Problem: Competing risks model

Eksempel

Problem

Let \(T_1\) and \(T_2\) be independent and exponentially distributed variables with \(f_{T_j} (t) = \lambda_j e^{ −\lambda_j t}\), \(t \geq 0\), and let \(T = \min\{T_1, T_2\}\) and \(D = 1\) if \(T_1 \leq T_2\), \(2\) if \(T_1 > T_2\). Assuming \(T_1\) and \(T_2\) to be the potential times until failure of two different causes, \(T\) is the time until failure and \(D\) is a variable specifying the reason for the failure. The given setup is therefore a competing risks model with \(k = 2\) possible causes for failure. The cause-specific rates are one way to describe a competing risks model. An alternative way for describing such a model is to consider the cumulative incidence functions, \(P(T \leq t, D = j)\) for \(j = 1, \ldots, k\).

(a)

Show that the cumulative incidence functions for the competing risks model specified above are \[P(T \leq t, D = j) = \frac{\lambda_j}{\lambda_1 + \lambda_2} \left(1 − e^{ −(\lambda_1+\lambda_2)t}\right),\]  for \(j = 1, 2\).

(b)

Show that the cause-specific hazard rates for the competing risks model specified above are \[\alpha_j (t) =\lambda_j.\]

(c)

Discuss the property that cause-specific hazard rates are not influenced by the distribution of the potential failures of other causes, while this is not true for the cumulative incidence function. Note: This is a general property for competing risks.

(d)

Choose reasonable values for \(\lambda_1\) and \(\lambda_2\) and simulate a number of realisations of \((T, D)\). Illustrate the result in (c) by using the simulated data to estimate the cause-specific hazard rates and cumulative incidence functions. Try with different sets of values for \((\lambda_1, \lambda_2)\), changing only \(\lambda_1\) or \(\lambda_2\) at a time, and see the effect.

Hint: See the code below for some inspiration for this task.

Kopier 
library(survival)
library(cmprsk)

# obstime <- INSERT WAY TO GENERATE Ts
# d <- INSERT WAY TO CALCULATE Ds

#Estimate and plot the cause-specific cumulative hazard rate
causespec1 <- survfit(Surv(obstime,d==1)~1,type="fh2")
plot(causespec1,fun="cumhaz",mark.time=F) 

#Estimate and plot the cumulative incidence functions:
cif <- cuminc(obstime,d)
plot(cif$`1 1`$time,cif$`1 1`$est,lty=1,type="s",ylim=c(0,1))
lines(cif$`1 2`$time,cif$`1 2`$est,lty=2,type="s")
Inspiration for simulations in R.

Hints: To get some hints for how to solve the problem see the link at the bottom of this page.