laplace transform of gamma distribution we obtain

11. Parametric gamma frailty model Population survival function (1)
Integrate conditional survival function
Population density function
Population hazard function

12. Parametric gamma frailty model Population survival function (2)
Applied to gamma distribution we have
Population hazard function

13. Graphically #Set parameters
condHR<-2;Ktau.list<-c(0.05,0.1,0.25,0.5,0.75)
Theta.list<-2*Ktau.list/(1-Ktau.list);Sij<-seq(0.999,0.001,-0.001);Fij<-1-Sij
#Plot population/conditional hazard
plot(Fij,(Sij)^(Theta.list[1]),xlab="Sx,f(t)",type="n",ylab="Population/conditional hazard",axes=F,ylim=c(0,1.7))
box();axis(1,at=seq(0,1,0.2),labels=seq(1,0,-0.2),lwd=0.5);axis(2,lwd=0.5)
lines(c(Fij,1),c((Sij)^(Theta.list[1]),0),lty=1,lwd=1)
lines(c(Fij,1),c((Sij)^(Theta.list[2]),0),lty=2,lwd=1)
lines(c(Fij,1),c((Sij)^(Theta.list[3]),0),lty=3,lwd=1)
lines(c(Fij,1),c((Sij)^(Theta.list[4]),0),lty=4,lwd=1)
lines(c(Fij,1),c((Sij)^(Theta.list[5]),0),lty=5,lwd=1)
legend(0,1.75,legend=c(expression(paste(tau,"=0.05, ",theta,"=0.105")),
expression(paste(tau,"=0.10, ",theta,"=0.222")),
expression(paste(tau,"=0.25, ",theta,"=0.500")),
expression(paste(tau,"=0.50, ",theta,"=2.000")),
expression(paste(tau,"=0.75, ",theta,"=6.000"))),
ncol=2,lty=c(1,2,3,4,5))

14. Parametric gamma frailty model Population vs conditional hazard

15. Parametric gamma frailty model Population hazard ratio
Using population hazard functions
For the gamma frailty distribution

16. Graphically
plot(Fij,(Sij^(-Theta.list[1]))/(1/condHR+Sij^(-Theta.list[1])-1)
,xlab="Sx,f(t)",type="n",ylab="Population hazard ratio",axes=F,ylim=c(1,2.5))
box()
axis(1,at=seq(0,1,0.2),labels=seq(1,0,-0.2),lwd=0.5)
axis(2,at=seq(1,2.5,0.5),labels=seq(1,2.5,0.5),srt=90,lwd=0.5)
lines(c(Fij,1),c((Sij^(-Theta.list[1]))/(1/condHR+Sij^(-Theta.list[1])-1),1),lty=1,lwd=1)
lines(c(Fij,1),c((Sij^(-Theta.list[2]))/(1/condHR+Sij^(-Theta.list[2])-1),1),lty=2,lwd=1)
lines(c(Fij,1),c((Sij^(-Theta.list[3]))/(1/condHR+Sij^(-Theta.list[3])-1),1),lty=3,lwd=1)
lines(c(Fij,1),c((Sij^(-Theta.list[4]))/(1/condHR+Sij^(-Theta.list[4])-1),1),lty=4,lwd=1)
lines(c(Fij,1),c((Sij^(-Theta.list[5]))/(1/condHR+Sij^(-Theta.list[5])-1),1),lty=5,lwd=1)
legend(0,2.5,legend=c(expression(paste(tau,"=0.05, ",theta,"=0.105")),
expression(paste(tau,"=0.10, ",theta,"=0.222")),
expression(paste(tau,"=0.25, ",theta,"=0.500")),
expression(paste(tau,"=0.50, ",theta,"=2.000")),
expression(paste(tau,"=0.75, ",theta,"=6.000"))),
ncol=2,lty=c(1,2,3,4,5))

17. Parametric gamma frailty model Population hazard ratio example

18. Parametric gamma frailty model The conditional frailty density (1)
Assuming no covariate information
which corresponds for gamma with
~Gamma( , )

19. Parametric gamma frailty model The conditional frailty density (1)

20. Quadruples of correlated event times
Cluster of fixed size 4
Example: Correlated infection times in 4 udder quarters

21. Exercise
Fit gamma frailty model with Weibull baseline hazard to time to infection data at udder quarter level

22. R Program gamma frailty model
setwd("c://docs//onderwijs//survival//Flames//notas//") udder <- read.table("udderinfect.dat", header = T,skip=2) library(parfm) cowid<-as.factor(udder$cowid);timeto<-udder$timek stat<-udder$censor;heifer<-udder$LAKTNR udder<-data.frame(cowid=cowid,timeto=timeto,stat=stat,heifer=heifer) parfm(Surv(timeto,stat)~heifer,cluster="cowid",data=udder,frailty="gamma")

