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In the Cox model, violation of proportional hazards suggests that the effect of an exposure is not constant over time. One way this can be handled is by including time varying coefficients in order to produce covariate - time interaction term(s). It seems that there is so much emphasis on checking whether there is the presence of such a covariate - time interaction; but what about interactions of covariates with other covariates? I'm guessing this is just as important. What is it about the covariate time (i.e. proportional hazards) assumption that makes this so special and perhaps more important than any other type of interaction between covariates?

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... what about interactions of covariates with other covariates? I'm guessing this is just as important.

Interactions among covariates can be more important than covariate-time interactions, and might remove the need for time-varying coefficients.

If you omit any outcome-associated predictor from a Cox model, you can get biased estimates of the coefficients of included predictors and even get an apparent violation of proportional hazards (PH) that could be fixed by proper specification of the model. Omitting an outcome-associated interaction term thus can lead to an apparent violation of PH.

The covariate-time interactions used to model time-varying coefficients have their place, but I think that they tend to be over-emphasized. Very often a "statistically significant" violation of PH is of little practical consequence, or it might best be described by showing the shape of the plot of scaled Schoenfeld residuals over time rather than by trying to find an analytic expression for the time dependence. It's also possible that a PH model isn't appropriate for the data so that a different approach is better. See Sections 6.5 or 6.6 of Therneau and Grambsch.

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