Suppose you are interested in the Survival modelling technique Cox Proportional Hazard, where we model the hazard as:
$$ \lambda(t \vert x) = \lambda_0(t) exp (\beta x) $$
An extension of this model is time-varying covariates i.e.
$$ \lambda(t \vert x(t)) = \lambda_0(t) exp (\beta x(t)) $$
Another extension is time-dependent effects i.e.:
$$ \lambda(t \vert x) = \lambda_0(t) exp (\beta(t)x) $$
I am interested in time-varying covariates and have strong prior beliefs over the coefficients of the covariates (in the partial hazard) as well as the baseline. I have seen examples of Bayesian survival modelling (with Cox PHD) where you have time-dependent effects (i.e. a sequence of regression coefficients), see this example using PyMC3 and Python:
However, in the situation where you have time varying covariates, I am yet to find an example or implementation of this in python (or R for that matter). My question is:
- Is there a reason why you do not see implementations of Bayesian time-varying covariates COX PHD?
- If no to the above, does anyone have a good example of an implementation?