High concordance in cox PH model even though PH assumption is violated

Question

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:

1. What is the main consequence hereof?
2. Is there still some value in the hazard ratios for predictors for which the PH assumption is invalidated?
3. How can the fit be so good despite a key assumption being violated?

Answer 1

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. 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. 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.

Concordance for a multiple regression 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. Think about the corresponding case of a linear-regression fit of data that are not linearly related.

That said, the HR for a variable that doesn't meet the PH assumption can still be interpreted as a type of time-averaged value. That might not be a terribly useful result, however, if the PH violation is severe.

1. The main consequence is that you should examine variables that don't meet the PH assumption in more detail.

For example, it's possible that an apparent violation of PH for a continuous predictor is due to choosing an improper functional form for it. Try a flexible fit, like with a regression spline. Decide whether the PH violation is large enough to matter in practice; a very large data set might show a "statistically significant" PH violation of a very small magnitude. If the PH violation does matter in practice, consider stratifying by the offending variables, or devising time-dependent models.