Multivariate Survival Analysis

Alternative approaches

Prof. L. DuchateauGhent University

Overview The different approaches

The marginal model The fixed effects model The stratified model The copula model The frailty model

Efficiency comparisons

The marginal model The marginal model approach consists of

two stages Stage 1: Fit the model without taking into

account the clustering Stage 2: Adjust for the clustering in the data

The ML estimate from the Independence Working Model (IWM)

is a consistent estimator for (Huster, 1989) More generally, the ML estimate ( and

baseline parameters) from the IWM is also a consistent estimator for

Parameter refers to the whole population

Consistency of marginal model parameter estimates

The variance estimate based on the inverse of the information matrix of is an inconsistent estimator of Var( )

One possible solution: jackknife estimation General expression of jackknife estimator

(Wu, 1986)

with N the number of observations and a the number of parameters

Adjusting the variance of IWM estimates

The grouped jackknife estimator For clustered observations: grouped jackknife estimator

with s the number of clusters

Reconstitution: jackknife#Jackknife estimatorbdel<-rep(NA,100)b1<- -survreg(Surv(timerec,stat)~trt,data=reconstitution,dist="exponential")$coeff[2]for (i in 1:100){temp<-reconstitution[reconstitution$cowid!=i,]bdel[i]<--survreg(Surv(timerec,stat)~trt,data=temp,dist="exponential")$coeff[2]}var.robust<-0.98*sum((bdel-b1)^2);stderr.robust<-sqrt(var.robust)var.robust;stderr.robust

Reconstitution: jackknife in R?#Jackknife estimator using cluster() and robust=T commandanalexp.jc<- survreg(Surv(timerec,stat)~trt+cluster(cowid),robust=T,dist="exponential")Error in score %*% vv : non-conformable arguments

Example marginal model with jackknife estimator Example: Time to reconstitution with drug

versus placebo Estimates from IWM model with time-

constant hazard rate assumption are given by

Grouped jackknife = approximation

Jackknife estimator Adjusts for clustering Reconstitution example: jackknife

estimator is smaller Time to diagnosis example?

Jackknife estimator Adjusts for clustering Reconstitution example: jackknife

estimator is smaller

What is then the picture? Simulation

Jackknife estimator-simulations(1)

Is jackknife estimate always smaller than estimate from unadjusted model?

Generate data from the frailty model with

We generate 2000 datasets, each of 100 pairs of two subjects for the settings1. Matched clusters, no censoring2. 20% of clusters 2 treated or untreated subjects,

no censoring3. Matched clusters, 20% censoring

Jackknife estimator-simulations(2)

The fixed effects model The fixed effects model is given by

with the fixed effect for cluster i,

Assume for simplicity

The fixed effects model: ML solution

General survival likelihood expression

For fixed effects model using assumptions

Reconstitution: fixed effects model#Fixed effects modelres.fixed<-survreg(Surv(timerec,stat)~trt+as.factor(cowid),dist="exponential",data=reconstitution)res.fixedsummary(res.fixed)

Treatment effect for reconstitution data using R-function survreg (loglin. model)

Output treatment effect

Parameter interpretation corresponds to constant hazard of

untreated udder quarter of cow 1

corresponds to constant

hazard of untreated udder quarter of cow i Cowid65 ≈ 0 Cowid100 exp(-21+18.8)=0.11

Treatment effect: HR=exp(0.185)=1.203 with 95% CI [0.83;1.75]

Parameter interpretation

Investigate cow characteristic: heifer#Fixed effects modelres.fixed<-survreg(Surv(timerec,stat)~heifer+as.factor(cowid),dist="exponential",data=reconstitution)res.fixedsummary(res.fixed)

Heifer effect for reconstitution dataintroducing heifer first in the model

Hazard ratio impossibly high

Output heifer effect

Add cow characteristic: heifer after cowid?#Fixed effects modelres.fixed<-survreg(Surv(timerec,stat)~trt+as.factor(cowid)+heifer,dist="exponential",data=reconstitution)res.fixedsummary(res.fixed)

Heifer effect for reconstitution dataintroducing cowid first in the model

Hazard ratio equal to 1

Example: between cluster covariate (2)

Exercise Investigate method and type of fracture in

diagnosis data

Note on overparametrisation and confounding

Cell means model: no overparametrisation Milk reduction as a function of low and

high inoculation dose

Factor effects model: overparametrisation Milk reduction as a function of low and

high inoculation dose

Confounding between temperature in F and C Effect of temperature on bacterial growth

