Workshop frailty models

Luc Duchateau, Rosemary Nguti and Paul Janssen

Contents

• The statistical package R• The data set: time to first insemination• Inference for shared gamma frailty models

– Theoretical considerations– Fitting parametric and semiparametric models

• More flexible shared gamma frailty models– Time-varying covariates– Smoothing splines

• Other approaches for frailty models– Choice of the frailty density– A Bayesian approach for frailty models

The statistical package R

• Freeware (unlike Splus)

• Powerful statistical package

• Data handling a bit less powerful

• Downloadable from the Internet– Base– Libraries

• From within R• Download library yourself

Installing R via Internet

• Go to www.r-project.org

• Choose option download

• Choose closest mirrorsite http://cran.za.r-project.org/

• Choose operating systemWindows (95 and later)

• Choose subdirectory base

• Choose rw1090.exe

Installing additional packages

• Automatically within R– Menu item « Packages»

• « Install Packages from CRAN … »• Choose « survival »

– New session, specify « library(survival) »

• Download zipped package from CRAN– Unzip it under directory

« c:\Program Files\R\rw1091\library »

The data set: Time to first insemination

• Database of regional Dairy Herd Improvement Association (DHIA)– Milk recording service– Artificial insemination– Select sample– Subset of 2567 cows from 49 dairy farms

Fixed covariates data setinsemfix.dat

Time-varying covariates data setinsemtvc.dat

Fitting parametric frailty models (1)

• Read the data– insemfix<-read.table("c://insemfix.dat",header=T)

• Create vectors

heifer<- herd stat timeto

insem$heifer110...11

1

1..

49..49

011...11

5332

201...

10338

Fitting parametric frailty models (2)

• Derive quantities n, Di and e– n<-length(levels(as.factor(herd)) 49

– Di<-aggregate(stat,by=list(herd),FUN=sum)[,2]

– e<-sum(Di)

1 2 3 … 49

11 23 … 34

Fitting parametric frailty models (3)

• Observable likelihood for constant hazard

Fitting parametric frailty models (4)

• Calculate observable likelihood for parameters– h=1/p[1], =1/p[2], =p[3]

• Cumulative hazard

• Hazard

Timeto*exp(heifer*p[3])/p[1]cumhaz

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

stat*log(exp(heifer*p[3])/p[1]))lnhaz

aggregate(lnhaz,by=list(herd),FUN=sum)[,2]lnhaz

Fitting parametric frailty models (5)

likelihood.exponential<-function(p){cumhaz<-(timeto*exp(heifer*p[3]))/p[1]cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2]lnhaz<-stat*log(exp(heifer*p[3])/p[1])lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2]lik<-e*log(1/p[2])-n*log(gamma(p[2]))+sum(log(gamma(Di+p[2])))-sum((Di+p[2])*log(1+cumhaz/p[2]))+sum(lnhaz)-lik}

Fitting parametric frailty models (6)

0.0317 theta-0.1965,beta ,0146.0 h

days 58))exp(/(2log:heiferson inseminati toeMedian tim betah

initial<-c(100,17,0)t<-nlm(likelihood.exponential,initial,print.level=2)h<-1/t$estimate[1]htheta<-1/t$estimate[2]thetabeta<-t$estimate[3]beta

days 5.47/2log:smultiparouon inseminati toeMedian tim h

Interpretation

• From , the heterogeneity parameter, to density for median time to event

calcm<-function(m){(log(2)/(theta*h))^(1/theta) *(1/m)^(1+1/theta)* (1/gamma(1/theta)) *exp(-log(2)/(theta*h*m))}med<-seq(1,100)fmed<-sapply(med,calcm)plot(med,fmed,type="l",xlab="Median time",ylab="Density")

Interpretation

Median time

De

nsi

ty

0 20 40 60 80 100

0.0

0.0

10

.02

0.0

30

.04

Fitting semiparametric models

library(survival)coxph(Surv(timeto,stat)~heifer+frailty(herd))

Call:coxph(formula = Surv(timeto, stat) ~ heifer + frailty(herd)) coef se(coef) se2 Chisq DF p heifer -0.24 0.0432 0.0430 30.9 1.0 2.7e-08frailty(herd) 205.3 40.9 0.0e+00