23. Parametric gamma frailty model Example – parameter estimates
Udder quarter infection data

24. Population and conditional hazards Exercise
Depict the population hazard together with the conditional hazards for frailties equal to the mean, median and the 25th and 95th percentile of the frailty density

25. Population and conditional hazards R program
lambda<-0.838;theta<-1.793;alpha<-1.979;beta< time<-seq(0,4,0.1) condhaz<-function(t){frail*alpha*lambda*t^(alpha-1)} marghaz<-function(t){(alpha*lambda*t^(alpha-1))/(1+theta*lambda*t^(alpha))} frail<-1;condhaz.frailm<-sapply(time,condhaz); marghaz.marg<-sapply(time,marghaz); lowfrail<-qgamma(0.25,shape=1/theta,rate=1/theta);upfrail<-qgamma(0.75,shape=1/theta,rate=1/theta) frail<-lowfrail;condhaz.fraill<-sapply(time,condhaz) frail<-upfrail;condhaz.frailu<-sapply(time,condhaz)

26. Population and conditional hazards Graph
miny<-min(condhaz.frailm,condhaz.fraill,condhaz.frailu) maxy<-max(condhaz.frailm,condhaz.fraill,condhaz.frailu) par(cex=1.2,mfrow=c(1,2)) plot(c(min(time),max(time)),c(miny,maxy),type='n',xlab='Time (year quarters)',ylab='hazard function') lines(time,condhaz.frailm,lty=1);lines(time,marghaz.marg,lty=1,lwd=3) lines(time,condhaz.fraill,lty=2);lines(time,condhaz.frailu,lty=3)

27. Population and conditional hazards Plot
Udder quarter infection data
Heifer
Multiparous cow

28. Population and conditional hazard ratio - Exercise
Depict the population and conditional hazard ratio as a function of the poulation survival function

29. Population and conditional hazard ratio - R-program
#Set parameters condHR<-exp(0.317);theta<-1.793;Sij<-seq(0.999,0.001,-0.001);Fij<-1-Sij par(mfrow=c(1,1)) #Plot population/conditional hazard ratio plot(Fij,(Sij^(-theta))/(1/condHR+Sij^(-theta)-1),xlab="Sx,f(t)",ylab="Population hazard ratio",type="n",axes=F,ylim=c(1,2.5)) box() axis(1,at=seq(0,1,0.2),labels=seq(1,0,-0.2),lwd=0.5) axis(2,at=seq(1,2.5,0.5),labels=seq(1,2.5,0.5),srt=90,lwd=0.5) lines(Fij,(Sij^(-theta))/(1/condHR+Sij^(-theta)-1)) segments(0,condHR,1,condHR)

30. Population and conditional hazard ratio - plot
Udder quarter infection data
Multiparous cow versus heifer

31. The frailty density mean and variance time evolution - Exercise
Depict the frailty density mean and variance time evolution

32. The frailty density mean and variance time evolution - R-program
#Plot E(u) plot(Fij,(Sij^(theta)),xlab="Sx,f(t)",ylab="Conditional mean",type="l")

33. The frailty density mean and variance time evolution – plot
Udder quarter infection data

34. Parametric gamma frailty model Kendall’s tau
Dependence measures developed for binary data. Take two random clusters i, k with event times
Position gives also covariate information
Kendall’s tau is
or alternatively

35. Parametric gamma frailty model Kendall’s tau for gamma frailties
Kendall’s tau can be expressed in terms of the Laplace transform (without proof)
Using the Laplace transform of the gamma frailty, we obtain

36. The cross ratio function a local version Kendall’s tau
We only consider bivariate data such as time to reconstitution
Consider the bivariate risk set for two pairs and
This bivariate risk set takes values between its maximal size s (number of clusters) and 2

37. The cross ratio function definition
We then define the local measure as
We can now consider this local dependence measure for different values of r, where r/s is a proxy for time in terms of survival for uncensored data

38. The cross ratio function estimation
Consider all pairs with particular value r = ra and take ratio of concordant and discordant pairs
Often, we rather take a group of adjacent ra’s due to low sample size
We will work this out of uncensored data, otherwise we need som further approximations

