In this vignette, we briefly introduce how to simulate survival and recurrent event data from stochastic process point of view with the reda package. The core function named simEvent() provides an intuitive and flexible interface for simulating survival and recurrent event times from one stochastic process. Another function named simEventData() is simply a wrapper that calls simEvent() internally and collects the event times and the covariates of a given number of processes into a data frame. Examples of generating random samples from common survival and recurrent models are provided. In fact, the function simEvent() and simEventData() may serve as the building blocks for simulating multitype event data including multiple type event data, recurrent events with termination, and competing risk data. The details of function syntax and the objects produced are available in the package manual and thus not covered in this vignette.

Introduction

Intensity function

We introduce the general form of hazard rate function considered and implemented in the function simEvent(), which can be generalized to two classes called the relative risk model, and the accelerated failure time model. The introduction is based on Section 2.2 and Section 2.3 of Kalbfleisch and Prentice (2002). Other helpful references include Aalen, Borgan, and Gjessing (2008), Kleinbaum and Klein (2011), among others.

Let’s consider \(n\) stochastic processes with the baseline hazard rate function \(\rho(t)\) of time \(t\). For the stochastic process \(i\), \(i\in\{1,\ldots,n\}\), let \(\mathbf{z}_i=(z_{i1},\ldots,z_{ip})^{\top}\) denote the covariate vector of length \(p\), and \(\boldsymbol{\beta}=(\beta_1,\ldots,\beta_p)^{\top}\) denote the covariate coefficients.

Relative risk model

Given the covariates \(\mathbf{z}_i\) (not including the intercept term), the intensity function of time \(t\) for the relative risk regression model can be specified as follows: \[\lambda(t \mid \mathbf{z}_i) = \rho(t)\,r(\mathbf{z}_i, \boldsymbol{\beta}),\] where \(r(\mathbf{z}_i, \boldsymbol{\beta})\) is the relative risk function. For the Cox model (Cox 1972) and the Andersen-Gill model (Andersen and Gill 1982), \(r(\mathbf{z}_i, \boldsymbol{\beta})= \exp\{\boldsymbol{\beta}^{\top}\mathbf{z}_i\}\). Other common choices include the linear relative risk function: \(r(\mathbf{z}_i, \boldsymbol{\beta}) = 1 + \boldsymbol{\beta}^{\top}\mathbf{z}_i\), and the excess relative risk function: \(r(\mathbf{z}_i, \boldsymbol{\beta}) = \prod_{j=1}^p(1 + \beta_j z_{ij})\), both of which, however, suffer from the drawback that the \(r(\mathbf{z}_i, \boldsymbol{\beta})\) is not necessarily positive. Therefore, the coefficient estimates must be restricted to guarantee that the relative risk function is positive for all possible covariates. The restriction disappears for the exponential relative risk function since it always returns positive values.

We may extend the model by considering a random frailty effect \(u_i\) (of expectation one) to account for heterogeneity between different processes or processes from different clusters. The intensity function becomes \[\lambda(t \mid \mathbf{z}_i, u_i) = u_i\,\rho(t)\,r(\mathbf{z}_i, \boldsymbol{\beta}).\] The common choices for distribution of the random frailty effect include Gamma distribution of mean one, and lognormal distribution of mean zero in logarithm scale.

Furthermore, both covariates and coefficients may be time-varying. So a general form of the intensity function for the relative risk model is given as follows: \[\lambda\bigl(t \mid \mathbf{z}_i(t), u_i\bigr) = u_i\,\rho(t)\,r\bigl(\mathbf{z}_i(t), \boldsymbol{\beta}(t)\bigr).\]

