Last updated: 2023-03-28

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1. Model:

\[ \begin{split} b&\sim N(0, \sigma^2)\\ h(t)&=h_0(t)\exp\{bx\}\\ \end{split} \] And we assume \(h_0(t) =h_0\), constant baseline hazard and independent censoring. \(y_i=\min(t_i,c_i)\), and \(\delta_i\) indicates censoring status. \(\delta_i=1\): the event time is observed and \(\delta_i=0\) indicates censoring.

library(survival)
source("./code/VI_exponential.R")

2. data simulation

\[ \begin{split} S(t)&=\exp\{-h(t)*t\}\\ F(t)&=1-S(t)\sim U[0,1]\\ \end{split} \] Using this relationship, we can simulate \(t_i\) by: \[ \begin{split} t_i=\log(1-u_i)/(-h_0\exp\{bx_i\}) \end{split} \] Check if the data simulation makes sense

set.seed(1)
n = 5000
censoring_lvl = 0
s2 = 1 #prior variance of b
h0 = 2
b = rnorm(1, mean = 0, sd = s2)
print(b)
[1] -0.6264538
seeds  = c(1:100)
bhat = rep(NA, 100)

for (seed in seeds){
  x = rnorm(n, 1)
  y <- log(1 - runif(n)) / (-h0*exp(x*b))
  # Status variable delta. delta = 1, outcome observed. delta = 0, censored. 
  d = rbinom(n, size = 1, prob = 1-censoring_lvl)
  y[!d] <- y[!d] * runif(sum(!d))   
  y.surv <- Surv(y, d)
  cox1 <- coxph(y.surv ~ x)
  bhat[seed] = coef(cox1)[1]
}
hist(bhat)
abline(v = mean(bhat), col = "red")
abline(v = b, col = "blue")

Version Author Date
bcdf196 yunqiyang0215 2023-03-27

3. Run variational approximation

# change parameter init vals
# the first param is m, the second is v2.
lower = c(-10, 1e-6)
upper = c(10, 10)
res1 <- update_q(x, y, d, h0 = 2, m = 1, v2 = 1, s2 = 10, maxiter = 50, tol = 0.1, lower, upper)
tail(res1)
           elbo       h0          m    v2        s2
[45,] -4558.438 2.034419 -0.6260834 1e-06 0.3919814
[46,] -4558.438 2.034419 -0.6260834 1e-06 0.3919814
[47,] -4558.438 2.034419 -0.6260834 1e-06 0.3919814
[48,] -4558.438 2.034419 -0.6260834 1e-06 0.3919814
[49,] -4558.438 2.034419 -0.6260834 1e-06 0.3919814
[50,] -4558.438 2.034419 -0.6260834 1e-06 0.3919814
# change parameter init vals
lower = c(-50, 1e-3)
upper = c(50, 50)
res2 <- update_q(x, y, d, h0 = 2, m = 1, v2 = 1, s2 = 10, maxiter = 50, tol = 0.1, lower, upper)
tail(res2)
           elbo       h0          m    v2        s2
[45,] -4559.926 2.034362 -0.6270516 0.001 0.3941937
[46,] -4559.926 2.034362 -0.6270516 0.001 0.3941937
[47,] -4559.926 2.034362 -0.6270516 0.001 0.3941937
[48,] -4559.926 2.034362 -0.6270516 0.001 0.3941937
[49,] -4559.926 2.034362 -0.6270516 0.001 0.3941937
[50,] -4559.926 2.034362 -0.6270516 0.001 0.3941937
# change parameter init vals
lower = c(-50, 1)
upper = c(50, 50)
res3 <- update_q(x, y, d, h0 = 10, m = 1, v2 = 1, s2 = 50, maxiter = 50, tol = 0.1, lower, upper)
tail(res3)
           elbo       h0         m v2       s2
[45,] -12776.24 1.124916 -1.687747  1 3.848488
[46,] -12776.24 1.124916 -1.687747  1 3.848488
[47,] -12776.24 1.124916 -1.687747  1 3.848488
[48,] -12776.24 1.124916 -1.687747  1 3.848488
[49,] -12776.24 1.124916 -1.687747  1 3.848488
[50,] -12776.24 1.124916 -1.687747  1 3.848488
# change parameter init vals
lower = c(-50, 1e-10)
upper = c(50, 50)
res4 <- update_q(x[1:50], y[1:50], d, h0 = 10, m = 1, v2 = 1, s2 = 50, maxiter = 50, tol = 0.1, lower, upper)
tail(res4)
           elbo       h0         m    v2       s2
[45,] -478714.8 562.3696 -2.115074 1e-10 4.473538
[46,] -478714.8 562.3696 -2.115074 1e-10 4.473538
[47,] -478714.8 562.3696 -2.115074 1e-10 4.473538
[48,] -478714.8 562.3696 -2.115074 1e-10 4.473538
[49,] -478714.8 562.3696 -2.115074 1e-10 4.473538
[50,] -478714.8 562.3696 -2.115074 1e-10 4.473538
grid = seq(1e-7, 5, by = 0.001)
res = rep(NA, length(grid))

for (i in 1:length(grid)){
  res[i] = partial_dv_v2(grid[i], x[1:100], y[1:100], s2 = 1, m = 0, h0 = 1)
}
# all less than 0. 
plot(grid, res, ylab = "value of partial derivative of v2")


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 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