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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?
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    $\begingroup$ Regarding 3) sometimes violating assumptions improves fit unrealistically. $\endgroup$ – Peter Flom - Reinstate Monica Nov 21 '13 at 12:07
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    $\begingroup$ Thanks for the reply @PeterFlom. I am confused how this may happen for this particular model type, though. I would expect that a good fit is only possible if the PH assumption holds. Clearly I have been proven wrong. $\endgroup$ – Marc Claesen Nov 21 '13 at 12:13
  • $\begingroup$ I don't have any strong intuition one way or the other on this one. But surely it's possible to invent a case where fit is perfect and PH assumption is violated. $\endgroup$ – Peter Flom - Reinstate Monica Nov 21 '13 at 12:15
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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.

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  • $\begingroup$ thank you for your explanation on concordance, I like your intuitive explanation of it at the end of the first paragraph. So, unlike the other global tests (Wald, Logrank, Likelihood Ratio, etc.) listed as a summary of a CoxPH model, concordance is assessing the predictive performacne of the model? Do you have any source / links on how this is calculated for CoxPH? I am happy to ask another question if you prefer. $\endgroup$ – Zhubarb Nov 26 '14 at 16:27
  • $\begingroup$ @Zhubarb, the R and C source codes in the survival package show how this is calculated; "concordance1.c", called by survConcordance.fit, shows how the death times (including ties) and predictors are handled, and code at the end of summary.coxph shows how concordance is displayed. Note that concordance is not the best way to assess predictive performance; see stats.stackexchange.com/questions/17480/… $\endgroup$ – EdM Nov 26 '14 at 17:20

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