Accelerated failure time model

The relative risk models incorporate the covariates and their coefficients through a relative risk function multiplied by the baseline hazard rate, which provides an intuitive interpretation. However, we may consider a direct relationship between the covraiates \(\mathbf{z}\) (including the intercept term) and the time to failure \(T>0\), \(\log(T) = \alpha + \sigma W\), where \(W\) is a standardized random error variable with density function \(f_W(w)\), suvival function \(S_W(w)\) and hazard function \(\rho(w) = f_W(w) / S_W(w)\), \(\sigma\) represents the standard error, and \(\alpha = \boldsymbol{\beta}^{\top}\mathbf{z}\). We assume that \(W\) is independent of \(\boldsymbol{\beta}\) given the covariates \(\mathbf{z}\). Taking exponentiation gives \(T=\exp(\boldsymbol{\beta}^{\top}\mathbf{z})\exp(W)\) with density \(f_T(t) = \frac{1}{\sigma t}f_W\bigl((\log(t) - \alpha)/\sigma\bigr)\), survival function \(S_T(t) = S_W\bigl((\log(t) - \alpha)/\sigma\bigr)\), and hazard function \[\lambda(t \mid \mathbf{z}_i) = \frac{1}{\sigma t}\rho\bigl((\log(t) - \alpha)/\sigma\bigr).\] Let \(\lambda_{\mathbf{z}}=\exp(- \alpha)\) and \(p = 1 / \sigma\), we may rewrite the hazard function as follows: \[\lambda(t \mid \mathbf{z}_i) = \frac{p}{t}\rho\bigl(p\log(\lambda_{\mathbf{z}} t)\bigr).\] The resulting model is called accelerated failure time (AFT) model.

For example, suppose \(W\) follows standard logistic distribution with density \(f_W(w) = e^w / (1 + e^w) ^ 2\), survival function \(S_W(w) = 1 / (1 + e^w)\), and hazard function \(\rho(w) = e^w / (1 + e^w)\). This leads to log-logistic model of \(T\) with hazard function \[\lambda(t \mid \mathbf{z}_i) = \frac{p (t\lambda_{\mathbf{z}})^p/t}{1 + (t\lambda_{\mathbf{z}})^p}.\]

Similarly, we may further consider frailty factor, time-variant covariates, and time-varying covariate coefficients. So a general form of the intensity function may be given as follows: \[\lambda\bigl(t \mid \mathbf{z}_i(t), u_i\bigr) = \frac{u_i\,p}{t}\,\rho\bigl( p \log(t) - p \boldsymbol{\beta}(t)^{\top}\mathbf{z}_i(t)\bigr).\]

Intensity function in general form

The function simEvent() and simEventData() allow users to specify an intensity function in an even more general form given below. \[\lambda\bigl(t \mid \mathbf{z}_i(t), u_i\bigr) = u_i\,\rho\bigl(t, \mathbf{z}_i(t), \boldsymbol{\beta}(t)\bigr) \,r\bigl(\mathbf{z}_i(t), \boldsymbol{\beta}(t)\bigr),\] where \(u_i\), \(\rho(\cdot)\), and \(r(\cdot)\) corresponds to the argument frailty, rho, and relativeRisk, respectively.

Sampling methods

The thinning method (Lewis and Shedler 1979) and the inversion method (Çınlar 1975) are implemented for sampling event times. It can be shown that both methods achieve the given hazard rate. For function simEvent(), the thinning method is the default method when the hazard rate function is bounded within follow-ups. Otherwise, the inversion method will be used. We may specify the sampling method via the argument method in function simEvent() and simEventData().

Getting started

library(reda)                    # attach reda package to the search path
packageVersion("reda")           # check the package version
## [1] '0.5.0'
options(digits = 3)              # set the number of significant digits to print
set.seed(123)                    # set random number seed

Poisson process

Homogeneous Poisson process

A homogeneous/stationary Poisson process (HPP) has a constant hazard rate over time with the interarrival times (between two successive arrivals/events) following exponential distribution. Two simple examples of simulating a homogeneous Poisson process using simEvent() are given as follows:

## HPP from time 1 to 5 of intensity 1 without covariates
simEvent(rho = 1, origin = 1, endTime = 5)
## 'simEvent' S4 class object:
## [1] 1.84 2.42 3.75 3.78 3.84 4.15 4.47 4.61
## HPP from 0 to 10 of baseline hazard rate 0.5 with two covariates
simEvent(z = c(0.2, 0.5), zCoef = c(0.5, - 0.1), rho = 0.5, endTime = 10)
## 'simEvent' S4 class object:
## [1] 0.717 1.076 2.692 5.666 6.577 7.701

The function simEventData() enable us to simulate multiple processes and collect the simulated event times into a survival or recurrent event data format.

