While studying about time-dependent receiver operating characteristic (ROC) using Cox regression & Kaplan-Meier, I stuck with some problem regarding the definition used for both method.
For Cox Model, the time-dependent ROC is commonly estimated as I/D definition (Incidence sensitivity/Dynamic specificity). The procedure to estimate time-dependent ROC under this definition is as follows:
- Fit the Cox Model
- Sensitivity & specificity are respectively computed through, $$ Se_t^I(x)=P(X>x|T=t) $$ $$ Sp_t^D(x)=P(X\le x|T>t) $$
The $t$ here is referring to time-to-event.
For Kaplan-Meier approach, the time-dependent ROC is commonly estimated as C/D definition (Cumulative sensitivity/Dynamic specificity). The procedure to estimate time-dependent ROC under this definition is as follows:
- Select time point to generate ROC curve
- Assign status of subject at the selected time point as alive or dead (e.g, if time-to-event is $T=5$, subject status = 0 (alive) if selected time point $<5$ but subject status = 1 (dead) if selected time point $\ge 5$).
- Sensitivity & specificity are respectively computed through, $$ Se_t^C(x)=P(X>x|T\le t) $$ $$ Sp_t^D(x)=P(X\le x|T>t) $$
The $t$ here is referring to time of observation.
Thus, if I want to apply C/D definition for time-dependent ROC estimated through the Cox model, do I need to use time-to-event or time of observation?
If $T$ represents the time of observation, does this means that I need to fit the Cox model multiple time at each selected time point (because at different time point as shown in figure below, I'll obtain different multiplicative effect $e^{x\beta}$)?
Thank you for the clarification. Do let me know in case my question is not clear.