(log(CFU))

Temperature in °C vs °F

Conversion from°F to °C

Infinite number of model representations

Confounding between blocks and block factors Cow factor is not confounded with

treatment factor

Fitting model with cow and treatment

Model with cow and treatment vs cow alone

Adding the heifer factor

Infinite number of model representations

Example: heifer - cowid confounded There is complete confounding between

fixed heifer effect and cowid

The stratified model Based on the Cox model

where now baseline hazard function unspecified Cox (1972) showed that if only order of events

matters, the survival likelihood reduces to the partial likelihood

Partial likelihood for the stratified model The stratified model is given by

Maximisation of partial likelihood

Reconstitution: stratified model#stratified Cox modellibrary(survival)res.strat<-coxph(Surv(timerec,stat)~trt+strata(cowid),data=reconstitution)res.stratsummary(res.strat)

Example for bivariate data The partial likelihood for reconstitution data

Estimates

Exercise Fit the stratified model for the diagnosis

data

The copula model The copula model is often considered to be

a two-stage model First obtain the population (marginal)

survival functions for each subject in a cluster.

The copula function then links these population survival functions to generate the joint survival function (Frees et al., 1996).

Example of copula model Time to diagnosis of being healed

Bivariate copula model likelihood Four different possible contributions of a

cluster

Estimated population survival functions are inserted, only copula parameters unknown

The Clayton copula The Clayton copula (Clayton, 1978) is

The Clayton copula corresponds to the family of Archimedean copulas, i.e.,

with in the Clayton copula case

Clayton copula likelihood Two censored observations

Observation j censored

No observations censored

Example Clayton copula (1) For diagnosis of being healed data, first fit

separate models for RX and US technique For instance, separate parametric models

Fitting the copula: two stage approach#Clayton copula for time to diagnosis timetodiag <- read.table("c:\\docs\\onderwijs\\survival\\flames\\diag.csv", header = T,sep=";")t1<-timetodiag$t1/30;t2<-timetodiag$t2/30;c1<-timetodiag$c1;c2<-timetodiag$c2;

surv1<-survreg(Surv(t1,c1)~1);l1<-exp(-surv1$coeff/surv1$scale);r1<-(1/surv1$scale)surv2<-survreg(Surv(t2,c2)~1);l2<-exp(-surv2$coeff/surv2$scale);r2<-(1/surv2$scale)

s1<-exp(-l1*t1^(r1));f1<-s1*r1*l1*t1^(r1-1)s2<-exp(-l2*t2^(r2));f2<-s2*r2*l2*t2^(r2-1)

loglikcon.gamma<-function(theta){P<-s1^(-theta)+ s2^(-theta)-1loglik<- -(1-c1)*(1-c2)*(1/theta)*log(P)+c1*(1-c2)*((1+1/theta)*log(P)+(theta+1)*log(s1)- log(f1))+c2*(1-c1)*((1+1/theta)*log(P)+(theta+1)*log(s2)-log(f2))+c1*c2*(log(1+theta)- (2+1/theta)*log(P)-(theta+1)*log(s1)+log(f1)-(theta+1)*log(s2)+log(f2))-sum(loglik)}

nlm(loglikcon.gamma,c(0.5))

Example Clayton copula (2) Estimates for marginal models are

Based on these estimates we obtain

which can be inserted in the likelihood expression which is then maximized for

Exercise Fit the copula model to the diagnosis data

as one-stage model

Fitting the copula: one stage approach

#Clayton copula for time to diagnosis – one stageloglikcon3.gamma<-function(param){theta<-param[1];l1<-param[2];l2<-param[3];r1<-param[4];r2<-param[5]s1<-exp(-l1*t1^(r1));f1<-s1*r1*l1*t1^(r1-1)s2<-exp(-l2*t2^(r2));f2<-s2*r2*l2*t2^(r2-1)P<-s1^(-theta)+ s2^(-theta)-1loglik<- -(1-c1)*(1-c2)*(1/theta)*log(P)+c1*(1-c2)*((1+1/theta)*log(P)+(theta+1)*log(s1)-log(f1))+c2*(1-c1)*((1+1/theta)*log(P)+(theta+1)*log(s2)-log(f2))+c1*c2*(log(1+theta)-(2+1/theta)*log(P)-(theta+1)*log(s1)+log(f1)-(theta+1)*log(s2)+log(f2))-sum(loglik)}nlm(loglikcon3.gamma, c(0.5,1,1,1,1))