## recurrent events from two processes with same covariates
simEventData(2, z = c(0.2, 0.5), zCoef = c(1, - 0.5), rho = 0.5, endTime = 5)
##   ID time event origin X.1 X.2
## 1  1 3.38     1      0 0.2 0.5
## 2  1 5.00     0      0 0.2 0.5
## 3  2 1.84     1      0 0.2 0.5
## 4  2 5.00     0      0 0.2 0.5

In the example given above, the number of process is explicitly specified to be two. However, if it is not specified, it will be the number of rows of the covariate matrix. See the example given below.

## recurrent events from two processes
## with different time-invariant covariates and time origins
simEventData(z = cbind(rnorm(2), 0.5), zCoef = c(1, - 0.5),
             rho = 0.2, origin = c(1, 0), endTime = c(10, 9))
##   ID  time event origin   X.1 X.2
## 1  1  4.91     1      1 0.838 0.5
## 2  1  6.31     1      1 0.838 0.5
## 3  1  7.04     1      1 0.838 0.5
## 4  1 10.00     0      1 0.838 0.5
## 5  2  3.44     1      0 0.153 0.5
## 6  2  6.68     1      0 0.153 0.5
## 7  2  9.00     0      0 0.153 0.5

We can also simulate survival data by taking the first event of each process. Setting recurrent = FALSE in function simEvent() (or function simEventData()) gives us the survival time(s) or the right censoring time(s). In the example given below, we specified endTime = "rnorm" and arguments = list(endTime = list(mean = 10)) for generating a random censoring times from normal distribution with mean ten and unit standard deviation. Also note that the specified origin is recycled for these ten processes.

## survival data by set 'recurrent = FALSE'
simEventData(z = cbind(rnorm(10), 1), zCoef = c(0.2, - 0.5), rho = 0.1,
             origin = c(0, 1), endTime = stats::rnorm, recurrent = FALSE,
             arguments = list(endTime = list(mean = 10)))
##    ID  time event origin    X.1 X.2
## 1   1  9.11     0      0 -0.772   1
## 2   2 10.38     0      1  0.287   1
## 3   3  4.55     1      0 -1.221   1
## 4   4 10.39     0      1  0.435   1
## 5   5  9.55     1      0  0.800   1
## 6   6  8.97     0      1 -0.164   1
## 7   7  8.25     1      0  1.243   1
## 8   8 10.96     0      1 -0.934   1
## 9   9 10.08     0      0  0.394   1
## 10 10 11.13     0      1  0.404   1

Nonhomogeneous Poisson process

In contrast to HPP, a nonhomogeneous Poisson process (NHPP) has a time-varying hazard rate function. In that case, we may specify the baseline hazard rate function rhoFun() to be a function object whose first argument represents the time variable. A quick example is given below, where the baseline hazard rate function \(\rho(t) = \sin(t) + 1\).

rhoFun <- function(x, b = 1) (sin(x) + 1) * b
simEvent(rho = rhoFun)
## 'simEvent' S4 class object:
## [1] 1.15

As demonstrated in the last example for HPP, other possible arguments of the function objects can be specified via the arguments. For example, that arguments = list(rho = list(b = 0.5)) specifies the baseline hazard rate function to be \(\rho(t) = 0.5(\sin(t) + 1)\).

simEventData(z = cbind(rexp(2), c(0, 1)), zCoef = c(0.1, - 0.5),
             rho = rhoFun, arguments = list(rho = list(b = 0.5)))
##   ID  time event origin    X.1 X.2
## 1  1 3.000     0      0 0.0687   0
## 2  2 0.715     1      0 0.5692   1
## 3  2 1.120     1      0 0.5692   1
## 4  2 2.208     1      0 0.5692   1
## 5  2 3.000     0      0 0.5692   1

Renewal processes

In the Poisson process, the interarrival times between two successive arrivals (or events) follow exponential distribution independently. We may generalize the distribution of interarrival times and consider more general renewal processes.

