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In an article that I have been reading, they have a simulation study:

In this simulation, we generate $T_i$ from the following group-specific linear transformation model: $$H(T_i) = \beta_{k,1} X_{i,1} + \beta_{k,2} X_{i,2} + \varepsilon_i, i = 1, 2, \ldots, n; \quad k = 1, 2 $$ where $ H(t) = \log\left(2(e^{4t} - 1)\right)$ and $\varepsilon_i $follows the standard extreme-value distribution. In this case, the linear transformation model is equivalent to the Cox proportional hazards model. We generate samples from a two-component Cox proportional hazards model with mixing weights $ \pi_1 = \frac{1}{3}$, $\pi_2 = \frac{2}{3}$, and $ \beta_1 = (-3, -2)^T$, $\beta_2 = (1, 1)^T$. The covariates $X_i$ are generated from a multivariate normal distribution with a mean of zero and a first-order autoregressive structure $ \Sigma = (\sigma_{st})$ with $ \sigma_{st} = 0.5^{|s - t|}$ for $ s, t = 1, 2$. The censoring time is generated from a uniform distribution on $[0, C]$, where $C$ is chosen to achieve censoring proportions of $5\%$ and $25\%$.

My question is, how would I generate the survival time? Here is my approach:

Transformation Functions: I start by inverting the transformation model to simulate survival time. $$ H(t) = \log(2(e^{4t} - 1)), \\ H^{-1}(y) = \frac{1}{4} \log\left(\frac{e^y}{2} + 1\right). $$

Model Parameters: The model involves two components with mixing weights $$\pi_1 = \frac{1}{3}, \quad \pi_2 = \frac{2}{3}, $$ and parameter vectors $$ \beta_1 = (-3, -2)^{\top}, \quad \beta_2 = (1, 1)^{\top}$$

Covariate Generation: $\Sigma$ is the covariance matrix of the covariate variables, and therefore, the covariate is generated from a multivariate normal (MVN) $$\Sigma = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix}, \\ X \sim \text{MVN}(\mu = (0, 0)^{\top}, \Sigma)$$

Group Assignment and Survival Time Simulation: $$\text{group}_i = \begin{cases} 1 & \text{if } U_i < \pi_1 \\ 2 & \text{otherwise} \end{cases}$$, where $$ U_i \sim \text{Uniform}(0, 1)$$

$$X\beta = \begin{cases} \beta_1 \cdot X_i + \epsilon_i & \text{if group}_i = 1 \\ \beta_2 \cdot X_i + \epsilon_i & \text{if group}_i = 2 \end{cases}. $$ where $$\epsilon_i \sim \text{Extreme Value Distribution}(0, 1)$$

Edit:

Generating survival times to simulate Cox proportional hazards models

To generate event times from the proportional hazards model, we can use the inverse probability method (Bender et al., 2005): if $V$ is uniform on $(0, 1)$ and if $S(\cdot \,|\, \mathbf{x})$ is the conditional survival function derived from the proportional hazards model, i.e. $$ S(t \,|\, \mathbf{x}) = \exp \left( -H_0(t) \exp(\mathbf{x}^\prime \mathbf{\beta}) \vphantom{\Big(} \right) $$ then it is a fact that the random variable $$ T = S^{-1}(V \,|\, \mathbf{x}) = H_0^{-1} \left( - \frac{\log(V)}{\exp(\mathbf{x}^\prime \mathbf{\beta})} \right) $$ has survival function $S(\cdot \,|\, \mathbf{x})$. This result is known as ``the inverse probability integral transformation''. Therefore, to generate a survival time $T \sim S(\cdot \,|\, \mathbf{x})$ given the covariate vector, it suffices to draw $v$ from $V \sim \mathrm{U}(0, 1)$ and to make the inverse transformation $t = S^{-1}(v \,|\, \mathbf{x})$.

In this article titled "On the linear transformation model for censored data" this is written:

Recently Cheng, Wei & Ying (1995, 1997) developed a class of simple inference procedures for semiparametric linear transformation models with censored survival data. Specifically, let $T$, $C$ and $Z$ denote the failure time, the censoring time and the $p \times 1$ covariate vector. Let $h(\cdot)$ be an unknown increasing function. A linear transformation model is \begin{equation} h(T) = Z^T\beta + \epsilon, \quad (1) \end{equation} where $\epsilon$ has a completely known density $f$ and distribution function $F$, and $\beta$ is the vector of unknown regression coefficients. If $F(s) = 1 - \exp \{ - \exp(s) \}$, an extreme value distribution, (1) is the proportional hazards model (Cox, 1972). Note that, if $S_Z(t)$ is the survival function of $T$ for given $Z$, then (1) can be rewritten as \begin{equation} g\{S_Z(t)\} = h(t) - Z^T\beta, \quad (2) \end{equation} where $g^{-1}(\cdot) = 1 - F(\cdot)$.

