# Cox regression covariate specific concordance independent of distribution of covariates in sample?

Concordance seems to be a very good metric for telling you how well the model allows you to order individuals for the particular sample that is being used. However, when applying a cox regression model with a single categorical variable (say male/female survival) to a population that is 99% female, the concordance will always come out at ~0.50 no matter how good the categorical variable (male/female) is at distinguishing people that are at high risk. Is there another way to measure how well a particular variable allows you to order two different groups, independent of the sample size that is used?

To illustrate the point, below is a small simulation to plot the concordance after fitting a cox proportional hazards model to some simulated data, where alpha is used to specify the proportion of individuals that are male/female. As noted, when alpha is small/large, the concordance is near 0.5.

conc <- c()
alphas <- seq(0,1,length.out=100)
for(alpha in alphas){
n <- 1000
x <- c(rep(0,round(n*alpha)),rep(1,round(n*(1-alpha))))
deaths <- rexp(1000,rate=exp(5*x))
# Assume no censoring
sdata <- data.frame(Y=deaths,d=1,x=x)
coxfit1 <- coxph(Surv(Y,d)~x,sdata)