In function simEvent() (and simEventData()), we may specify the distribution of the interarrival times via the argument interarrival, which takes the function stats::rexp() for generating interarrival times following exponential distribution by default. In general, the argument interarrival takes a function with at least one argument named rate for generating random (nonnegative) interarrival times from a certain distribution at the given arrival rate. A quick example of generating the interarrival times following Gamma distribution of scale one is given as follows:

set.seed(123)
simEvent(interarrival = function(n, rate) rgamma(n, shape = 1 / rate))
## 'simEvent' S4 class object:
## [1] 0.182 1.878

If the specified function has an argument named n, the function simEvent() will assume that the function can generate n number of random interarrival times at one time and take advantage of the vectorization for a potentially better performance. However, it is optional. The example given below produces an equivalent result.

set.seed(123)
simEvent(interarrival = function(rate) rgamma(n = 1, shape = 1 / rate))
## 'simEvent' S4 class object:
## [1] 0.182 1.878

Beyond time-invariance

Time-variant covariates

In practice, some covariates such as patients’ age, automobile’s mileage may vary over time. The argument z in the function simEvent() and simEventData() may take a function of time that returns a vector of covariates for generating event times with the time-varying covariates. Let’s consider an example of generating recurrent event times with three covariates, where two of which are time-variant.

set.seed(123)
zFun1 <- function(time) cbind(time / 10 + 1, as.numeric(time > 1), 0.5)
simEventData(z = zFun1, zCoef = c(0.1, 0.5, - 0.5))
##   ID  time event origin  X.1 X.2 X.3
## 1  1 0.971     1      0 1.10   0 0.5
## 2  1 1.880     1      0 1.19   1 0.5
## 3  1 1.902     1      0 1.19   1 0.5
## 4  1 1.940     1      0 1.19   1 0.5
## 5  1 2.156     1      0 1.22   1 0.5
## 6  1 2.371     1      0 1.24   1 0.5
## 7  1 2.471     1      0 1.25   1 0.5
## 8  1 3.000     0      0 1.30   1 0.5

In the example given above, the covariate vector is \(\mathbf{z}(t)=(0.1t+1, \boldsymbol{1}(t > 1), 0.5)^{\top}\). If the covariate function has more arguments, we may specify them by a named list in arguments. The example given below produces the equivalent results.

set.seed(123)
zFun2 <- function(x, a, b) cbind(x / 10 + a, as.numeric(x > b), 0.5)
simEventData(z = zFun2, zCoef = c(0.1, 0.5, - 0.5),
             arguments = list(z = list(a = 1, b = 1)))
##   ID  time event origin  X.1 X.2 X.3
## 1  1 0.971     1      0 1.10   0 0.5
## 2  1 1.880     1      0 1.19   1 0.5
## 3  1 1.902     1      0 1.19   1 0.5
## 4  1 1.940     1      0 1.19   1 0.5
## 5  1 2.156     1      0 1.22   1 0.5
## 6  1 2.371     1      0 1.24   1 0.5
## 7  1 2.471     1      0 1.25   1 0.5
## 8  1 3.000     0      0 1.30   1 0.5

Notice that in the examples given above, if we generate event times for more than one process, the time-varying covariate function will remain the same for different processes, which may not be the case in practice. A more realistic situation is that the time-variant covariate functions are different among different processes but coming from a common function family. Let’s consider the Stanford heart transplant data (Crowley and Hu 1977) as an example (the heart data available in the survival package). The covariate transplant indicating whether the patient has already received a heart transplant before time t is time-dependent and can be represented by a indicator function family, \(\boldsymbol{1}(t > b)\), where \(b\) is a known parameter that may differ among patients. For a particular patient \(i\), \(b = b_i\) is a known constant. In that case, we specify the function parameters inside the quote function as follows:

zFun3 <- function(time, a, b) cbind(time / 10 + a, as.numeric(time > b))
(simDat <- simEventData(nProcess = 3, z = zFun3, zCoef = c(- 0.1, 0.5),
                        arguments = list(z = list(a = quote(rpois(1, 10) / 10),
                                                  b = quote(runif(1, 1, 3))))))
##    ID  time event origin   X.1 X.2
## 1   1 0.104     1      0 1.710   0
## 2   1 0.328     1      0 1.733   0
## 3   1 1.335     1      0 1.833   0
## 4   1 3.000     0      0 2.000   1
## 5   2 0.583     1      0 1.258   0
## 6   2 2.327     1      0 1.433   1
## 7   2 3.000     0      0 1.500   1
## 8   3 0.804     1      0 0.680   0
## 9   3 0.838     1      0 0.684   0
## 10  3 1.472     1      0 0.747   0
## 11  3 3.000     0      0 0.900   1

