#### Transcript Dia 1

```Semantics and History of the
term frailty
Luc Duchateau
Ghent University, Belgium
Semantics of term frailty
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Medical field: gerontology
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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
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Frailty often at higher aggregation level (e.g.
hospital in multicenter clinical trial)
Frailty: random effect, time constant,
estimable
History of term frailty - Beard (1)
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Introduced by Beard (1959) in univariate setting
to improve population mortality modelling by
allowing heterogeneity
Beard (1959) starts from Makeham’s law (1868)
with
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the constant hazard and with
the hazard increases with time
Longevity factor is added to model
History of term frailty- Beard (2)
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Beard’s model
Population survival function
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Population hazard function
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Hazard at time t for
subject with frailty u
Survival at time t for
subject with frailty u
History of term frailty - Vaupel (1)
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
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Frailty – two subpopulations (1)
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Vaupel and Yashin (1985) studied
heterogeneity due to two subpopulations
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Population 1:
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Population 2:
Frailty – two subpopulations (2)
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Smokers:high and low recidivism rate
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)
Frailty – two subpopulations (3)
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Reliability engineering
Frailty – two subpopulations (4)
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Two hazards increasing at different rates
Frailty – two subpopulations (5)
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Two parallel hazards (at log scale)
Exercise
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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 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)
```