Last updated: 2023-06-28
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Knit directory: survival-susie/
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The standard BF for comparing the hypothesis of having one predictor vs. the null model: \[ \frac{P(D|H_1)}{P(D|H_0)}=\frac{\int\int p(y,\delta|x,b, h_0)p(b)dbdh_0}{\int p(y,\delta|x,b=0,h_0)dh_0} \]
Based on the paper, “A Bayesian Justification of Cox’s Partial Likelihood” by Sinha, D. and others, we can safely use the partial likelihood to compute Bayes factor. That is,
\[ \begin{split} \frac{P(D|H_1)}{P(D|H_0)}&=\frac{\int Lp(b)p(b)db}{Lp(b=0)}\\ Lp(b)&=\prod_{i=1}^n\{\frac{\exp(x_ib)}{\sum_{j\in R(y_i)}\exp(x_jb)} \}^{\delta_i} \end{split} \]
# 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)
}library(survival)
library(cubature)
source("./code/surv_susie_helper.R")set.seed(1)
# simulate 2 variables
n = 100
X = cbind(rbinom(n, size = 2, prob = 0.3), rbinom(n, size = 1, prob = 0.3))
# the first element of b is for intercept
b = c(1, 1, 0)
censor_lvl = 0.7
dat <- sim_surv_exp(b, X, censor_lvl)
survT <- Surv(dat$y[,1], dat$y[,2])# @param b: the effect size
# @param surT: a Surv() object, containing time and status
# @param x: covariate
get_partial_lik <- function(b, survT, x){
partial.loglik = logLik(coxph(survT ~ x, init = c(b), control=list('iter.max'= 0, timefix=FALSE)))
partial.lik = exp(partial.loglik) # handling ties...
return(as.numeric(partial.lik))
}
integrand <- function(b, survT, x, prior_sd){
val = get_partial_lik(b, survT, x)*dnorm(b, mean = 0, sd = prior_sd)
return(val)
}prior_sd = 1
res = cubintegrate(f = integrand, survT = survT, x = dat$X[,1], prior_sd = 1, lower = -Inf, upper = Inf, method = "hcubature")
res$integral
[1] 3.867775e-27
$error
[1] 6.23873e-27
$neval
[1] 15
$returnCode
[1] 0prob.h0 = get_partial_lik(b = 0, survT, dat$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)$lbflbf.numerical[1] 6.984264abf[1] 7.086837
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_1.0.1 tibble_3.1.5 pkgconfig_2.0.3 rlang_1.1.1
[17] Matrix_1.5-3 cli_3.1.0 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_0.3.8 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