Extended Kaplan-Meier for survival curves from Cox PH model with time-varying covariates?

Question

I am interested in using a proportional hazards model with external time-varying covariates for the purpose of predicting future event probabilities. Since extracting survival curves with time-varying data is not as simple as the time-independent case, I was wondering if the following procedure makes sense:

• Fit the time-varying PH model
• Discretize/bin the fitted hazard rates at each time step
• Estimate extended Kaplan-Meier survival curves using the binned hazards as specified in this paper: http://amstat.tandfonline.com/doi/abs/10.1198/000313005X70371

Answer 1

I can't seem to access the link (it seems the website has some temporary technical difficulties), but your approach makes sense.

In KM it is assumed that this jumps only at event times, so if the event times are denoted as $t_1 < t_2 < ...$ then $$\hat S(t) = \exp(-\sum_{i | t \geq t_i} \hat h(t_i))$$ If there is an external time-varying covariate, say $x(t)$ with an estimated effect $\beta$. Then the "baseline" group corresponds to the hazard $h(t)$ and the "exposed" gorup to $h(t) \exp(x(t) \beta)$. You can see that, to estimate the $S(t)$ in the "exposed" group you only need the values at the event time points of this time-varying covariate.