Explanatory variables that are monotonic functions of time in Cox proportional hazards models?

Question

I am fitting a Cox PH model where the main explanatory variable is a monotonic function of time. Is this violating any assumptions of the Cox model?

Answer 1

The assumption of proportional hazards is not about the time-dependency of the covariate but about the one of its effect on the hazard.

Say you have a covariate $X$ which does not vary over time.

The Cox model is a model based on the hazard and is defined as:

$$ h(t \mid X(t) ) = h_0(t) \exp(\beta X)\ $$ The terming proportional hazards comes from the property that the ratio of hazards for two values of $X$, $x_1$ and $x_2$ say, does not depends on time:

$$ \frac{h(t \mid x_1)}{h(t \mid x_2)} = \exp(\beta(x_1-x_2)) $$

Now, assume the covariate $X$ is time dependent, $X= \{X(t), t>0\}$.
We can still define a model that somehow ressemble the Cox model (often called the extented Cox model): $$ h(t \mid X) = h_0(t) \exp(\beta X(t)) $$ In this model, the hazard at time $t$ relies on the whole path of $X$ only through the "current" value $X(t)$ (this assumption can be relaxed). What varies over time is the value of the covariates and not its effect on the hazard function.

This model is a proportional hazard model in that $\beta$ is still time-independent.
The hazard ratio at time $t$ between two values of $X$, $x_1(t)$ and $x_2(t)$ is simply $$ \exp(\beta(x_1(t) - x_2(t)) $$

The coefficient $\beta$ can still be estimated by partial likelihood maximisation.

The proportional hazard assumption made by the Cox model would be violated is the true model was:

$$ h(t \mid X) = h_0(t) \exp(\beta(t) X) $$ that is the effect of the covariate varies over time.