basics of the parametric frailty model luc duchateau ghent university, belgium

Basics of the parametric frailty

model

Luc DuchateauGhent University, Belgium

Overview

Frailty distributions The parametric gamma frailty model The parametric positive stable frailty

model The parametric lognormal frailty model

Frailty distributions

Power variance function family Gamma Inverse Gaussian Positive stable General PVF

Compound Poisson Lognormal

Parametric gamma frailty modelFrailty density function (1)

Two-parameter gamma density

One-parameter gamma density

Parametric gamma frailty modelFrailty density function examples

One-parameter gamma density

Parametric gamma frailty modelLaplace transform of frailty density

Characteristic function

Moment generating function

Laplace transform for positive r.v.

Parametric gamma frailty modelLaplace transf. generates moments

Generate nth moment Use nth derivative of Laplace transform

Evaluate at s=0

Parametric gamma frailty modelGamma Laplace transform

Gamma Laplace transform

Parametric gamma frailty modelJoint survival function (1) Joint survival function in conditional model

Now use notation

For cluster with covariates

Parametric gamma frailty modelJoint survival function (2) Applied to Laplace transform of gamma

distribution we obtain

Parametric gamma frailty modelPopulation survival function (1) Integrate conditional survival function

Population density function

Population hazard function

Parametric gamma frailty modelPopulation survival function (2) Applied to gamma distribution we have


Graphically

#Set parameterscondHR<-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 hazardplot(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))

Parametric gamma frailty modelPopulation vs conditional hazard

Parametric gamma frailty modelPopulation hazard ratio Using population hazard functions

For the gamma frailty distribution

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))

Parametric gamma frailty modelPopulation hazard ratio example

Parametric gamma frailty modelThe conditional frailty density (1)

Assuming no covariate information

which corresponds for gamma with

~Gamma( , )

Parametric gamma frailty modelThe conditional frailty density (1)

~Gamma( , )

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

quarters

Exercise Fit gamma frailty model with Weibull

baseline hazard to time to infection data at udder quarter level

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

Parametric gamma frailty modelExample – parameter estimates

Udder quarter infection data

Population and conditional hazardsExercise 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

Population and conditional hazardsR programlambda<-0.838;theta<-1.793;alpha<-1.979;beta<-0.317time<-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)

Population and conditional hazardsGraphminy<-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)

Population and conditional hazardsPlot

Udder quarter infection dataHeifer Multiparous cow

Population and conditional hazard ratio - Exercise Depict the population and conditional

hazard ratio as a function of the poulation survival function

Population and conditional hazard ratio - R-program #Set parameterscondHR<-exp(0.317);theta<-1.793;Sij<-seq(0.999,0.001,-0.001);Fij<-1-Sijpar(mfrow=c(1,1))#Plot population/conditional hazard ratioplot(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)

Population and conditional hazard ratio - plot


Multiparous cow versus heifer

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

variance time evolution

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")

The frailty density mean and variance time evolution – plot


Parametric gamma frailty modelKendall’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

Parametric gamma frailty modelKendall’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

The cross ratio functiona 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

The cross ratio functiondefinition

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

The cross ratio functionestimation

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

The cross ratio functionR programme

#Read datatimetodiag<-read.table("timetodiag.csv",header=T,sep=";")timetodiag<-timetodiag[timetodiag$c2!=0,];t1<- timetodiag$t1;t2<- timetodiag$t2numobs<-length(t1);limit.low<-(seq(0,10)*10)+1;limit.up<- limit.low+9numpairs<-choose(numobs,2)res<-cbind(limit.low,limit.up,NA);results<-matrix(NA,nrow=numpairs,ncol=8)

The cross ratio functionR program

#Put values pairwise in results sectioniter<-0for (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] }}


#determine the size of the risk set for each pairfor (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}


#determine the cross ratio function for each group of ra valuesfor (i in 1:10){low<- limit.low[i]up<- limit.up[i]temp<-results[results[,6]>= low & results[,6]<= up,]conc<-0;discord<-0for (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}

The cross ratio functionPlot and add model based g(r)

resrplot((resr[,1]+ resr[,2])/(2*numobs),resr[,3],xlim=c(1,0),xlab="Estimated survival function",ylab="Cross ratio function")

theta<-1.793cr<-theta+1segments(0,cr,1,cr)

Parametric gamma frailty modelCross 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

Parametric gamma frailty modelCross ratio function example

Cross ratio for gamma density is constant

For the reconstitution data, we have

=0.47

=2.793

Parametric positive stable (PS) frailty model

The positive stable distribtion

Laplace transform

Infinite mean!

Parametric PS frailty modelFrailty density function examples

Positive stable density functions

Parametric PS frailty modelJoint survival function Joint survival function

Parametric PS frailty modelMarginal 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


Derivatives of Laplace transforms


Marginal likelihood expression cluster i

Parametric PS frailty modelPopulation survival function Integrate conditional survival function

Population density function

Parametric PS frailty modelPopulation hazard function


Ratio population/conditional hazard

Parametric PS frailty modelPopulation vs conditional hazard

Parametric PS frailty modelPopulation hazard ratio Using population hazard functions

For the PS frailty distribution

Parametric PS frailty modelPopulation hazard ratio example

Parametric PS frailty modelR programme

Parametric PS frailty modelExample – parameter estimates


Cond. HR=

Pop. HR=

Parametric PS frailty modelThe conditional frailty density

Assuming no covariate information, conditional density not PS, still PVF

Parametric PS frailty modelDependence measures

Kendall’s tau is given by Cross ratio function =0.47

Parametric PS frailty modelDependence measures

Cross ratio function – two dimensional

Parametric lognormal frailty model

Introduced by McGilchrist (1993) as

Therefore, for frailty we have

Parametric lognormal frailty modelFrailty density function examples

Lognormal density functions

Parametric lognormal frailty modelLaplace transform

No explicit expression for Laplace transform … difficult to compare

Maximisation of the likelihood is based on numerical integration of the normally distributed frailties

Parametric lognormal frailty modelExample 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

Parametric frailty model udder infection: lognormal/gamma

basics of the parametric frailty model luc duchateau ghent university, belgium

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