workshop frailty models luc duchateau, rosemary nguti and paul janssen

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

http://www.r-project.org/

http://cran.za.r-project.org/

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


• 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


• Observable likelihood for constant hazard


• 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


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}


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')


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


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

workshop frailty models luc duchateau, rosemary nguti and paul janssen

Documents

frailty models choice

semiparametric models

frailty density

dairy farms slide

paul janssen slide

fixed covariates data

data insemfix

statistical package