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