cox-modelregressionsurvival
What are the consequences of violating the Proportional Hazards assumption in a Cox Model? I've got a Model where two factors are highly significative, but all the estimated betas associated to the levels of these factors violate the PH assumption according to the Schoenefeld Goodness of Fit test.
Here are the beta vs time plot for some of the betas:
There are several options: I'd recommend examining the impact of the assumption on your hazard ratio estimates as a next step, rather than relying on a test statistic (or even the log-log graphs -- from a determining the impact perspective.) The two I'd initially suggest:
explicitly add in a time * covariate interaction to examine how the hazard ratio for your covariate changes over time.
add in a "heaviside" interaction term that explicitly models a hazard ratio for an "early" period and a "late" period. This is probably only sensible when you have an a priori cut-off for defining early/late (e.g. we've used it when modelling rescreening times in breast cancer, where there are defined screening provider targets for a 27 month rescreen interval: so we defined the break point for the heaviside function at 27 months.)
Obviously stratification by the covariate in question isn't an option... since there's only one covariate and it's the one you're principally interested in!
Link for some options in Survival analysis: a self learning text
and also see the following presentation (which is on the mathematical side) course notes from National University of Singapore
I think you've presented your question very well. I often use the extended Cox model (with time-varying covariates) and I've been struck by the same obstacles several times.
Like you pointed out:
cox.zph(model)
rho chisq p
s 0.2066 12.536 0.000399
CTRUE 0.0453 2.212 0.136984
l 0.0461 0.835 0.360842
GLOBAL NA 14.684 0.002108
Your model violates the PH assumption; the model as a whole violates it due to the marked violation of the s variable.
Then you show us this plot:
I rarely bother to assess the plots if my cox.zph() is alright. However, I guess that the plot y axis presents beta coefficient of s and it appears to vary during follow-up. The proportional hazards assumption states that the hazard of any variables must be constant throughout the study period. That is, hazard of s should not fluctuate (vary) with time. This plot shows that the hazard associated with s is less pronounced between 200 and 700 days. To me thats the graphical evidence for the p=0.000399 in the cox.zph().
Why did that happen? Subject matter will be your guide to decide why this occurred. I have not seen violations being as sudden as this one; the patterns are often more successive than your graph presents.
More importantly, you did the right thing to include the interaction term between s and stop. See John Fox example here.
Some would argue that you should be satisfied with that (without rechecking cox.zph(). I would recommend a recheck, however.
Best Answer
I think it depends on who you ask.
According to this article