standardize_effect_coxph

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Last updated: 2023-05-17

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---

Description:

check if standardizing will change the coefficient of variables in coxph model. Let’s still use the exponential model from https://yunqiyang0215.github.io/survival-susie/sim_survival.html

The exponential model models the mean of exponential as: \[ \begin{split} E(T_i) &= \mu_i\\ \log\mu_i &= \beta_0 + X\beta \end{split} \] The hazard in exponential model is \(\lambda_i = 1/\mu_i=\frac{1}{\exp\{\beta_0+X\beta\}}=\lambda_0\exp\{X\alpha\}\). Therefore, \(\alpha\) is the output from coxph model and \(\alpha=-\beta\).

Conclusion:

1. Operations only on x:

1. Mean center won’t change the coefficient in coxph model.

2. The coefficient will be multiplied by \(s\) if the variable is divided by \(s\).

2. Standardize y: the beta estimate is the same as the original data without standardize y.

library(mvtnorm)
library(survival)

# Here we use parametric model to simulate data with survival time,
# assuming survival time is exponentially distributed. 
# We first simulate the mean of exponential from linear combinations
# of variables, and then simulate survival time. 
# T\sim 1/u*exp(-t/u), and the true model is:
# log(T) = \mu + e = b0 + Xb + e
# @param b: vector of length (p+1) for true effect size, include intercept.
# @param X: variable matrix of size n by p. 
# @param status: censoring status. 1 = censored, 2 = event observed. 
sim_dat <- function(b, X){
  n = nrow(X)
  p = ncol(X)
  mu <- exp(cbind(rep(1,n), X) %*% b)
  surT <- rexp(n, rate = 1/mu)
  dat <- data.frame(cbind(surT, X))
  x.name <- unlist(lapply(1:p, function(i) paste0("x", i)))
  names(dat) = c("surT", x.name)
  dat$status <- rep(2, n)
  return(dat)
}

set.seed(1)
n = 200
b = c(1,0.1)
X = as.matrix(rnorm(n, mean = 3, sd = 2))
dat = sim_dat(b, X)

x.c = scale(X, center = TRUE, scale = FALSE)
x.sd = scale(X, center = TRUE, scale = TRUE)

hist(dat$surT, breaks = 20)

Past versions of unnamed-chunk-4-1.png

Version	Author	Date
62ab36f	yunqiyang0215	2023-02-13

Effect of mean center/standardize X

## Create  survival object. status == 2 is death
dat$y <- with(dat, Surv(surT, status == 2))
# Fit cox ph. Cox ph model with select multiple significant predictors..
cox1 <- coxph(y ~ x1, data =  dat)
cox2 <- coxph(dat$y ~ x.sd)
cox3 <- coxph(dat$y ~ x.c)

coef(cox1)

         x1 
-0.09671461 

coef(cox2)

      x.sd 
-0.1797146 

coef(cox3)

        x.c 
-0.09671461 

summary(cox1)

Call:
coxph(formula = y ~ x1, data = dat)

  n= 200, number of events= 200 

       coef exp(coef) se(coef)      z Pr(>|z|)  
x1 -0.09671   0.90782  0.04239 -2.282   0.0225 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

   exp(coef) exp(-coef) lower .95 upper .95
x1    0.9078      1.102    0.8354    0.9865

Concordance= 0.566  (se = 0.02 )
Likelihood ratio test= 5.27  on 1 df,   p=0.02
Wald test            = 5.21  on 1 df,   p=0.02
Score (logrank) test = 5.21  on 1 df,   p=0.02

summary(cox2)

Call:
coxph(formula = dat$y ~ x.sd)

  n= 200, number of events= 200 

         coef exp(coef) se(coef)      z Pr(>|z|)  
x.sd -0.17971   0.83551  0.07877 -2.282   0.0225 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     exp(coef) exp(-coef) lower .95 upper .95
x.sd    0.8355      1.197     0.716     0.975

Concordance= 0.566  (se = 0.02 )
Likelihood ratio test= 5.27  on 1 df,   p=0.02
Wald test            = 5.21  on 1 df,   p=0.02
Score (logrank) test = 5.21  on 1 df,   p=0.02

summary(cox3)

Call:
coxph(formula = dat$y ~ x.c)

  n= 200, number of events= 200 

        coef exp(coef) se(coef)      z Pr(>|z|)  
x.c -0.09671   0.90782  0.04239 -2.282   0.0225 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    exp(coef) exp(-coef) lower .95 upper .95
x.c    0.9078      1.102    0.8354    0.9865

Concordance= 0.566  (se = 0.02 )
Likelihood ratio test= 5.27  on 1 df,   p=0.02
Wald test            = 5.21  on 1 df,   p=0.02
Score (logrank) test = 5.21  on 1 df,   p=0.02

Effect of standardize y.

## Create  survival object. status == 2 is death
surT = scale(dat$surT, center = TRUE, scale = TRUE)
status = rep(2, length(surT))
y <- Surv(surT, status)

# Fit cox ph.
cox4 <- coxph(y ~ dat$x1)
summary(cox4)

Call:
coxph(formula = y ~ dat$x1)

  n= 200, number of events= 200 

           coef exp(coef) se(coef)      z Pr(>|z|)  
dat$x1 -0.09671   0.90782  0.04239 -2.282   0.0225 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

       exp(coef) exp(-coef) lower .95 upper .95
dat$x1    0.9078      1.102    0.8354    0.9865

Concordance= 0.566  (se = 0.02 )
Likelihood ratio test= 5.27  on 1 df,   p=0.02
Wald test            = 5.21  on 1 df,   p=0.02
Score (logrank) test = 5.21  on 1 df,   p=0.02

Session information

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 mvtnorm_1.1-3   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