A hazard ratio above one does not necessarily mean higher risk, when using a time-dependent cox model

The Cox model is one of the most used statistical models in medical research. It models the hazard rate of an event and its association with covariates through hazard ratios. In the simple setting without competing risks nor time-dependent covariates, there exists a one-to-one mathematical connection between the hazard rate and the risk of experiencing the event within any given time period (e.g., 5 years). This makes it possible to conclude that a covariate associated with a hazard ratio above one is associated with a higher risk of event. Although it is becoming widely known that this connection is lost in the presence of competing risks, it seems that fewer users of the Cox model are aware that this connection is also lost when using time-dependent covariates. In other words, it seems still widely unknown that, when using a time-dependent Cox model, a hazard ratio estimated above one does not necessarily mean that there is a higher risk. Hence, this note aims to clarify why this is not the case with a detailed pedagogical example.