---
title: "Simulate Survival and Recurrent Event Data with reda"
author: Wenjie Wang
date: "`r Sys.Date()`"
output:
  rmarkdown::html_document:
    toc: true
    toc_depth: 2
    toc_float: true
bibliography: ../inst/bib/reda.bib
vignette: >
  %\VignetteIndexEntry{Simulate Survival and Recurrent Event Data with reda}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


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 {#intro}

## Intensity function {#intensity}

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 @kalbfleisch2002wiley.  Other helpful
references include @aalen2008springer, @kleinbaum2011springer, 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 [@cox1972jrssb] and the Andersen-Gill
model [@andersen1982aos], $r(\mathbf{z}_i, \boldsymbol{\beta})=
\exp\{\boldsymbol{\beta}^{\top}\mathbf{z}_i\}$. 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 [@lewis1979nrlq] and the inversion method
[@cinlar1975printice] 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 {#getting-started}

```{r setup}
library(reda)                    # attach reda package to the search path
packageVersion("reda")           # check the package version
options(digits = 3)              # set the number of significant digits to print
set.seed(123)                    # set random number seed
```

# Poisson process {#poisson-process}

## Homogeneous Poisson process {#hpp}

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:

```{r hpp}
## HPP from time 1 to 5 of intensity 1 without covariates
simEvent(rho = 1, origin = 1, endTime = 5)
## 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)
```

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

```{r hpp-data-1}
## 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)
```

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.

```{r hpp-data-2}
## 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))
```

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.

```{r hpp-data-3}
## 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)))
```

## Nonhomogeneous Poisson process {#nhpp}

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 $\rho(t) = \sin(t) + 1$.

```{r nhpp}
rhoFun <- function(x, b = 1) (sin(x) + 1) * b
simEvent(rho = rhoFun)
```

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 $\rho(t) = 0.5(\sin(t) + 1)$.

```{r nhpp-data}
simEventData(z = cbind(rexp(2), c(0, 1)), zCoef = c(0.1, - 0.5),
             rho = rhoFun, arguments = list(rho = list(b = 0.5)))
```

# Renewal processes {#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:

```{r rgamma-wi-n}
set.seed(123)
simEvent(interarrival = function(n, rate) rgamma(n, shape = 1 / rate))
```

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.

```{r rgamma-wo-n}
set.seed(123)
simEvent(interarrival = function(rate) rgamma(n = 1, shape = 1 / rate))
```

# 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.

```{r time-varying-covariates-1}
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))
```

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.

```{r time-varying-covariates-2}
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)))
```

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 [@crowley1977jasa] 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:

```{r time-varying-covariates-3}
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))))))
```

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:

```{r check-a}
## check the values of parameter `a` for different processes
with(simDat, unique(cbind(ID, a = X.1 - time / 10)))
```


## 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:

```{r time-varying-coef-1}
zCoefFun <- function(time, shift) cbind(sqrt(time / 10), sin(time + shift), 0.1)
simEventData(z = zFun1, zCoef = zCoefFun,
             arguments = list(zCoef = list(shift = 1)))
```

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}

## Frailty for individual processes {#individual-frailty}

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:

```{r frailty-gamma}
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)))
```

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.

```{r frailty-lognormal-1}
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)))
```

```{r frailty-lognormal-2}
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)))
```

```{r frailty-lognormal-3}
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)))
```

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 {#shared-frailty}

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.

```{r shared-frailty-1}
## 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))
```

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]`).

```{r shared-frailty-2}
set.seed(123)
simEventData(nProcess = 4, z = zFun1, zCoef = c(0.1, 0.5, - 0.5),
             frailty = frailtyEffect)
```


# 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:

```{r weibull-ph}
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)))
```

## 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:

```{r gompertz}
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)))
```


## Log-logistic model

As discussed in Section [accelerated failure time
model](#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:

```{r loglogistic}
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)))
```

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](#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:

```{r log-normal}
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)))
```

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
