Performance of goodness-of-fit tests for the Cox proportional hazards model with time-varying covariates

There are few readily-implemented tests for goodness-of-fit for the Cox proportional hazards model with time-varying covariates. Through simulations, we assess the power of tests by Cox (J R Stat Soc B (Methodol) 34(2):187-220, 1972), Grambsch and Therneau (Biometrika 81(3):515-526, 1994), and Lin et al. (Biometrics 62:803-812, 2006). Results show that power is highly variable depending on the time to violation of proportional hazards, the magnitude of the change in hazard ratio, and the direction of the change. Because these characteristics are unknown outside of simulation studies, none of the tests examined is expected to have high power in real applications. While all of these tests are theoretically interesting, they appear to be of limited practical value.