## Survival Analysis: Competing Risk Theory

Hazard function Regression:

The hazard function, describes the instantaneous rate of occurrence of the event of interest in subjects who are still at risk of the event. In a setting in which the outcome was all-cause mortality, the hazard function at a given point in time would describe the instantaneous rate of death in subjects who were alive at that point in time.

The Cox proportional hazards regression model relates the hazard function to a set of covariates. In the absence of competing events, the Cox proportional hazards regression model can be written as:

h(t)=h_0(t)e^{X\beta} \\
\textit{;where} h_0(t) \textit{denotes the baseline hazard function }

The Cox model relates the covariates to the hazard function of the outcome of interest (and not directly to the survival times themselves). The covariates have a relative effect on the hazard function because of the use of the logarithmic transformation. The regression coefficients are interpreted as log-hazard ratios. The hazard ratio is equal to the exponential of the associated regression coefficient. The hazard ratio denotes the relative change in the hazard function associated with a 1-unit increase in the predictor variable. Although the regression coefficients from the Cox model describe the relative effect of the covariates on the hazard of the occurrence of the outcome, the following relationship also holds in the absence of competing risks:

S(t)=S_0(t)e^{X\beta} \\
\textit{;where} S_0(t) \textit{denotes the baseline survival function }

Thus, the relative effect of a given covariate on the hazard of the outcome is equal to the relative effect of that covariate on the logarithm of the survival function. Therefore, in the absence of competing risks, making inferences about the effect of a covariate on the hazard function permits one to make equivalent inferences about the effect of that covariate on prognosis or survival.

This direct correspondence between the hazard function and incidence in the absence of competing risks may be used to conclude that a given risk factor or variable increased the risk of an event, without specifying whether risk denoted the hazard of an event (ie, the rate of the occurrence of the event in those still at risk of the event) or the incidence of the event (ie, the probability of the occurrence of the event).

Competing risks implies that a subject can experience one of a set of different events or outcomes. In this case, two different types of hazard functions are of interest:

• the cause-specific hazard function and
• the subdistribution hazard function.