How the concordance index is calculated in Cox model if the actual event times are not predicted? I am new to the field of survival analysis. I was reading about the interpretation of C-index and realized it only cares about the sequence of predictions. I was always using the sci-kit survival package and never deeply though how the C-index is calculated if the actual survival times are not predicted in Cox proportional hazard model. I would appreciate if someone simply explain this to me.
 A: Below is my attempt to answer this question.
Concordance index is a measure of how discriminant your model is.
For survival analysis, say you have a covariate $X$ and a survival time $T$.
Assume that higher values of $X$ imply shorter value for $T$ (thus $X$ has a deleterious effect on $T$).
Discrimination means that you are able to say, with high reliability, that between two patients which one will have a shorter survival time.
For a perfectly discriminative model, if you pick two sujects at random $(X_1,T_1)$ and $(X_2,T_2)$ then the one with the largest value of $X$ will have, with probability $1$, a shorter survival time:
$$
c=\mathbb P( T_1 < T_2 \mid X_1 \geq X_2) = 1
$$
In your dataset if you pick two patients at random, there is 4 cases:

*

*$X_1 \geq X_2$ and $T_1 < T_2$ : There is corcordance $(C)$

*$X_1 \geq X_2$ and $T_1 > T_2$  : Discordance $(D)$

*$X_1 = X_2$ : Equal risks $(R)$

*$T_1 = T_2$ : Equal times

The last case is not taken into account to estimate the concordance (at least I think so).
In case $3$, since the two patients have the same risk, the best you can do to say which one will have the shorter survival time is to toss a fair coin.
The estimated concordance index based on your data is:
$$
\hat c= \frac{C+\frac{R}{2}}{C+D+R}
$$
where $C$, $D$ are the total number of concordant, discordant couples, $R$ the total number of couple with the exact same risk.
The $\frac{R}{2}$ at the numerator comes from the coin toss.
When there is censoring (as often with survival data) the computation of $\hat c$ is modified but the idea and interpretation of $c$ remains the same.
Example
Say you have $8$ patients with data:
\begin{array}{c| c|c}
\text{Id} & \text{Time} \ (T) & X \\ \hline
1 & 1 & 1 \\
2 & 2 & 3 \\
3 & 3 & 2 \\
4 & 12 & 10 \\
5 & 17 & 15 \\
6 & 27 & 40 \\
7 & 36 & 60 \\
8 & 55 & 80 \
\end{array}
In that case, we see that larger values of $X$ imply larger values of $T$.  Thus a couple is concordant if $X_1 < X_2$ and  $T_1 < T_2$.
There are $\binom{8}{2}=28$ choices of couples of patients, among those only the couple $(2,3)$ is discordant (since $X_2 > X_3$ but $T_2 < T_3$).  There is no couple with equal risk thus $R=0$.
Then the estimated concordance index is $\frac{27}{28} \approx 0.964$.
You can check this with the R package survival (sorry I'm not used to survival analysis with Python):
require(survival)
time<-c(1,2,3,12,17,27,36,55)
X<-c(1,3,2,10,15,40,60,80)
data<-data.frame(matrix(c(time,X),ncol=2,8,byrow = F))
mod<-coxph(Surv(data[,1],rep(1,8))~data[,2])
mod$concordance #~0.964

So to answer your question about predicted times, you can see that neither the values of $T$ or $X$ change the estimation of $c$: it's only a matter of ordering between predictor and survival times. You can change the value in the previous example without breaking the number of concordant/discordant couples and still have the same estimated concordance.
In which direction should I look for the covariate $X$?
Is a couple concordant if  $X_1 > X_2$ and  $T_1 < T_2$ or if $X_1 < X_2$ and  $T_1 < T_2$?
For the Cox model, it depends on the estimated hazard-ratio. If the ratio, $e^\beta$ is $>1$ then larger values of $X$ imply larger risks thus shorter times.  So if $e^\beta > 1$ a couple is concordant if  $X_1 > X_2$ and  $T_1 < T_2$, and if $e^\beta < 1$ a couple is concordant if $X_1 < X_2$ and  $T_1 < T_2$.
Finally in the case of a vector of covariates, I think the procedure remain the same but instead of using the vector $X$ we use the predicted risk $\hat \beta X$ with $\hat \beta$ estimated from the Cox model.
A: You are correct that time is not the default output of a Cox model. However, for any given unit with its covariate pattern, the model gives a relative hazard. By definition, units with higher hazard ratios should have shorter time to event. The censored c-index compares the estimated hazard ratio to both the actual event status and actual time to event (or censoring time) to produce its estimate.