In the example given above, the covariate X.2 is simulated from the indicator function famliy, \(\boldsymbol{1}(t > b)\), where parameter \(b\) follows uniform distribution between 1 and 3. Internally, the parameters specified in arguments were evaluated for each process. We may check the values of the parameter a from the generated covariate X.1 for different processes as follows:

## check the values of parameter `a` for different processes
with(simDat, unique(cbind(ID, a = X.1 - time / 10)))
##      ID   a
## [1,]  1 1.7
## [2,]  2 1.2
## [3,]  3 0.6
## [4,]  3 0.6

Time-variant covariate coefficients

The assumption of time-invariance on the covariate coefficients can be hard to justify in practice. We may simulate event times with time-varying covariate coefficients by specifying the argument zCoef to be a function of time that returns a vector of coefficients at the input time point. How we may specify the argument zCoef for time-varying coefficients is very similar to the way we may specify the argument z for time-varying covariates introduced in last section. For example, we may generate event times with both covariates and their coefficients being time-variant as follows:

zCoefFun <- function(time, shift) cbind(sqrt(time / 10), sin(time + shift), 0.1)
simEventData(z = zFun1, zCoef = zCoefFun,
             arguments = list(zCoef = list(shift = 1)))
##   ID time event origin  X.1 X.2 X.3
## 1  1 1.12     1      0 1.11   1 0.5
## 2  1 1.34     1      0 1.13   1 0.5
## 3  1 2.31     1      0 1.23   1 0.5
## 4  1 2.43     1      0 1.24   1 0.5
## 5  1 3.00     0      0 1.30   1 0.5

As demonstrated in the example given above, we may similarly specify the other arguments of the time-varying coefficient function via a named list in arguments.

Frailty models

Frailty for individual processes

Let’s consider frailty factors for individual processes first, where each process \(i\) has its own frailty effect \(w_i\). A popular choice of the frailty distribution is the one-parameter gamma distribution of mean one, which often leads to an explicit marginal likelihood in a relatively simple expression. Similar to the argument z, zCoef, and rho, the argument frailty may take a function as input for simulating the frailty effect. For example, we may simulate the recurrent event times for one process with frailty factor following gamma(2, 0.5) via frailty = "rgamma" and arguments = list(frailty = list(shape = 2, scale = 0.5)) as follows:

set.seed(123)
simEventData(z = zFun1, zCoef = c(0.1, 0.5, - 0.5), frailty = stats::rgamma,
             arguments = list(frailty = list(shape = 2, scale = 0.5)))
##   ID  time event origin  X.1 X.2 X.3
## 1  1 0.135     1      0 1.01   0 0.5
## 2  1 0.620     1      0 1.06   0 0.5
## 3  1 1.102     1      0 1.11   1 0.5
## 4  1 1.324     1      0 1.13   1 0.5
## 5  1 3.000     0      0 1.30   1 0.5

The named list list(shape = 2, scale = 0.5) was passed to the function rgamma (from the stats package). The random number seed was reset so that we might compare the results with the first example of time-variant covariates. Note that it is users’ job to make sure the specified distribution of the frailty factor has mean one (or makes sense in a certain way). The function simEvent() and simEventData() only check the sign of the simulated frailty effect. An error will be thrown out if the generated frailty effect is not positive.

To demonstrate how to specify other distribution of the frailty factor, we simulate the recurrent event times for one process by three slightly different but equivalent approaches in the following examples, where the frailty effect follows log-normal distribution of mean one.

set.seed(123)
## use function `rlnorm` from the stats package
simEvent(z = zFun1, zCoef = c(0.1, 0.5, - 0.5), frailty = stats::rlnorm,
         arguments = list(frailty = list(sdlog = 1)))
## 'simEvent' S4 class object:
## [1] 1.59 1.63 1.70 2.08 2.45 2.63
set.seed(123)
## use a customized function with argument `n` and `sdlog`
logNorm1 <- function(n, sdlog) exp(rnorm(n = n, mean = 0, sd = sdlog))
simEvent(z = zFun1, zCoef = c(0.1, 0.5, - 0.5), frailty = logNorm1,
         arguments = list(frailty = list(sdlog = 1)))
## 'simEvent' S4 class object:
## [1] 1.59 1.63 1.70 2.08 2.45 2.63
set.seed(123)
## use a customized function with argument `sdlog` only
logNorm2 <- function(sdlog) exp(rnorm(n = 1, mean = 0, sd = sdlog))
simEvent(z = zFun1, zCoef = c(0.1, 0.5, - 0.5), frailty = logNorm2,
         arguments = list(frailty = list(sdlog = 1)))
## 'simEvent' S4 class object:
## [1] 1.59 1.63 1.70 2.08 2.45 2.63

If the function specified for frailty has an argument named n, that n = 1 will be specified internally by the function simEvent(), which is designed for using the functions generating random numbers, such as rgamma() and rlnorm() from the stats package.

