# Cause-specific survival function in survival analysis

In survival analysis, when there are competing risks, it is well-known that although the cause-specific hazard function, $$\lambda_j^\#(t)$$, is interpretable, $$S_j^\#(t) = e^{-\Lambda_j^\#(t)}$$ may not be, because "the cause-specific survival function will always be greater than or equal to the overall survival function $$S(t)$$ if there are any competing events" (Edwards, 2016).

I don't understand why this is true. Could someone help me with the proof that $$S_j^\#(t) \geq S(t)$$?

The overall survival function $$S(t)$$ is associated with the event time corresponding to the minimum of the respective cause-specific event times; another way of saying this is that the overall survival is always smaller than the cause-specific survival.
$$S(t) = P\left(\min \left(T_1, \ldots, T_d\right) > t\right) = P\left(T_1 > t, \ldots, T_d > t\right) \leq P(T_1 > t) = S_1(t)$$
• @bob I mixed up the sign of the inequality, sorry. The event $\min (T_1, \ldots, T_d) > t$ is included inside every event $T_1 > t$. Oct 23, 2020 at 15:42