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My question is that in what condition that the Cox-PH assumption said to be satisfied after including the time-varying covariates with time interaction in the Cox-PH model? is that must all variables in the model should not be statistically significant or it is enough only the time-varying covariates should not be statistically significant?

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  • $\begingroup$ Please give an example of just what you mean by "including the time-varying covariates with time interaction." There might be a couple of ways to interpret that phrase, and the answer could differ depending on what you specifically mean. $\endgroup$
    – EdM
    Commented Oct 15, 2021 at 15:30
  • $\begingroup$ I mean that after "including the time-varying covariates with time interaction" to test the proportional hazard of Cox-PH model based on P-value. My question is that Is it a must that all variables (Time-independent and Time-dependent variables) must not statistically significant or only the time-varying variables must not be statistically significant if the proportional hazard assumption said to be satisfied? $\endgroup$ Commented Oct 22, 2021 at 6:14

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Strictly speaking, if the coefficient for any predictor violates proportional hazards (PH) then your model does not meet a critical assumption of Cox models. Quoting from Section 3.5 of the main R survival vignette:

A key simplifying assumption of the model is that all of the coefficients except $\beta_0$ (the baseline hazard) are constant over time.

It doesn't matter whether the predictor itself is constant or varies with time. The example used in Section 3.5 of that vignette to illustrate violation of PH has no time-varying covariates. The question is whether the coefficient for the predictor varies over time. If it does, then PH is violated.

The question is what to do about it. With very large data sets, you might have a "statistically significant" violation of PH that is of relatively small magnitude. On the other hand, with small data sets you might not have the power to find important violations of PH. So all work with Cox models requires some care in evaluating PH.

Many threads on this site suggest things to try next. Violation by a categorical predictor not of primary interest but that needs to be accounted for can be dealt with by stratification. Violaton by a continuous predictor might be fixed by transformation. You might construct a model that explicitly includes a time-varying coefficient for the predictor, as explained in the R time-dependence vignette. You might need to abandon a PH model and use a different type of model.

It also might be possible to acknowledge the PH violation and work with it nevertheless, if it's small in magnitude. The coefficient reported by a Cox model when PH doesn't hold is a type of event-averaged value that might be good enough in practice.

Finally, make sure that you start by evaluating the PH assumption correctly. Although including "covariates with time interaction" in the model is one way to do that, it's easy to make a mistake in how the data underlying that interaction terms need to be constructed. The score test implemented in the R cox.zph() function, explained in Section 3.5.2 of the main survival vignette, is easier to use and can provide plots that demonstrate the magnitude and shape of the deviation from PH.

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