1. Semantics and History of the term frailty Luc Duchateau Ghent University, Belgium

2. Semantics of term frailty  Medical field: gerontology Frail people higher morbidity/mortality risk Determine frailty of a person (e.g. Get-up and Go test) Frailty: fixed effect, time varying, surrogate  Modelling: statistics Frailty often at higher aggregation level (e.g. hospital in multicenter clinical trial) Frailty: random effect, time constant, estimable

3.  Introduced by Beard (1959) in univariate setting to improve population mortality modelling by allowing heterogeneity  Beard (1959) starts from Makeham’s law (1868) with the constant hazard and with the hazard increases with time  Longevity factor is added to model History of term frailty - Beard (1)

4.  Beard’s model  Population survival function  Population hazard function History of term frailty- Beard (2) Survival at time t for subject with frailty u Hazard at time t for subject with frailty u

5.  Term frailty first introduced by Vaupel (1979) in univariate setting to obtain individual mortality curve from population mortality curve  For the case of no covariates History of term frailty - Vaupel (1)

6.  Vaupel and Yashin (1985) studied heterogeneity due to two subpopulations Population 1: Population 2: Frailty – two subpopulations (1)

7.  Smokers:high and low recidivism rate Frailty – two subpopulations (2)

8. R program age<-seq(0,75) mu1.1<-rep(0.06,76);mu1.2<-rep(0.08,76) pi1.0<-0.8 pi1<-(pi1.0*exp(-age*mu1.1))/(pi1.0*exp(-age*mu1.1)+(1-pi1.0)*exp(- age*mu1.2)) mu1<-pi1*mu1.1+(1-pi1)*mu1.2 plot(age,mu1,type="n",xlab="Time(years)",ylab="Hazard",axes=F,ylim=c(0. 05,0.09)) box();axis(1,lwd=0.5);axis(2,lwd=0.5) lines(age,mu1);lines(age,mu1.1,lty=2);lines(age,mu1.2,lty=2)

9.  Reliability engineering Frailty – two subpopulations (3)

10.  Two hazards increasing at different rates Frailty – two subpopulations (4)

11.  Two parallel hazards (at log scale) Frailty – two subpopulations (5)

12. Exercise  Assume that the population of heroine addicts consists of two subpopulations. The first subpopulation (80%) has a constant monthly hazard of quitting drug use of 0.10, whereas the second subpopulation (20%) has a constant monthly hazard of quitting drug use of 0.20.  What is the hazard of the population after 2 years?  Make a picture of the hazard function of the population as a function of time

13. Hazard after two years

14. R programme time<-seq(0,4,0.1) mu1.1<-rep(0.1,length(time));mu1.2<-rep(0.2, length(time)) pi1.0<-0.8 pi1<-(pi1.0*exp(-time*mu1.1))/(pi1.0*exp(-time*mu1.1)+(1-pi1.0)*exp(- time*mu1.2)) mu1<-pi1*mu1.1+(1-pi1)*mu1.2 plot(time,mu1,type="n",xlab="Time(years)",ylab="Hazard",axes=F,ylim=c(0. 09,0.21)) box();axis(1,lwd=0.5);axis(2,lwd=0.5) lines(time,mu1);lines(time,mu1.1,lty=2);lines(time,mu1.2,lty=2)