Example Clayton copula (3) For parametric marginal models, the

likelihood can also be maximized simul-taneously for all parameters leading to

Thus, for small sample sizes, the two-stage approach can differ substantially from the one-stage approach

Example Clayton copula (4) Alternatives can be used for marginal

survival functions Nonparametric Semiparametric

leading to

The frailty model The ‘shared’ frailty model is given by

with the frailty An alternative formulation is given by

with

The gamma frailty model Gamma frailty distribution is easiest choice

with and

Marginal likelihood for the gamma frailty model Start from conditional (on frailty) likelihood

with containing the baseline hazard

parameters, e.g., for Weibull

Marginal likelihood: integrating out the frailties … Integrate out frailties using distribution

with

Closed form expression for marginal likelihood Integration leads to (homework)

and taking log and summing over s clusters

Maximisation of marginal likelihood leads to estimates Marginal likelihood no longer contains

frailties. By maximisation estimates of are obtained Furthermore, the asymptotic variance-

covariance matrix can be obtained as the inverse of the observed information matrix

with the Hessian matrix with entries

Entries of Hessian matrix from marginal likelihood

As an example, the entry of the Hessian matrix for is given by

Example for the parametric gamma frailty model Consider time to first insemination data Assume Weibull distributed event times and

model the heifer effect We have the following conditional functions

R program: read the data

#read data

setwd("c://docs//onderwijs//survival//Flames//notas//")

insemfix<-read.table("insemfix.csepv", header=T,sep=",")

#Create four column vectors, four different variables

herd<-insemfix$herdnr;timeto<-(insemfix$end*12/365.25)

stat<-insemfix$score;heifer<-insemfix$par2

#Derive some values

n<-length(levels(as.factor(herd)));

di<-aggregate(stat,by=list(herd),FUN=sum)[,2];r<-sum(di)

R program: the function #Observable likelihood weibull

#l=exp(p[1]), theta=exp(p[2]), beta=p[3], rho=exp(p[4])

#r=No events,di=number of events by herd

likelihood.weibul<-function(p){

cumhaz<-exp(heifer*p[3])*(timeto^(exp(p[4])))*exp(p[1])

cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2]

lnhaz<-stat*(heifer*p[3]+log((exp(p[4])*timeto^(exp(p[4])-1))*exp(p[1])))

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)+

sum(sapply(di,function(x) ifelse(x==0,0,log(prod(x+1/exp(p[2])-seq(1,x))))))

-lik}

R program: the output

res<-nlm(likelihood.weibul,c(log(0.128),log(0.39),0.15,log(1.76)), hessian=T)

lambda<-exp(res$estimate[1])

theta<-exp(res$estimate[2])

beta<-res$estimate[3]

rho<-exp(res$estimate[4])

Time to first insemination: effect of heifer with herd as cluster ML

Monthly hazard rate

Monotone increasing

Variance of frailties

Within herd heifer effect

Hazard ratio with 95 % CI

Using parfm librarylibrary(parfm)#Create four column vectors, four different variablesherd<-as.factor(insemfix$herdnr);timeto<-(insemfix$end*12/365.25)stat<-insemfix$score;heifer<-insemfix$par2insem<-data.frame(herd=herd,timeto=timeto,stat=stat,heifer=heifer)parfm(Surv(timeto,stat)~heifer,cluster="herd",data=insem,frailty="gamma")

Interpretation of frailty variance The parameter refers to the variability at

the hazard level: difficult to interprete! Maybe plot the hazard function for subjects

with a particular frailty

Plotting hazard of insemination for multiparous cows #Interpretation of parameterslambda<-0.174;theta<-0.394;rho<-1.769lambda<-lambda*((365.25/12)^(-rho))

time<-seq(1,350);timet<-time+29.5h1f0<-lambda*rho*time^(rho-1)h1f05<-qgamma(0.05,1/theta,1/theta)*lambda*rho*time^(rho-1)h1f95<-qgamma(0.95,1/theta,1/theta)*lambda*rho*time^(rho-1)

