# Generate survival data with predefined censoring rates for proportional hazards models `surv.simulate` simulates one or multiple (right-censored) survival datasets for proportional hazards models by simultaneously incorporating a baseline hazard function from three different survival distributions (exponential, Weibull and Gompertz), a random censoring time generated from a uniform distribution with an known/unknown upper limit, and a set of baseline covariates. When the upper limit of the uniform censoring time distribution is unknown, `surv.simulate` can be used separately to obtain the upper limit with a predefined censoring rate. ```r surv.simulate(L = 1, Z, beta, a, b, u1 = 0, u2, cen_rate, gen_data_from = c("exp", "weibul", "gomp"), only_u2 = FALSE, n.rep = 100, Trace = FALSE) ``` ## Arguments - `L`: the number of datasets to be generated. Default is `L = 1`. - `Z`: a list of `L` design matrices of dimension $n_\ell \times p$, where $n_\ell$ is the number of samples observed for the $\ell^{\text{th}}$ dataset and $p$ is the number of covariables. When `L = 1`, `Z` can be a matrix. - `beta`: the vector of the (true) coefficients values, with a length of $p$ (the number of covariates). - `a`: scale parameter, which should be non-negative. See Details for the form of the cumulative hazard that can be used. - `b`: shape/location parameter, which should be non-negative. It is not used when `gen_data_from =`exp . See Details for the form of the cumulative hazard that can be used. - `u1`: a known non-negative lower limit of the uniform distribution for generating random censoring time. Default is `u1 = 0`. If `cen_rate` is not equal to 0, then `u1` does not need to be defined. - `u2`: an non-negative upper limit of the uniform random censoring time distribution. The upper limit can be unknown (`u2 = NULL`, the default), or predefined. When this argument is assumed to be unknown, `u2 = NULL`, it is calculated by the algorithm within `surv.simulate()`. However, if the argument `u2` is known, the censoring rate cannot be predefined (meaning there is no control over it) and is calculated based on the generated dataset. See Details and References . - `cen_rate`: a value representing the proportion of observations in the simulated survival data that are censored. The range of this argument is from 0 to 1. When the upper limit is known, `cen_rate` can nor be predefined. If there is no censoring (`cen_rate = 0`), the lower (`u1`) and upper (`u2`) limits of the uniform distribution do not need to be specified. - `gen_data_from`: a description of the distribution from which the time to event is generated. This is a character string and can be `exponential` (exp ), `Weibull` (weibul ), or `Gompertz` (gomp ). Can be abbreviated. By default, the `exponential` distribution is used. - `only_u2`: logical flag for calculating only the upper limit of the uniform censoring time distribution. If `only_u2 = TRUE`, the dataset(s) are not generated. If `only_u2 = TRUE`, the arguments `Z` and `u2` do not need to be specified, and `cen_rate` should not be set to $0$. Default is `only_u2 = FALSE`. - `n.rep`: a scalar specifying the number of iterations. This argument is exclusively used in the case of the `Gompertz` distribution. Default is 100. - `Trace`: logical flag indicating whether the output of the desired `u2` and the censoring proportion for different datasets should be produced for each iteration. It works `gen_data_from =`gomp . ## Details `surv.simulate` function generates $L$ simulated right-censored survival datasets from exponential, Weibull, or Gompertz distributions, incorporating the covariates, `Z`, distributed according to a multivariate `normal` distribution, with censoring time generated from a uniform distribution `Uniform(u1, u2)`, where `u1` is known but `u2` can be either known or unknown. `surv.simulate()` can also be used to calculate the unknown upper limit of the uniform distribution, `u2`, with a predefined censoring rate. To do this, set `u2 = NULL` and `only_u2 = TRUE`. In this case, the datasets are