High concordance in cox PH model even though PH assumption is violated I am using Cox proportional hazards models for survival analysis. The specific reason I am interested in them is because they give a nice way to quantify effect size between groups via the hazard ratio, assuming the PH assumption is not violated. 
I use R's survival package for the modeling. The models I am making have no interaction terms between predictors. I am currently in the following situation for some models:


*

*the model is a good fit, with concordance over 80% (obtained via summary of the coxph fit). 

*the PH assumption is violated for one or more predictors (e.g. very small $p$-values for some predictors in cox.zph).


This leads me to a couple of questions:


*

*What is the main consequence hereof? 

*Is there still some value in the hazard ratios for predictors for which the PH assumption is invalidated?

*How can the fit be so good despite a key assumption being violated?

 A: In reverse order:
3) The concordance is simply the proportion of pairs of cases in which the case with the higher-risk predictor had an event before the case with the lower-risk predictor. With a single numeric predictor, the concordance will be the same for any monotone transformation of the predictor even though the Cox model fits may be substantially different. Crudely put, concordance shows your ability to predict who of a pair will die sooner, but not necessarily how much sooner or what proportion of the variance of event times is explained by the model.
Concordance for a multivariate model uses the combined linear predictor from the Cox regression as the numeric predictor for each case. So if variables with non-proportional hazards have small-magnitude coefficients compared with other variables, or if their relations to outcome are strong enough despite non-proportionality, the rankings of combined linear predictors may be well correlated with the rankings of event times--which is all that concordance tells you.
2) Absent the PH assumption, HRs aren't strictly valid and can be highly misleading. Think about the corresponding case of a linear-regression fit of data that are not linearly related.
1) Main consequence is that you should examine variables that don't meet the PH assumption in more detail. Consider stratifying by those variables, or devising time-dependent models.