Shared frailty for clusters

When different processes come from several clusters, we may consider a same frailty effect shared among processes within a cluster. The case we considered in last section where frailty factors are different among individual processes is a special case when the cluster size is one.

In the function simEvent() (and simEventData()), the argument frailty may take a numeric number (vector) as input for specific shared frailty effect for clusters. For instance, we may simulate the recurrent event times for four processes coming from two clusters with shared gamma frailty within cluster, where the first two processes come from one cluster while the remaining two come from another cluster.

## shared gamma frailty for processes from two clusters
frailtyEffect <- rgamma(2, shape = 2, scale = 0.5)
simEventData(nProcess = 4, z = zFun1, zCoef = c(0.1, 0.5, - 0.5),
             frailty = rep(frailtyEffect, each = 2))
##    ID  time event origin  X.1 X.2 X.3
## 1   1 0.572     1      0 1.06   0 0.5
## 2   1 1.625     1      0 1.16   1 0.5
## 3   1 1.947     1      0 1.19   1 0.5
## 4   1 2.345     1      0 1.23   1 0.5
## 5   1 3.000     0      0 1.30   1 0.5
## 6   2 1.695     1      0 1.17   1 0.5
## 7   2 2.776     1      0 1.28   1 0.5
## 8   2 3.000     0      0 1.30   1 0.5
## 9   3 1.702     1      0 1.17   1 0.5
## 10  3 2.876     1      0 1.29   1 0.5
## 11  3 3.000     0      0 1.30   1 0.5
## 12  4 0.276     1      0 1.03   0 0.5
## 13  4 2.443     1      0 1.24   1 0.5
## 14  4 3.000     0      0 1.30   1 0.5

If the length of the specified frailty vector is less than the number of processes, the vector will be recycled internally. In the example given below, the process 1 and process 3 shared a same frailty effect (frailtyEffect[1L]). Similarly, the process 2 and process 4 shared a same frailty effect (frailtyEffect[2L]).

set.seed(123)
simEventData(nProcess = 4, z = zFun1, zCoef = c(0.1, 0.5, - 0.5),
             frailty = frailtyEffect)
##    ID  time event origin  X.1 X.2 X.3
## 1   1 0.956     1      0 1.10   0 0.5
## 2   1 1.851     1      0 1.19   1 0.5
## 3   1 1.873     1      0 1.19   1 0.5
## 4   1 1.910     1      0 1.19   1 0.5
## 5   1 2.124     1      0 1.21   1 0.5
## 6   1 2.335     1      0 1.23   1 0.5
## 7   1 2.433     1      0 1.24   1 0.5
## 8   1 3.000     0      0 1.30   1 0.5
## 9   2 0.315     1      0 1.03   0 0.5
## 10  2 0.472     1      0 1.05   0 0.5
## 11  2 1.181     1      0 1.12   1 0.5
## 12  2 2.486     1      0 1.25   1 0.5
## 13  2 2.886     1      0 1.29   1 0.5
## 14  2 3.000     0      0 1.30   1 0.5
## 15  3 1.000     1      0 1.10   1 0.5
## 16  3 1.908     1      0 1.19   1 0.5
## 17  3 2.695     1      0 1.27   1 0.5
## 18  3 3.000     0      0 1.30   1 0.5
## 19  4 0.499     1      0 1.05   0 0.5
## 20  4 2.308     1      0 1.23   1 0.5
## 21  4 2.731     1      0 1.27   1 0.5
## 22  4 2.948     1      0 1.29   1 0.5
## 23  4 3.000     0      0 1.30   1 0.5

Common parametric survival models

In this section, we present examples for generating event times with time-invariant covariates for several common parametric survival models.

