If I have a categorial variable (let's say $X$) and a continuos variable (let's say $Y$). IF I fitted a standard cox model, it will result in a violation of the constant HR over time (based of Schoenfeld residuals). So, I fitted again with interaction (with time). In R, this can be written as $$Surv(start,stop,event) = X + Y + X:stop + Y:stop$$

And, the interaction time is significant for both variables. My question is, how to correctly interpret this ? Is it correct to say that the hazard ratio of $X$ and $Y$ is increasing (or decreasing) with time ? Since the interaction is significant...


1 Answer 1


Setting up time-interactions

First you need to split your dataset in order to do the time interaction. Then you set the start-time as the interaction term as that is independent of the outcome.


Yes, it is either decreasing or increasing with time although note that you also need to take into account the raw variable. There are two basic situations:

  • If $\beta_X < 0$ and $\beta_{X:stop} < 0$ or $\beta_X > 0$ and $\beta_{X:stop} > 0$ then the difference increases over time.
  • If $\beta_X > 0$ and $\beta_{X:stop} < 0$ or $\beta_X < 0$ and $\beta_{X:stop} > 0$ then the difference is diminishing until the time point $\beta_X = Time*\beta_{X:stop}$ where the risks start to differ.

An important issue is that you are all the time assuming linearity with this construction. You should also check if there is a non-linear relationship for the time interaction. I use this together with the contrast function in the rms package in order to visualize this effect. I


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