A Goodness-of-Fit Test for Log-Linearity in Cox Proportional Hazards Model Under Monotonic Covariate Effects

The Cox proportional hazards (PH) model is widely used because it models the covariates to the hazard through a log-linear effect. However, exploring flexible effects becomes desirable within the Cox PH framework when only a monotonic relationship between covariates and the hazard is assumed. This work proposes a partial-likelihood-based goodness-of-fit (GOF) test to assess the log-linear effect assumption in a univariate Cox PH model. Rejection of log-linearity suggests the need to incorporate monotonic and non-log-linear covariate effects on the hazard. Our simulation studies show that the proposed GOF test controls type I error rates and exhibits consistency across various scenarios. We illustrate the proposed GOF test with two datasets, breast cancer data and lung cancer data, to assess the presence of log-linear effects in the Cox PH model.