So based on the above Cox model can be written as \begin{equation} \log[-\log \{S_Z(t)\}] = h(t) + Z^T\beta, \end{equation}

Now, if I were to find $S_{Z}^{-1}$, \begin{equation} S^{-1}(u) = h^{-1}(\log(-\log(u)) - Z^T\beta) \end{equation}

In my question notition, the survival time will be generated as, \begin{equation} T=S^{-1}(V|X) = H^{-1}(\log(-\log(V)) - X^T\beta) \end{equation} where $H^{-1}(y)= \frac{1}{4} \log\left(\frac{e^y}{2} + 1\right)$ and $V \sim U(0,1)$

Update Thank you, @Lukas Lohse, for your answer. Also, thank you, @EdM, for taking a look at the question. I found an article that used a similar model for a different reason, and R code for the simulation is available online (which is in the second simulation setup). You see in their simulation how survival time is set up. I am wondering how they came up with this setup and it seems more accurate.

library(survival)
library(mvtnorm)

H_inv <- function(y){
  1/4 * (log(exp(y)/2 + 1))
}

b1 <- c(-3, -2)
n <- 10^5
replications <- 1000

First:

coefficients_m1 <- matrix(NA, nrow = replications, ncol = 2)
for (i in 1:replications) {
  X <- rmvnorm(n, sigma = rbind(c(1, 0.5), c(0.5, 1)))
  lin_pred <- as.vector(X %*% b1)
  error <- log(-1*log(runif(n)))
  times <- H_inv(error - lin_pred)
  
  m1 <- coxph(Surv(time = times, event = rep(TRUE, n)) ~ X[, 1] + X[, 2])
  coefficients_m1[i, ] <- coefficients(m1)
}

mean_coefficients_m1 <- colMeans(coefficients_m1)
bias_m1 <- mean_coefficients_m1 - b1

list(mean_coefficients = mean_coefficients_m1, bias = bias_m1)

results

$mean_coefficients
[1] -2.913310 -1.942282

$bias
[1] 0.08669012 0.05771758

Second :

coefficients_m2 <- matrix(NA, nrow = replications, ncol = 2)
for (i in 1:replications) {
  X <- rmvnorm(n, sigma = rbind(c(1, 0.5), c(0.5, 1)))
  lin_pred <- as.vector(X %*% b1)
  temp = rexp(n)
  times = as.numeric(0.5*log(2*temp*exp(-lin_pred)+1.0))
  
  m2 <- coxph(Surv(time = times, event = rep(TRUE, n)) ~ X[, 1] + X[, 2])
  coefficients_m2[i, ] <- coefficients(m2)
}

mean_coefficients_m2 <- colMeans(coefficients_m2)
bias_m2 <- mean_coefficients_m2 - b1

list(mean_coefficients = mean_coefficients_m2, bias = bias_m2)

results

 $mean_coefficients

 [1] -2.99789 -1.99855

 $bias
 [1] 0.002110146 0.001449510
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  • $\begingroup$ To make sure there is no confusion, could you please edit the question to include a link to the "article that I have been reading" where you first mention it, in the first sentence? Sometimes there are things elsewhere in an article that help clarify a specific quote. $\endgroup$
    – EdM
    Feb 2 at 19:39
  • $\begingroup$ sciencedirect.com/science/article/abs/pii/S0304407622001841 $\endgroup$
    – ADAM
    Feb 2 at 21:44
  • 2
    $\begingroup$ $H(t)$ is defined differently in the two papers, probably accounting for the slight difference in bias in recovering the assumed coefficients. The second approach samples directly from the minimum extreme value distribution instead of using the inverse probability method. $\endgroup$
    – EdM
    Feb 4 at 9:36

1 Answer 1

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While I'm interested in seeing what @EdM has to say, this question seems like a straightforward yes, your approach works. I have coded up a version, limited to one group, in R and both the coefficients get recovered and we can find $H(t)$ in the baseline hazard.

library(survival)
library(mvtnorm)

H_inv <- function(y){
  1/4 * (log(exp(y)/2 + 1))
}
b1 <- c(-3, -2)
# large n for the simulation so we can check the coefficients
n <- 10^5
# simulate
X <- rmvnorm(n, sigma = rbind(c(1, 0.5), c(0.5, 1)))
cov(X)
colMeans(X)

lin_pred <- as.vector(X %*% b1)
error <- log(-1*log(runif(n)))
times <- H_inv(error - lin_pred)

m1 <- coxph(Surv(time = times, event = rep(TRUE, n)) ~ X[, 1] + X[, 2])
summary(m1) 

result:

            coef exp(coef)  se(coef)      z Pr(>|z|)    
X[, 1] -2.912249  0.054353  0.007981 -364.9   <2e-16 ***
X[, 2] -1.943424  0.143213  0.005986 -324.7   <2e-16 ***

Looking at the baseline hazard:

# H(t) is the (log) cumulative baseline-Hazard,
bhz <- basehaz(m1)
plot(bhz[, 2], log(bhz[, 1]))
curve(log(2*(exp(4*x) - 1)), add = T, col = 2)

result: enter image description here

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  • $\begingroup$ Nice demonstration that the proposed method can work (+1). I might have approached this by sampling directly from the extreme value distribution (see this page), but that's not what the OP requested. $\endgroup$
    – EdM
    Feb 3 at 15:03
  • $\begingroup$ @EdM I have made some update on the question section, please check them out. Thank you! $\endgroup$
    – ADAM
    Feb 3 at 21:53

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