Last updated: 2023-09-15
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Knit directory: survival-susie/
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Assess difference in Bayes factor computed by numerical integration vs. ABF. The example of using numerical integration to compute BF is available at: https://yunqiyang0215.github.io/survival-susie/numerical_integration.html
We assess three scenarios under relatively large sample size, \(n= 10,000\).
Extremely high censoring rate, censoring rate = 0.99
Low allele frequency, MAF = 0.001
Both.
Result shows that low allele frequency makes the difference between BF and ABF larger.
# Function to simulate survival time under exponential model. That is,
# assuming survival time is exponentially distributed.
# lambda(t) = lambda*exp(b0 + Xb). S(t) = exp(-lambda*t),
# F(t) = 1- S(t) \sim Unif(0,1). Therefore, t = log(1-runif(0,1))/-exp(b0+Xb).
# For censored objects, we simulate the censoring time by rescale the actual survival time.
# @param b: vector of length (p+1) for true effect size, include intercept.
# @param X: variable matrix of size n by p.
# @param censor_lvl: a constant from [0,1], indicating the censoring level in the data.
# @return dat: a dataframe that contains `y`, `x` and `status`.
# `status`: censoring status: 0 = censored, 1 = event observed. See Surv() in library(survival)
sim_surv_exp <- function(b, X, censor_lvl){
n = nrow(X)
p = ncol(X)
dat = list()
status <- ifelse(runif(n) > censor_lvl, 1, 0)
lambda <- exp(cbind(rep(1,n), X) %*% b)
surT <- log(1 - runif(n)) /(-lambda)
# rescale censored subject to get observed time
surT[status == 0] = surT[status == 0] * runif(sum(status == 0))
y = cbind(surT, status)
colnames(y) = c("time", "status")
colnames(X) <- unlist(lapply(1:p, function(i) paste0("x", i)))
dat[["X"]] = X
dat[["y"]] = y
return(dat)
}# @param b: the effect size
# @param surT: a Surv() object, containing time and status
# @param x: covariate
get_partial_lik <- function(b, survT, x){
ll.partial = logLik(coxph(survT ~ x, init = c(b), control=list('iter.max'= 0, timefix=FALSE)))
max.ll.partial = logLik(coxph(survT ~ x))
lik.partial.normed = exp(ll.partial - max.ll.partial) # partial likelihood / max(partial likelihood)
return(as.numeric(lik.partial.normed))
}
integrand <- function(b, survT, x, prior_sd){
val = get_partial_lik(b, survT, x) * dnorm(b, mean = 0, sd = prior_sd)
return(val)
}library(survival)
library(cubature)
source("./code/surv_susie_helper.R")seeds = c(1:5)
result = matrix(NA, ncol = 3, nrow = length(seeds))
colnames(result) = c("diff", "integral", "error")
for (seed in seeds){
set.seed(seed)
# simulate 1 variable
n = 1e4
X = cbind(rbinom(n, size = 2, prob = 0.2))
# the first element of b is for intercept
b = c(1, 1)
censor_lvl = 0.99
dat <- sim_surv_exp(b, X, censor_lvl)
survT <- Surv(dat$y[,1], dat$y[,2])
prior_sd = 1
res = cubintegrate(f = integrand, survT = survT, x = dat$X[,1], prior_sd = 1, lower = -100, upper = 100, method = "hcubature")
max.ll.partial = logLik(coxph(survT ~ dat$X[,1]))
prob.h1 = res$integral*exp(max.ll.partial)
prob.h0 = get_partial_lik(b = 0, survT, dat$X[,1])
