Survival Analysis: Quantifying Survival Differences when Proportional Hazards is Violated

Question

I am comparing survival between two groups distinguished by a single factor. The survival curves converge at both ends (near time 0 and near the maximum time) but never cross, and R's survival::cox.zph() test also indicates that the proportional hazards assumption does not hold.

• Given the lack of proportional hazards, is the difference in median survival (calculated with R's survival::survfit()) still a valid measure quantifying the differences in survival?

• Since the differences converge at both ends, far from the median, would the best measure of the survival difference be the hazard ratio expressed as a function of time t?

• What measure would you recommend using to quantify this difference for nontechnical stakeholders?

Thanks so much!

Answer 1

With only 2 groups distinguished by a single categorical factor, the score test in a Cox regression is the same as the log-rank test used to evaluate the difference between two Kaplan-Meier curves. See the Wikipedia entry on the log-rank test. That's true whether or not the proportional hazard (PH) assumption holds. So you have a reliable way to estimate whether the survival curves differ.

The choice of how to present the difference in survival curves absent PH depends on your and your audience's understanding of the subject matter. I'd probably want to display both entire survival curves as the best representation of the results.

If you need a single value, the median survival time is pretty easy to understand. Furthermore, if your data are better represented by an accelerated failure time model than a PH model, as your description might suggest, then the ratio between median survival times nicely summarizes the differential "acceleration" of the time scale between the 2 conditions.