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.
What if the SNP is very rare and censoring rate is extremely high?
As long as the total sample size is large, even there is only 1 event, the fit is not bad.
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=0.04In this extreme case, the fit is not bad.
set.seed(1)
# simulate 2 variables
n = 10000
X = cbind(rbinom(n, size = 2, prob = 5e-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)table(dat$y[,2], dat$X[,1])
0 1
0 9810 95
1 94 1surT <- Surv(dat$y[,1], dat$y[,2])
fit1 <- coxph(surT ~ dat$X[,1])
summary(fit1)Call:
coxph(formula = surT ~ dat$X[, 1])
n= 10000, number of events= 95
coef exp(coef) se(coef) z Pr(>|z|)
dat$X[, 1] 1.380 3.974 1.010 1.366 0.172
exp(coef) exp(-coef) lower .95 upper .95
dat$X[, 1] 3.974 0.2516 0.5492 28.76
Concordance= 0.511 (se = 0.013 )
Likelihood ratio test= 1.25 on 1 df, p=0.3
Wald test = 1.87 on 1 df, p=0.2
Score (logrank) test = 2.18 on 1 df, p=0.1
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