When using real-life datasets, hazard rates are hardly constant nor
do they follow a prescribed distribution. Change-point hazard functions
allow researchers to model more complex time-to-event patterns.
Functions with more than one change-point allow for additional
flexibility. The cpsurvsim package allows users to simulate
time-to-event data from the exponential (or piecewise constant) hazard
function and the Weibull hazard function. Two simulation methods are
provided: the inverse CDF method and a memoryless method exploiting the
memoryless property of survival data.
In survival analysis, the probability that an individual experiences an event at time \(t\) is independent of the probability that they experience an event up to that point. The memoryless method of simulation uses that assumption in the sense that the probability an event occurs after a change-point is independent of the probability of an event occuring before the change-point. In this way, data in between change-points are simulated from independent exponential or Weibull hazard distributions with scale parameters \(\theta\) corresponding to each time interval.
For multiple change-points, the exponential change-point hazard function (also known as the piecewise constant hazard function) for \(K\) change-points is
\(\begin{eqnarray} h(t)&=&\begin{cases} \theta_1 & 0\leq t<\tau_1\\ \theta_2 & \tau_1\leq t <\tau_2\\ \vdots & \vdots \\ \theta_{K+1} & t\geq\tau_K \end{cases} \end{eqnarray}\).
The CDF method implemented in exp_cdfsim draws on the
work of Walke1, using the relationship between the CDF and
cumulative hazard function, \(1-F(t) =
\exp(−H(t))\), in order to simulate data. Specifically, we
generate values (\(x\)) from the
exponential distribution and substitute them into the inverse hazard
function \(\begin{eqnarray}
H^{-1}(x)&=&\begin{cases} \frac{x}{\theta_1} & 0\leq
x<A\\ \frac{x-A}{\theta_2}+\tau_1 & A\leq x<A+B\\
\frac{x-A-B}{\theta_3}+\tau_2 & A+B\leq x <A+B+C\\
\frac{x-A-B-C}{\theta_4}+\tau_3 & A+B+C\leq x<A+B+C+D\\
\frac{x-A-B-C-D}{\theta_5}+\tau_4 & x\geq A+B+C+D \end{cases}
\end{eqnarray}\)
where \(A=\theta_1\tau_1\), \(B=\theta_2(\tau_2-\tau_1)\), \(C=\theta_3(\tau_3-\tau_2)\), and \(D=\theta_4(\tau_4-\tau_3)\) in order to get
simulated event times \(t\). The
function exp_cdfsim allows for up to 4 change-points.
The exp_memsim function implements the memoryless method
to simulate data for each time interval between change-points from
independent exponential distributions, using the inverse CDF function
\(F^{-1}(u)=(-\log(1-u))/\theta\). This
inverse CDF is implemented in the function exp_icdf.
The Weibull change-point hazard function for \(K\) change-points is
\(\begin{eqnarray} h(t)&=&\begin{cases} \theta_1 t^{\gamma-1} & 0\leq t<\tau_1\\ \theta_2 t^{\gamma-1} & \tau_1\leq t<\tau_2 \\ \vdots & \vdots \\ \theta_{K+1} t^{\gamma-1} & t\geq\tau_K \end{cases} \end{eqnarray}\).
We derive the inverse hazard function for four change-points as
\(\begin{eqnarray} H^{-1}(x)&=&\begin{cases} (\frac{\gamma}{\theta_1}x)^{1/\gamma} & 0\leq x<A\\ [\frac{\gamma}{\theta_2}(x-A)+\tau_1^{\gamma}]^{1/\gamma} & A\leq x<A+B\\ [\frac{\gamma}{\theta_3}(x-A-B)+\tau_2^\gamma]^{1/\gamma} & A+B\leq x<A+B+C\\ [\frac{\gamma}{\theta_4}(x-A-B-C)+\tau_3^\gamma]^{1/\gamma} & A+B+C\leq x<A+B+C+D\\ [\frac{\gamma}{\theta_5}(x-A-B-C-D)+\tau_4^\gamma]^{1/\gamma} & x\geq A+B+C+D \end{cases} \end{eqnarray}\)
where \(A=\frac{\theta_1}{\gamma}\tau_1^{\gamma}\),
\(B=\frac{\theta_2}{\gamma}(\tau_2^\gamma-\tau_1^\gamma)\),
\(C=\frac{\theta_3}{\gamma}(\tau_3^\gamma-\tau_2^\gamma)\),
and \(D=\frac{\theta_4}{\gamma}(\tau_4^\gamma-\tau_3^\gamma)\).
In the function weib_cdfsim, we simulate values (\(x\)) from the exponential distribution and
plug them into this function to get simulated event times \(t\). weib_cdfsim allows for up
to 4 change-points.
The function weib_memsim simulates data between
change-points from independent Weibull distributions using the inverse
CDF function \(F^{-1}(u)=(-\gamma/\theta
\log(1-u))^{1/\gamma}\). This inverse CDF is implemented in the
function weib_icdf.
Rainer Walke. Example for a piecewise constant hazard data simulation in R. Max Planck Institute for Demographic Research, 2010. https://www.demogr.mpg.de/papers/technicalreports/tr-2010-003.pdf↩︎