Weibull model

The Weibull model is one of the most widely used parametric survival models. Assume the event times of the process \(i\), \(i\in\{1,\ldots,n\}\), follow Weibull model with hazard function \(h_i(t) = \lambda_i p t^{p-1}\), where \(p>0\) is the shape parameter, and \(\lambda_i\) can be reparametrized with regression coefficients. When \(p=1\), the Weibull model reduces to the exponential model, whose hazard rate is a constant over time.

One common reparametrization is \(\lambda_i = \exp(\beta_0 + \boldsymbol{\beta}^{\top}\mathbf{z}_i)\), which results in Weibull proportional hazard (PH) model. Let \(\lambda_0 = \exp(\beta_0)\) and we may rewrite the hazard function \(h_i(t) = \rho(t) \exp(\boldsymbol{\beta}^{\top}\mathbf{z}_i)\), where \(\rho(t) = \lambda_0 p t^{p-1}\) is the baseline hazard function. For example, we may simulate the survival data of ten processes from Weibull PH model as follows:

nProcess <- 10
rho_weibull_ph <- function(x, lambda, p) lambda * p * x ^ (p - 1)
simEventData(z = cbind(rnorm(nProcess), rbinom(nProcess, 1, 0.5)),
             zCoef = c(0.5, 0.2), endTime = rnorm(nProcess, 10),
             recurrent = FALSE, rho = rho_weibull_ph,
             arguments = list(rho = list(lambda = 0.01, p = 2)))
##    ID  time event origin    X.1 X.2
## 1   1  7.48     1      0  0.887   1
## 2   2  4.24     1      0 -0.151   1
## 3   3  6.81     1      0  0.330   0
## 4   4 10.59     1      0 -3.227   0
## 5   5  4.73     1      0 -0.772   1
## 6   6 10.28     0      0  0.287   0
## 7   7  6.93     1      0 -1.221   1
## 8   8  4.35     1      0  0.435   0
## 9   9  3.78     1      0  0.800   0
## 10 10  8.93     0      0 -0.164   1

Gompertz model

The baseline hazard function of the gompertz model is \(\rho(t) = \lambda\exp(\alpha t)\), where \(\lambda > 0\) is the scale parameter and \(\alpha\) is the shape paramter. So the logrithm of the baseline hazard function is linear in time \(t\).

Similar to the example for the Weibull model given in last section, we may simulate the survival data of ten processes as follows:

rho_gompertz <- function(time, lambda, alpha) lambda * exp(alpha * time)
simEventData(z = cbind(rnorm(nProcess), rbinom(nProcess, 1, 0.5)),
             zCoef = c(0.5, 0.2), endTime = rnorm(nProcess, 10),
             recurrent = FALSE, rho = rho_gompertz,
             arguments = list(rho = list(lambda = 0.1, alpha = 0.1)))
##    ID time event origin    X.1 X.2
## 1   1 8.95     0      0 -1.462   1
## 2   2 8.74     0      0  0.688   0
## 3   3 4.92     1      0  2.100   0
## 4   4 9.43     1      0 -1.287   1
## 5   5 5.41     1      0  0.788   1
## 6   6 3.35     1      0  0.769   0
## 7   7 7.29     1      0  0.332   0
## 8   8 4.48     1      0 -1.008   1
## 9   9 7.40     1      0 -0.119   1
## 10 10 9.78     0      0 -0.280   0

Log-logistic model

As discussed in Section accelerated failure time model, the hazard function of the log-logistic model is \[\lambda(t \mid \mathbf{z}_i) = \frac{p (t\lambda_{\mathbf{z}})^p/t}{1 + (t\lambda_{\mathbf{z}})^p}.\] So we may simulate the survival data of ten processes from the log-logistic model as follows:

rho_loglogistic <- function(time, z, zCoef, p) {
    lambda <- 1 / parametrize(z, zCoef, FUN = "exponential")
    lambda * p * (lambda * time) ^ (p - 1) / (1 + (lambda * time) ^ p)
}
simEventData(z = cbind(1, rnorm(nProcess), rbinom(nProcess, 1, 0.5)),
             zCoef = c(0.3, 0.5, 0.2), end = rnorm(nProcess, 10),
             recurrent = FALSE, relativeRisk = "none", rho = rho_loglogistic,
             arguments = list(rho = list(p = 1.5)))
##    ID   time event origin X.1     X.2 X.3
## 1   1  0.206     1      0   1  0.9625   1
## 2   2  1.350     1      0   1  0.6843   0
## 3   3  1.972     1      0   1 -1.3953   1
## 4   4  3.480     1      0   1  0.8496   1
## 5   5  0.834     1      0   1 -0.4466   0
## 6   6 11.054     0      0   1  0.1748   1
## 7   7  0.916     1      0   1  0.0746   1
## 8   8  1.664     1      0   1  0.4282   1
## 9   9  3.238     1      0   1  0.0247   1
## 10 10  0.279     1      0   1 -1.6675   1

