Basics of the parametric frailty model Luc Duchateau Ghent University, Belgium.
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Transcript of 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
Population hazard function
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
Udder quarter infection data
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
Udder quarter infection data
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$t2numobs<-length(t1);limit.low<-(seq(0,10)*10)+1;limit.up<- limit.low+9numpairs<-choose(numobs,2)res<-cbind(limit.low,limit.up,NA);results<-matrix(NA,nrow=numpairs,ncol=8)
The cross ratio functionR program
#Put values pairwise in results sectioniter<-0for (i in 1:(numobs-1)){ for (j in (i+1):numobs){ iter<-iter+1 results[iter,1]<-t1[i] results[iter,2]<-t2[i] results[iter,3]<-t1[j] results[iter,4]<-t2[j] }}
The cross ratio functionR program
#determine the size of the risk set for each pairfor (iter in 1:numpairs){ minval1<-min(results[iter,1],results[iter,3])minval2<-min(results[iter,2],results[iter,4])temp<-timetodiag[t1>=minval1 & t2>=minval2,]m<-length(temp$t1)results[iter,6]<-m}
The cross ratio functionR program
#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
Parametric PS frailty modelMarginal likelihood (2)
Derivatives of Laplace transforms
Parametric PS frailty modelMarginal likelihood (3)
Marginal likelihood expression cluster i
Parametric PS frailty modelPopulation survival function Integrate conditional survival function
Population density function
Parametric PS frailty modelPopulation hazard function
Population 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
Udder quarter infection data
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