Fine-Gray subdistribution hazards regression models the cumulative incidence of the event of interest when competing events can preclude it.
Standard survival analysis (K-M, Cox PH) assumes censoring is non-informative. When a 'competing' event happens — death from another cause when the outcome of interest is cancer-specific death, for example — that subject is no longer at risk for the event of interest. Treating competing events as censoring biases K-M upward (overestimates incidence) and biases Cox PH in subtle ways.
Fine-Gray models the hazard of the *subdistribution* — the hazard of the event of interest, with subjects who experienced a competing event kept in the risk set indefinitely with downweighting. The result is a hazard ratio for the cumulative incidence function (CIF), which is the right population-level summary in the competing-risks setting.
Cox-on-cause-specific-hazard is the alternative: model the event of interest's hazard with competing events as censoring. Use it when you want the etiologic interpretation ('among those still at risk'); use Fine-Gray when you want the predictive / risk-management interpretation ('what fraction will experience this event by time t').
Time + multi-state event (0 = censored, 1 = event of interest, ≥ 2 = competing) + predictors.
Subdistribution HR + CI + p per predictor.
SDHR > 1 ⇒ predictor increases the CIF of the event of interest. Unlike cause-specific HR, this is the right effect for absolute-risk communication.
If you also fit Cox-on-cause-specific-hazard for the same outcome, the two effects can differ — Fine-Gray attributes some of the apparent effect to changes in the competing-event hazard.