Iterations: 10 outer, 23 Newton-Raphson Variance of random effect= 0.123 I-likelihood = -16953

Degrees of freedom for terms= 1.0 40.9 Likelihood ratio test=281 on 41.9 df, p=0 n= 2579

HR=exp(-0.24)=0.79

Time-varying covariates data

Contribtion to the denominator:

tij=10 : 2.29tij=55 : 2.61

Fitting Cox models withtime-varying covariates (1)

#Read datainsemtvc<-read.table("c://insemtvc.dat",header=T)#Create four column vectors, four different variablesherd<-insemtvc$herdbegin<-insemtvc$beginend<-insemtvc$endstat<-insemtvc$statureum<-insemtvc$ureumheifer<-insemtvc$heiferlibrary(survival)coxph(Surv(begin,end,stat)~heifer+ureum+frailty(herd))

Fitting Cox models withtime-varying covariates (2)

Call:coxph(formula = Surv(begin, end, stat) ~ heifer + ureum + frailty(herd)) coef se(coef) se2 Chisq DF p heifer -0.22820 0.0466 0.0464 23.99 1.0 9.7e-07ureum -0.00349 0.0366 0.0358 0.01 1.0 9.2e-01frailty(herd) 177.64 39.4 0.0e+00

Iterations: 10 outer, 23 Newton-Raphson Variance of random effect= 0.116 I-likelihood = -14401.2 Degrees of freedom for terms= 1.0 1.0 39.4 Likelihood ratio test=251 on 41.3 df, p=0 n= 23076

HR=exp(-0.00349)=0.997

Fitting Cox models withsmoothing splines (1)

#Read datainsemtvc<-read.table("c://insemtvc.dat",header=T)#Create four column vectors, four different variablesherd<-insemtvc$herdbegin<-insemtvc$beginend<-insemtvc$endstat<-insemtvc$statureum<-insemtvc$ureumheifer<-insemtvc$heiferlibrary(survival)fit<-coxph(Surv(begin,end,stat)~heifer+pspline(ureum,nterm=3,theta=0.5))temp<-order(ureum)plot(ureum[temp],predict(fit,type='terms')[temp,2],type='l',xlab='Ureum',ylab='Risk score')

Fitting Cox models withsmoothing splines (2)

Call:coxph(formula = Surv(begin, end, stat) ~ heifer + pspline(ureum, nterm = 3, theta = 0.5))

coef se(coef) se2 Chisq DF p heifer -0.2284 0.0457 0.0457 24.97 1.00 5.8e-07pspline(ureum, nterm = 3, 0.0094 0.0379 0.0349 0.06 1.00 8.0e-01pspline(ureum, nterm = 3, 3.31 1.18 8.7e-02

Iterations: 1 outer, 3 Newton-Raphson Theta= 0.5 Degrees of freedom for terms= 1.0 2.2 Likelihood ratio test=30.2 on 3.18 df, p=1.56e-06 n= 23076

Fitting Cox models withsmoothing splines (3)

Ureum

Ris

k sc

ore

1 2 3 4 5 6 7

-0.2

5-0

.20

-0.1

5-0

.10

-0.0

50

.0

Fitting the shared normal frailty model (1)

#Read datainsemfix<-read.table("c://docs//onderwijs//frailtymodels//insemfix.dat",header=T)#Create four column vectors, four different variablesherd<-insemfix$herdtimeto<-insemfix$timeto-min(insemfix$timeto)stat<-insemfix$statheifer<-insemfix$heifercoxph(Surv(timeto,stat)~heifer+frailty(herd,distribution="gaussian"))

Fitting the shared normal frailty model (2)

Call:coxph(formula = Surv(timeto, stat) ~ heifer + frailty(herd, distribution = "gaussian"))

coef se(coef) se2 Chisq DF p heifer -0.241 0.0431 0.043 31.1 1.0 2.4e-08frailty(herd, distributio 199.9 38.6 0.0e+00

Iterations: 6 outer, 14 Newton-Raphson Variance of random effect= 0.0916 Degrees of freedom for terms= 1.0 38.6 Likelihood ratio test=276 on 39.5 df, p=0 n= 2579