sim_survival
Yunqi Yang
2/1/2023
Last updated: 2023-02-23
Checks: 7 0
Knit directory: survival-susie/
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Description:
Simulate time-to-event data based on exponential model. And fit proportional hazard model to data. Let’s first simulate data without censoring.
The exponential regression (AFT: accelerated failure time):
It assumes survival time \(T\) follows exponential distribution. Under this assumption, the hazard is constant over time. \[ \begin{split} T&\sim \exp(\mu)\\ f(t)&=\frac{1}{\mu}\exp\{-t/\mu\}\\ \lambda(t)&=1/\mu \end{split} \]
Remember in exponential distribution, \(E(T)=\mu\). So we model the \(\log\mu\) part by linear combinations of variables. \[ \begin{split} \log(T_i) &= \log(E(T_i)) + \epsilon_i\\ &=\beta_0 + X_i^T\beta+\epsilon_i \end{split} \]
Simulate under 4 simple scenarios, 50 variables are available.
The null model, time \(T_i\) is simulated from the model that only has intercept. High correlation among all predictors.
Single effect model without correlation. Time \(T_i\) depends on \(x_1\) only, and no correlation between \(x_1\) and other variables.
Single effect model with correlation. Time \(T_i\) depends on \(x_1\) only, and high correlation between \(x_1\) and other variables.
\(\log T_i = \beta_0+\beta_1x_{i1} + \beta_2x_{i2}+\epsilon_i\), and high correlation among all variables.
library(mvtnorm)
library(survival)# Function to construct correlation matrix among predictors.
# Diagonal elements are all 1s, the off-diagonal elements = corr
# @param p: number of predictors
# @param corr: correlation
cov_simple_het = function(p, corr){
for(i in 1:length(corr)){
cov = matrix(corr, nrow=p,ncol=p)
diag(cov) <- 1
}
return(cov)
}
# Here we use parametric model to simulate data with survival time,
# assuming survival time is exponentially distributed.
# We first simulate the mean of exponential from linear combinations
# of variables, and then simulate survival time.
# T\sim 1/u*exp(-t/u), and the true model is:
# log(T) = \mu + e = b0 + Xb + e
# @param b: vector of length (p+1) for true effect size, include intercept.
# @param X: variable matrix of size n by p.
# @param status: censoring status. 1 = censored, 2 = event observed.
sim_dat <- function(b, X){
n = nrow(X)
p = ncol(X)
mu <- exp(cbind(rep(1,n), X) %*% b)
surT <- rexp(n, rate = 1/mu)
dat <- data.frame(cbind(surT, X))
x.name <- unlist(lapply(1:p, function(i) paste0("x", i)))
names(dat) = c("surT", x.name)
dat$status <- rep(2, n)
return(dat)
}Scenario 1: null model, X no correlation
set.seed(1)
n <- 100
p <- 50
b <- c(1, rep(0, 50))
sigma <- cov_simple_het(p, corr = 0)
X<- rmvnorm(n, sigma = sigma)
dat1 <- sim_dat(b, X)hist(dat1$surT, breaks = 20)
Scenario 2: null model, X correlation = 0.9
set.seed(1)
n <- 100
p <- 50
b <- c(1, rep(0, 50))
sigma <- cov_simple_het(p, corr = 0.9)
X<- rmvnorm(n, sigma = sigma)
dat2 <- sim_dat(b, X)hist(dat2$surT, breaks = 20)
Scenario 3: single effect model with independent predictors
set.seed(1)
n <- 100
p <- 50
b <- c(1, 3, rep(0, p-1))
sigma <- cov_simple_het(p, corr = 0)
X<- rmvnorm(n, sigma = sigma)
dat3 <- sim_dat(b, X)hist(dat2$surT, breaks = 20)
Scenario 4: single effect model with highly correlated predictors
set.seed(1)
n <- 100
p <- 50
b <- c(1, 3, rep(0, p-1))
sigma <- cov_simple_het(p, corr = 0.9)
X<- rmvnorm(n, sigma = sigma)
dat4 <- sim_dat(b, X)hist(dat3$surT, breaks = 20)
Scenario 5: two effects model with highly correlated variables
set.seed(1)
n <- 100
p <- 50
b <- c(1, 3, 1.5, rep(0, p-2))
sigma <- cov_simple_het(p, corr = 0.9)
X<- rmvnorm(n, sigma = sigma)
dat5 <- sim_dat(b, X)hist(dat5$surT, breaks = 20)
sim_dat_simple <- list(dat1, dat2, dat3, dat4, dat5)
saveRDS(sim_dat_simple, "./data/sim_dat_simple.rds")
sessionInfo()R version 4.1.1 (2021-08-10)
Platform: x86_64-apple-darwin20.6.0 (64-bit)
Running under: macOS Monterey 12.0.1
Matrix products: default
BLAS: /usr/local/Cellar/openblas/0.3.18/lib/libopenblasp-r0.3.18.dylib
LAPACK: /usr/local/Cellar/r/4.1.1_1/lib/R/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] survival_3.2-11 mvtnorm_1.1-3 workflowr_1.6.2
loaded via a namespace (and not attached):
[1] Rcpp_1.0.8.3 highr_0.9 pillar_1.6.4 compiler_4.1.1
[5] bslib_0.4.1 later_1.3.0 jquerylib_0.1.4 git2r_0.28.0
[9] tools_4.1.1 digest_0.6.28 lattice_0.20-44 jsonlite_1.7.2
[13] evaluate_0.14 lifecycle_1.0.1 tibble_3.1.5 pkgconfig_2.0.3
[17] rlang_1.0.6 Matrix_1.5-3 cli_3.1.0 rstudioapi_0.13
[21] yaml_2.2.1 xfun_0.27 fastmap_1.1.0 stringr_1.4.0
[25] knitr_1.36 fs_1.5.0 vctrs_0.3.8 sass_0.4.4
[29] grid_4.1.1 rprojroot_2.0.2 glue_1.4.2 R6_2.5.1
[33] fansi_0.5.0 rmarkdown_2.11 magrittr_2.0.1 whisker_0.4
[37] splines_4.1.1 promises_1.2.0.1 ellipsis_0.3.2 htmltools_0.5.2
[41] httpuv_1.6.3 utf8_1.2.2 stringi_1.7.5 cachem_1.0.6
[45] crayon_1.4.1