lbf.numerical = log(res$integral) - log(prob.h0)
abf <-surv_uni_fun(x = dat$X[,1], y = survT, o = rep(0, n), prior_variance = prior_sd^2)$lbf
result[seed, 1] = lbf.numerical - abf
result[seed, 2] = res$integral
result[seed, 3] = res$error
}seeds = c(1:5)
result = matrix(NA, ncol = 3, nrow = length(seeds))
colnames(result) = c("diff", "integral", "error")
for (seed in seeds){
set.seed(seed)
# simulate 1 variable
n = 1e4
X = cbind(rbinom(n, size = 2, prob = 0.001))
# the first element of b is for intercept
b = c(1, 1)
censor_lvl = 0.6
dat <- sim_surv_exp(b, X, censor_lvl)
survT <- Surv(dat$y[,1], dat$y[,2])
prior_sd = 1
res = cubintegrate(f = integrand, survT = survT, x = dat$X[,1], prior_sd = 1, lower = -100, upper = 100, method = "hcubature")
max.ll.partial = logLik(coxph(survT ~ dat$X[,1]))
prob.h1 = res$integral*exp(max.ll.partial)
prob.h0 = get_partial_lik(b = 0, survT, dat$X[,1])
lbf.numerical = log(res$integral) - log(prob.h0)
abf <-surv_uni_fun(x = dat$X[,1], y = survT, o = rep(0, n), prior_variance = prior_sd^2)$lbf
result[seed, 1] = lbf.numerical - abf
result[seed, 2] = res$integral
result[seed, 3] = res$error
}result diff integral error
[1,] 0.10664739 0.3084903 1.524437e-06
[2,] 0.05312745 0.1639938 6.036238e-08
[3,] 0.08580469 0.1176151 6.913447e-08
[4,] 0.12090607 0.2083243 1.052610e-06
[5,] 0.05529886 0.2403162 3.301337e-09seeds = c(1:3)
result = matrix(NA, ncol = 3, nrow = 3)
colnames(result) = c("diff", "integral", "error")
for (seed in seeds){
set.seed(seed)
# simulate 1 variable
n = 1e4
X = cbind(rbinom(n, size = 2, prob = 0.001))
# the first element of b is for intercept
b = c(1, 1)
censor_lvl = 0.99
dat <- sim_surv_exp(b, X, censor_lvl)
survT <- Surv(dat$y[,1], dat$y[,2])
prior_sd = 1
res = cubintegrate(f = integrand, survT = survT, x = dat$X[,1], prior_sd = 1, lower = -100, upper = 100, method = "hcubature")
max.ll.partial = logLik(coxph(survT ~ dat$X[,1]))
prob.h1 = res$integral*exp(max.ll.partial)
prob.h0 = get_partial_lik(b = 0, survT, dat$X[,1])
lbf.numerical = log(res$integral) - log(prob.h0)
abf <-surv_uni_fun(x = dat$X[,1], y = survT, o = rep(0, n), prior_variance = prior_sd^2)$lbf
result[seed, 1] = lbf.numerical - abf
result[seed, 2] = res$integral
result[seed, 3] = res$error
}result diff integral error
[1,] -0.03289383 0.9676211 3.969705e-08
[2,] -0.07537664 0.9273741 4.480898e-08
[3,] -0.08586509 0.9176952 4.704538e-08
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] cubature_2.1.0 survival_3.2-11 workflowr_1.6.2
loaded via a namespace (and not attached):
[1] Rcpp_1.0.8.3 pillar_1.6.4 compiler_4.1.1 bslib_0.4.1
[5] later_1.3.0 jquerylib_0.1.4 git2r_0.28.0 tools_4.1.1
[9] digest_0.6.28 lattice_0.20-44 jsonlite_1.7.2 evaluate_0.14
[13] lifecycle_1.0.3 tibble_3.1.5 pkgconfig_2.0.3 rlang_1.1.1
[17] Matrix_1.5-3 cli_3.6.1 rstudioapi_0.13 yaml_2.2.1
[21] xfun_0.27 fastmap_1.1.0 stringr_1.4.0 knitr_1.36
[25] fs_1.5.0 vctrs_0.6.3 sass_0.4.4 grid_4.1.1
[29] rprojroot_2.0.2 glue_1.4.2 R6_2.5.1 fansi_0.5.0
[33] rmarkdown_2.11 magrittr_2.0.1 whisker_0.4 splines_4.1.1
[37] promises_1.2.0.1 ellipsis_0.3.2 htmltools_0.5.5 httpuv_1.6.3
[41] utf8_1.2.2 stringi_1.7.5 cachem_1.0.6 crayon_1.4.1