# Proportional hazards assumption and time-dependent covariates

Is there a way to check that the proportional hazards assumption is correct for a Cox model with time-varying covariates ?

## 3 Answers

If we add time-dependent covariates or interactions with time to the Cox proportional hazards model, then it is not a “proportional hazards” model any longer.

See this presentation: http://ms.uky.edu/~mai/sta635/Cox%20model.pdf

or this lecture notes: http://www.math.ucsd.edu/~rxu/math284/slect7.pdf

But this is a widely known feature.

The answer of Serpico is true for time-dependent coefficients but not if the model uses time-dependant covariates. If this was the case, I don't know the correct answer.

• This is a confusing answer. – Michael Chernick Oct 9 '17 at 14:16
• Time-dependent covariates are measured values that change along the study. Time-dependents coefficients imply that some $\beta$ is not constant across time. Philosophically speaking, there's not much difference - since I guess there's always a way to convert between products $\beta(t) x$ and $\beta x(t)$, if we can choose any shape of time-dependency. – juod Oct 9 '17 at 16:23
• @juod Your comment would therefore indicate that we must not verify the proportional hazards assumption with a model with time-dependent covariates. However, the answer for this question: stats.stackexchange.com/questions/246488/… appears to indicate otherwise – Emmanuel.W Oct 9 '17 at 16:49
• @Emmanuel.W I agree with you that such a model still assumes proportional hazards given same covariate levels. I was referring to the discrepancy between your answer and Serpico's references - they say that the term $\beta x t$ has a time-dependent covariate, while I'd say it's the coefficient that is time-dependent there. – juod Oct 10 '17 at 8:06

As Serpico suggests, the Cox model with time-dependent covariates is no longer a proportional hazards model. Here is a quote from David Collett's book, Modelling Survival Data in Medical Research (2nd ed., 2003, p. 253), that may provide some further clarification:

It is important to note that in the model given in equation $h_i(t) = \exp \left\{ \sum_{j=1}^p \beta_j x_{ji}(t) \right\} h_o(t)$, the values of the variables $x_{ji}(t)$ depend on the time $t$, and so the relative hazard $h_i(t)/h_0(t)$ is also time-dependent. This means that the hazard of death at time $t$ is no longer proportional to the baseline hazard, and the model is no longer a proportional hazards model.