not generated; only `u2` is. `surv.simulate()` uses a root-finding algorithm to select the censoring parameter that achieves predefined censoring rates in the simulated survival data. When `gen_data_from =`exp : * the cumulative baseline hazard function is considered as $\Lambda_0=a t$, * the event time for the $\ell^{\text{th}}$ dataset, $T_\ell$, is computed by $ - log(u) \ exp(- Z_\ell \boldsymbol{\beta}) / a$, where $u$ follows a standard uniform distribution; For `gen_data_from =`weibul : * the cumulative hazard function is as $\Lambda_0=a t ^ b$, * the event time is computed by $T_\ell= (- log(u) \ exp(- Z_\ell \boldsymbol{\beta}) / a)^{1/b}$, where $u$ follows a standard uniform distribution; For `gen_data_from =`gomp : * the cumulative hazard function is as $\Lambda_0=a (exp(b t) - 1) / b$, * the event time is computed by $T_\ell= \log(1- log(u) \ exp(- Z_\ell \boldsymbol{\beta}) b / a) / b$, where $u$ follows a standard uniform distribution; Finally the survival time is obtained by $\tilde{T}_\ell=\min\{T_\ell , C_\ell \}$. The function will be updated for `gen_data_from =`gomp . ## Returns `surv.simulate` returns a list containing the following components: - **D**: a list of $L$ data frames, with dimension $n_\ell \times (p+2)$. The first and second columns, named `time` and `status`, contain the simulated survival time and the censoring indicator, respectively, where $0$ means censored and $1$ means uncensored; - **censor_propor**: the vector of censoring proportions in the simulated datasets `D`, containing $L$ elements; - **u1**: the lower limit of the uniform distribution used to generate random censoring times with a predefined censoring rate. Sometimes this output is less than the value entered by the user, as it is adjusted to achieve the desired amount of censoring rate; - **u2**: the upper limit of the uniform distribution used to generate random censoring times. If `u2 = NULL`, this output will be the estimated upper limit necessary to achieve the desired censoring rate across the $L$ datasets. ## References Pazira H., Massa E., Weijers J.A.M., Coolen A.C.C. and Jonker M.A. (2025b). **Bayesian Federated Inference for Survival Models**, **Journal of Applied Statistics (Accepted)**. ## Author(s) Hassan Pazira Maintainer: Hassan Pazira hassan.pazira@radboudumc.nl ## See Also `MAP.estimation` ## Examples ```r # Setting a seed for reproducibility set.seed(1123) #------------------------- # Simulating Survival data #------------------------- N <- c(7, 10, 13) # the sample sizes of 3 datasets beta <- 1:4 p <- length(beta) L <- 3 # Define a function to generate multivariate normal samples mvrnorm_new <- function(n, mu, Sigma) { pp <- length(mu) e <- matrix(rnorm(n * pp), nrow = n) return(crossprod(t(e), chol(Sigma)) + matrix(mu, n, pp, byrow = TRUE)) } Z <- list() for (z in seq_len(L)) { Z[[z]] <- mvrnorm_new(n = N[z], mu = rep(0, p), Sigma = diag(rep(1, p),p)) colnames(Z[[z]]) <- paste0("Z_",seq_len(ncol(Z[[z]]))) } # One simulated dataset from exponential distribution with no censoring: surv_data <- surv.simulate(Z = Z[[1]], beta = beta, a = exp(-.9), cen_rate = 0, gen_data_from = "exp") surv_data surv_data$D[[1]][,1:2] # The simulated survival data # Calculate only 'u2' with a predefined censoring rate of 0.4: u2_new <- surv.simulate(Z = Z[1:2], beta = beta, a = exp(-.9), b = exp(1.8), u1 = 0.1, only_u2 = TRUE, cen_rate = 0.4, gen_data_from = "weibul")$u2 u2_new # Two simulated datasets with a known 'u2': # Using 'u2_new' to help control over censoring rate (was chosen 0.4) surv.simulate(Z = Z[1:2], beta = beta, a = exp(-.9), b = exp(1.8), u1 = 0.05, u2 = u2_new, gen_data_from = "weibul") # Three simulated datasets from 'weibul' with an unknown 'u2': surv.simulate(Z = Z, beta = beta, a = exp(-1), b = exp(1), u1 = 0.01, cen_rate = 0.3, gen_data_from = "weibul") # Two simulated datasets from 'gomp' with unknown 'u2' and censoring rate of 0.3: surv.simulate(Z = Z[2:3], beta = beta, a = exp(1), b = exp(2), u1 = 0.1, cen_rate = 0.3, gen_data_from = "gomp", Trace = TRUE) ```

surv.simulate function