Log rank test for a time-dependent variable?

Question

I am studying the association between a time-dependent treatment variable and death. The treatment is time-dependent because patients can switch groups during the follow-up. I followed [this paper] (https://www.tandfonline.com/doi/abs/10.1198/000313005X70371) to create extended Kaplan Meier curves using counting process data. Then tried to compare the curves using the log-rank test. My question is if the log-rank test can be used in the case of a time-dependent variable. I searched but did not find much discussion about this topic. Thanks for any input.

Answer 1

The abstract of the paper you cite notes:

The Cox regression model is easily extended to the case of time-varying covariates; however, there is no clear approach for similarly extending the standard Kaplan-Meier estimator.

You can handle a situation like this with a Cox regression, which is "easily extended to the case of time-varying covariates," to test the association of the treatment with outcome. In the simplest case with fixed treatment groups, the log-rank test on Kaplan-Meier curves is the same as the score test for the corresponding Cox model, anyway.

You can then try to use the methods described in that paper to illustrate the results in adjusted Kaplan-Meier plots, which some might find more intelligible.

I'm not sure if there is a way to do a log-rank test directly on such adjusted Kaplan-Meier survival curves; it's not clear how one would then get the number of estimated events needed to compare against the observed number of events when treatment group assignments are changing. Even if there is a way to do that, there wouldn't seem to be any advantages over Cox regression with time-varying covariates for inference.

One warning as you proceed: these methods for handling time-varying covariates implicitly assume that only the current value of the covariate helps to determine the hazard of an event. In your situation these methods wouldn't take the history of prior treatment for an individual into account unless you generated some corresponding covariate that incorporated that history along with the current treatment.