Plotting hazard of inseminationfor multiparous cows #Hazardspar(mfrow=c(1,2)); par(adj=0.5);par(cex=1.2)plot(c(0,360),c(min(h1f05,h2f05),max(h1f95,h2f95)),type='n',xlab="Time after calving (days)",ylab="hazard")lines(timet,h1f0,lty=1,lwd=3);lines(timet,h1f05,lty=1,lwd=1)lines(timet,h1f95,lty=1,lwd=1)par(adj=0);text(1,0.14,"Multiparous cows")

Exercise

Plot hazard of insemination for heifers

Plotting hazard of insemination for heifers #Interpretation of parameterslambda<-0.174;theta<-0.394;rho<-1.769;beta<--0.153lambda<-lambda*((365.25/12)^(-rho))

h2f0<-lambda*rho*exp(beta)*time^(rho-1)h2f05<-qgamma(0.05,1/theta,1/theta)*lambda*rho*exp(beta)*time^(rho-1)h2f95<-qgamma(0.95,1/theta,1/theta)*lambda*rho*exp(beta)*time^(rho-1)

Plotting hazard of insemination #Hazardspar(mfrow=c(1,2)); par(adj=0.5);par(cex=1.2)plot(c(0,360),c(min(h1f05,h2f05),max(h1f95,h2f95)),type='n',xlab="Time after calving (days)",ylab="hazard")lines(timet,h1f0,lty=1,lwd=3);lines(timet,h1f05,lty=1,lwd=1)lines(timet,h1f95,lty=1,lwd=1)par(adj=0);text(1,0.14,"Multiparous cows")par(adj=0.5)plot(c(0,360),c(min(h1f05,h2f05),max(h1f95,h2f95)),type='n',xlab="Time after calving (days)",ylab="hazard")lines(timet,h2f0,lty=1,lwd=3);lines(timet,h2f05,lty=1,lwd=1)lines(timet,h2f95,lty=1,lwd=1)par(adj=0);text(1,0.14,"Heifers")

Interpretation of frailty variance The parameter refers to the variability at

the hazard level: difficult to interprete!

Multiparouscows

Heifers

Transformation to median Density function of transformation of

random variable

Median of Weibull distribution To find the median survival time for cluster i, put = 0.5

Density of median for Weibull distribution The density function is then

with

and

Density of median for Weibull distribution Leading to

Plotting density function of median for multiparous cowslambda<-0.174;theta<-0.394;rho<-1.769;beta<--0.153;lambda<-lambda*((365.25/12)^(-rho))#Medianscalcm<-function(m){rho * (log(2)/(theta*lambda))^(1/theta) * (1/m)^(1+rho/theta) * (1/gamma(1/theta)) *exp(-log(2)/(theta*lambda*m^(rho)))}timedens<-seq(1,200,1)densmd1<-sapply(timedens,calcm)plot(c(0,230),c(min(densmd1),max(densmd1)),type='n',xlab="Median time to first insemination (days)",ylab="Density function median")lines(timedens+29.5,densmd1,lty=1,lwd=3)

Exercise Plot density of median for Heifers

Plotting density function of median for multiparous cows and heiferslambda<-0.174;theta<-0.394;rho<-1.769;beta<--0.153;lambda<-lambda*((365.25/12)^(-rho))#Medianscalcm<-function(m){rho * (log(2)/(theta*lambda))^(1/theta) * (1/m)^(1+rho/theta) * (1/gamma(1/theta)) *exp(-log(2)/(theta*lambda*m^(rho)))}timedens<-seq(1,200,1)densmd1<-sapply(timedens,calcm)lambda<-lambda*exp(beta)densmd2<-sapply(timedens,calcm)plot(c(0,230),c(min(densmd1,densmd2),max(densmd1,densmd2)),type='n',xlab="Median time to first insemination (days)",ylab="Density function median")lines(timedens+29.5,densmd1,lty=1,lwd=3);lines(timedens+29.5,densmd2,lty=2,lwd=3)legend(130,0.015,legend=c("Multiparous","Heifer"),lty=c(1,2))

Variability of median time to first insemination between herds

Exercise Derive the density function for the

percentage survivan at a particular time t

Transformation to percentage survival

The percentage in cluster i with first insemination at time t is given by

Thus

and

Interpretation of frailty variance in terms of % events at time t The density function is then obtained by

and thus

Variability of % first insemination at time t between herds

Multiparouscows

Heifers

Efficiency comparisons in the reconstitution data example Estimates (se) for reconstitution data