39. The cross ratio function R programme
#Read data
timetodiag<-read.table("timetodiag.csv",header=T,sep=";")
timetodiag<-timetodiag[timetodiag$c2!=0,];t1<- timetodiag$t1;t2<- timetodiag$t2
numobs<-length(t1);limit.low<-(seq(0,10)*10)+1;limit.up<- limit.low+9
numpairs<-choose(numobs,2)
res<-cbind(limit.low,limit.up,NA);
results<-matrix(NA,nrow=numpairs,ncol=8)

40. The cross ratio function R program
#Put values pairwise in results section
iter<-0
for (i in 1:(numobs-1)){
for (j in (i+1):numobs){
iter<-iter+1
results[iter,1]<-t1[i]
results[iter,2]<-t2[i]
results[iter,3]<-t1[j]
results[iter,4]<-t2[j] }

41. The cross ratio function R program
#determine the size of the risk set for each pair
for (iter in 1:numpairs){
minval1<-min(results[iter,1],results[iter,3])
minval2<-min(results[iter,2],results[iter,4])
temp<-timetodiag[t1>=minval1 & t2>=minval2,]
m<-length(temp$t1)
results[iter,6]<-m }

42. The cross ratio function R program
#determine the cross ratio function for each group of ra values
for (i in 1:10){
low<- limit.low[i]
up<- limit.up[i]
temp<-results[results[,6]>= low & results[,6]<= up,]
conc<-0;discord<-0
for (j in 1:length(temp[,1])){
signcomp<-sign((temp[j,1]-temp[j,3])* (temp[j,2]-temp[j,4]))
if (signcomp==1) conc<-conc+1
if (signcomp==-1) discord<-discord+1}
res[i,3] <-conc/discord}

43. The cross ratio function Plot and add model based g(r)
resr
plot((resr[,1]+ resr[,2])/(2*numobs),resr[,3],xlim=c(1,0),xlab="Estimated survival function",ylab="Cross ratio function")
theta<-1.793
cr<-theta+1
segments(0,cr,1,cr)

44. Parametric gamma frailty model Cross ratio function from model
The cross ratio function, a local measure:
Interpretation: time to recovery from mastitis
Positive experience: constitution at time t2
For positively correlated data, we assume that hazard in numerator>hazard in denominator

45. Parametric gamma frailty model Cross ratio function example
Cross ratio for gamma density is constant
For the reconstitution data, we have
=0.47
=2.793

46. Parametric positive stable (PS) frailty model
The positive stable distribtion
Laplace transform
Infinite mean!

47. Parametric PS frailty model Frailty density function examples
Positive stable density functions

48. Parametric PS frailty model Joint survival function

49. Parametric PS frailty model Marginal likelihood (1)
Example: Udder quarter infections, quadruples, clusters of size 4
Five different types of contributions, according to number of events in cluster
Order subjects, first uncensored (1, …, l)
Contribution of cluster i is equal to =0
>0

50. Parametric PS frailty model Marginal likelihood (2)
Derivatives of Laplace transforms

51. Parametric PS frailty model Marginal likelihood (3)
Marginal likelihood expression cluster i

52. Parametric PS frailty model Population survival function
Integrate conditional survival function
Population density function

53. Parametric PS frailty model Population hazard function
Ratio population/conditional hazard

54. Parametric PS frailty model Population vs conditional hazard

55. Parametric PS frailty model Population hazard ratio
Using population hazard functions
For the PS frailty distribution

56. Parametric PS frailty model Population hazard ratio example

57. Parametric PS frailty model R programme

58. Parametric PS frailty model Example – parameter estimates
Udder quarter infection data
Cond. HR=
Pop. HR=

59. Parametric PS frailty model The conditional frailty density
Assuming no covariate information, conditional density not PS, still PVF

60. Parametric PS frailty model Dependence measures
Kendall’s tau is given by
Cross ratio function
=0.47

61. Parametric PS frailty model Dependence measures
Cross ratio function – two dimensional

62. Parametric lognormal frailty model
Introduced by McGilchrist (1993) as
Therefore, for frailty we have

63. Parametric lognormal frailty model Frailty density function examples
Lognormal density functions

64. Parametric lognormal frailty model Laplace transform
No explicit expression for Laplace transform … difficult to compare
Maximisation of the likelihood is based on numerical integration of the normally distributed frailties

65. Parametric lognormal frailty model Example udder quarter infection (1)
Numerical integration using Gaussian quadrature (nlmixed procedure)
Difficult to compare with previous results as mean of frailty no longer 1
Convert results to density function of median event time

66. Parametric frailty model udder infection: lognormal/gamma