Last updated: 2023-06-16
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Conclusion:
Extremely high censoring rate probably won’t affect effect size estimate as long as there are moderate number of events.
If we draw a contigency table (\(2\times 2\)). \(\delta = 0,1\) indicates whether the outcome is observed, and \(x=0,1\) indicates genotypes. As long as there are a couple observations in each cell, the effect size estimate is fine. Otherwise, coefficient tends to be infinite.
library(survival)# 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)
}set.seed(1)
# simulate 2 variables
n = 100
X = cbind(rbinom(n, size = 1, prob = 0.3), rbinom(n, size = 1, prob = 0.1))
# 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)# rownames are unique y[,2] values, delta. delta = 1 indicates event happened.
# colnames are X values, genotype.
table(dat$y[,2], dat$X[,1])
0 1
0 53 26
1 15 6surT <- Surv(dat$y[,1], dat$y[,2])
fit1 <- coxph(surT ~ dat$X[,1])
summary(fit1)Call:
coxph(formula = surT ~ dat$X[, 1])
n= 100, number of events= 21
coef exp(coef) se(coef) z Pr(>|z|)
dat$X[, 1] 2.035 7.656 0.587 3.468 0.000525 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
dat$X[, 1] 7.656 0.1306 2.423 24.19
Concordance= 0.686 (se = 0.058 )
Likelihood ratio test= 10.26 on 1 df, p=0.001
Wald test = 12.02 on 1 df, p=5e-04
Score (logrank) test = 15.51 on 1 df, p=8e-05Select sub-samples where individuals who have event have x = 1. And individuals with x = 0 are all censored.
# let's just select sub-samples where individuals who have event have x = 1.
# And individuals with x = 0 are all censored.
indx = which(dat$y[,2] == dat$X[,1])
y = dat$y[indx, ]
x = dat$X[indx, 1]
table(y[,2], x) x
0 1
0 53 0
1 0 6surT <- Surv(y[,1], y[,2])
fit2 <- coxph(surT ~ x)Warning in coxph.fit(X, Y, istrat, offset, init, control, weights = weights, :
Loglik converged before variable 1 ; coefficient may be infinite.summary(fit2)Call:
coxph(formula = surT ~ x)
n= 59, number of events= 6
coef exp(coef) se(coef) z Pr(>|z|)
x 2.410e+01 2.936e+10 2.077e+04 0.001 0.999
exp(coef) exp(-coef) lower .95 upper .95
x 2.936e+10 3.406e-11 0 Inf
Concordance= 0.964 (se = 0.015 )
Likelihood ratio test= 29.39 on 1 df, p=6e-08
Wald test = 0 on 1 df, p=1
Score (logrank) test = 60.57 on 1 df, p=7e-15In fit3, we add 2 samples to data in fit2. Doesn’t help in this case.
In fit4, we add 4 samples to data in fit2. It helped a lot.
indx2 = which(dat$y[,2] != dat$X[,1])[c(1,4)]
y2 = rbind(y, dat$y[indx2, ])
x2 = c(x, dat$X[,1][indx2])
table(y2[,2], x2) x2
0 1
0 53 1
1 1 6surT <- Surv(y2[,1], y2[,2])
fit3 <- coxph(surT ~ x2)Warning in coxph.fit(X, Y, istrat, offset, init, control, weights = weights, :
Loglik converged before variable 1 ; coefficient may be infinite.summary(fit3)Call:
coxph(formula = surT ~ x2)
n= 61, number of events= 7
coef exp(coef) se(coef) z Pr(>|z|)
x2 2.335e+01 1.379e+10 1.443e+04 0.002 0.999
exp(coef) exp(-coef) lower .95 upper .95
x2 1.379e+10 7.25e-11 0 Inf
Concordance= 0.919 (se = 0.036 )
Likelihood ratio test= 28.75 on 1 df, p=8e-08
Wald test = 0 on 1 df, p=1
Score (logrank) test = 56.01 on 1 df, p=7e-14indx2 = which(dat$y[,2] != dat$X[,1])[c(1:4)]
y2 = rbind(y, dat$y[indx2, ])
x2 = c(x, dat$X[,1][indx2])
table(y2[,2], x2) x2
0 1
0 53 3
1 1 6surT <- Surv(y2[,1], y2[,2])
fit4 <- coxph(surT ~ x2)
summary(fit4)Call:
coxph(formula = surT ~ x2)
n= 63, number of events= 7
coef exp(coef) se(coef) z Pr(>|z|)
x2 3.996 54.381 1.099 3.635 0.000278 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
x2 54.38 0.01839 6.304 469.1
Concordance= 0.904 (se = 0.04 )
Likelihood ratio test= 21.26 on 1 df, p=4e-06
Wald test = 13.21 on 1 df, p=3e-04
Score (logrank) test = 38.8 on 1 df, p=5e-10set.seed(1)
# simulate 2 variables
n = 1000
X = cbind(rbinom(n, size = 1, prob = 0.3), rbinom(n, size = 1, prob = 0.1))
# the first element of b is for intercept
b = c(1, 1, 0)
censor_lvl = 0.99
dat <- sim_surv_exp(b, X, censor_lvl)In this case,
table(dat$y[,2], dat$X[,1])
0 1
0 690 301
1 6 3surT <- Surv(dat$y[,1], dat$y[,2])
fit1 <- coxph(surT ~ dat$X[,1])
summary(fit1)Call:
coxph(formula = surT ~ dat$X[, 1])
n= 1000, number of events= 9
coef exp(coef) se(coef) z Pr(>|z|)
dat$X[, 1] 1.4241 4.1541 0.7576 1.88 0.0601 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
dat$X[, 1] 4.154 0.2407 0.9411 18.34
Concordance= 0.699 (se = 0.09 )
Likelihood ratio test= 3.03 on 1 df, p=0.08
Wald test = 3.53 on 1 df, p=0.06
Score (logrank) test = 4.09 on 1 df, p=0.04
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 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_1.0.6
[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