Time-dependent covariates are those that may change in value over the course of observation. While it’s simple to modify Cox’s model to allow for time-dependent covariates, the computation of the resulting partial likelihood is much more time consuming, and the practical issues surrounding the implementation of the procedure can be quite complex. It’s easy to make mistakes without realizing it, so be sure you know what you’re doing.
To modify the model in equation (5.2) to include time-dependent covariates, all we need to do is write (t) after the xs that are time dependent. For a model with one fixed covariate and one time-dependent covariate, we have
log hi(t) = α(t) + β1xi1 + β2xi2(t).
This says that the hazard at ...