Semantics and History of the term

frailtyLuc Duchateau

Ghent University, Belgium

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

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)

Beard’s model Population survival function

Population hazard function

History of term frailty- Beard (2)

Survival at time t forsubject with frailty u

Hazard at time t forsubject with frailty u

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)

Vaupel and Yashin (1985) studied heterogeneity due to two subpopulations Population 1: Population 2:

Frailty – two subpopulations (1)

Smokers:high and low recidivism rate

Frailty – two subpopulations (2)

R programage<-seq(0,75)

mu1.1<-rep(0.06,76);mu1.2<-rep(0.08,76)pi1.0<-0.8pi1<-(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)

Reliability engineering

Frailty – two subpopulations (3)

Two hazards increasing at different rates

Frailty – two subpopulations (4)

Two parallel hazards (at log scale)

Frailty – two subpopulations (5)

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

Hazard after two years

R programmetime<-seq(0,4,0.1)

mu1.1<-rep(0.1,length(time));mu1.2<-rep(0.2, length(time))pi1.0<-0.8pi1<-(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)