I am performing backward selection for the Cox Proportional Hazards model with several variables (it is required that I do backward selection). One of these variables (let's call it "A") does not meet the proportionality assumption in the full model. So I would like to consider the possibility of stratifying on variable A. How should I perform backward selection in this scenario?
I am considering performing backward selection on the full model while stratifying on A (call the model selected "model B"). I will then compare fit statistics between model B with stratification and model B without stratification. If I see that the stratification is providing only a marginally better fit, then I will perform another backward selection on model B without stratification.
- Does this seem reasonable or is there a better way?
- How do I formally test if model B without stratification is better than model B with stratification? A likelihood ratio test would not work because the models are not nested and comparing AIC or BIC doesn't make sense because there are the same number of variables in each model.
Edit: any general advice on model selection in COX PH is appreciated as well!