Understanding Dependent Competing Risks: A Simulation Study to Illustrate the Relationship Between Cause-Specific Hazard and Marginal Hazard

In the analysis of competing risk data, the observed effect of a covariate can be obtained via a Fine and Gray sub-distribution hazard ratio. Sometimes, it is also desirable to obtain the virtual effect of a covariate as if the competing risks were non-existent. Under the latent failure time scenario, when the event of interest and the competing risk event are independent, the cause-specific hazard ratio obtained from the Cox model where the competing events are censored represents the ratio of the marginal hazards and can be interpreted as the virtual effect of the covariate. However, when the two events are not independent, the cause-specific hazard ratio is not the ratio of the marginal hazards as the ratio depends not only on the marginal hazards but also on the correlation between the competing risk and the event of interest. Using simulation, we investigated the degree to which the cause-specific hazard ratio changes relative to the marginal hazard with this correlation. It was found that the discrepancy between the cause-specific hazard ratio and the theoretical marginal hazard ratio increased as the proportion of competing risk events and the correlation between the events increased (>0.2). Depending on the direction of the correlation, the cause-specific hazard ratio can overor under-estimate the marginal hazard ratio. Using real-life datasets, we show how these results can be used to make inferences on the virtual effects.