Notice that in the function rho_loglogistic() for the hazard function of the log-logistic model, we wrapped the parametrization of the covariates and covariate coefficients with the function parametrize and specified that FUN = "exponential". In addition, we specified relativeRisk = "none" when calling simEventData() for AFT models.

Log-normal model

By following the discussion given in Section accelerated failure time model, it is not hard to obtain the hazard function of the log-normal model, \[\lambda(t \mid \mathbf{z}_i) = \frac{p}{t} \rho\bigl(p(\log(t) - \boldsymbol{\beta}^{\top}\mathbf{z}_i)\bigr),\] where \(\rho(w) = \phi(w) / \bigl(1 - \Phi(w)\bigr)\), \(\phi(w)\) and \(\Phi(w)\) is the density and cumulative distribution function of random variable following standard normal distribution.

The example of simulating survival times of ten processes from the log-normal model is given as follows:

rho_lognormal <- function(time, z, zCoef, p) {
    foo <- function(x) dnorm(x) / pnorm(x, lower.tail = FALSE)
    alpha <- parametrize(z, zCoef, FUN = "linear") - 1
    w <- p * (log(time) - alpha)
    foo(w) * p / time
}
simEventData(z = cbind(1, rnorm(nProcess), rbinom(nProcess, 1, 0.5)),
             zCoef = c(0.3, 0.5, 0.2), end = rnorm(nProcess, 10),
             recurrent = FALSE, relativeRisk = "none",
             rho = rho_lognormal, method = "inversion",
             arguments = list(rho = list(p = 0.5)))
##    ID    time event origin X.1    X.2 X.3
## 1   1  0.0907     1      0   1  0.433   1
## 2   2  3.7646     1      0   1  0.510   0
## 3   3  8.8247     0      0   1  1.196   1
## 4   4  0.1732     1      0   1 -1.084   0
## 5   5  0.2221     1      0   1 -1.145   1
## 6   6  8.8738     0      0   1  0.155   1
## 7   7  2.6047     1      0   1 -0.492   1
## 8   8 10.5684     0      0   1  2.367   1
## 9   9  0.1656     1      0   1 -1.434   1
## 10 10  0.2125     1      0   1 -1.611   0

Notice that the time origin was set to be zero by default. So the time variable \(t\) in the denominator of the hazard function \(\lambda(t \mid \mathbf{z}_i)\), may result in undefined value when \(t=0\). Therefore, we specified method = "inversion" for the inversion sampling method for the log-normal model.

Reference

Aalen, Odd, Ornulf Borgan, and Hakon Gjessing. 2008. Survival and Event History Analysis: A Process Point of View. Springer Science & Business Media.

Andersen, Per Kragh, and Richard David Gill. 1982. “Cox’s Regression Model for Counting Processes: A Large Sample Study.” The Annals of Statistics 10 (4): 1100–1120.

Çınlar, Erhan. 1975. Introduction to Stochastic Processes. Englewood Cliffs, NJ: Printice-Hall.

Cox, David R. 1972. “Regression Models and Life-Tables.” Journal of the Royal Statistical Society. Series B (Methodological) 34 (2): 187–220.

Crowley, John, and Marie Hu. 1977. “Covariance Analysis of Heart Transplant Survival Data.” Journal of the American Statistical Association 72 (357): 27–36.

Kalbfleisch, John D, and Ross L Prentice. 2002. The Statistical Analysis of Failure Time Data. Vol. 360. John Wiley & Sons.

Kleinbaum, D G, and M Klein. 2011. Survival Analysis: A Self-Learning Text. New York: Springer.

Lewis, P A, and G S Shedler. 1979. “Simulation of Nonhomogeneous Poisson Processes by Thinning.” Naval Research Logistics Quarterly 26 (3): 403–13.