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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 $S_j^\#(t) \geq S(t)$?

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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.

$$ S(t) = P\left(\min \left(T_1, \ldots, T_d\right) > t\right) = P\left(T_1 > t, \ldots, T_d > t\right) \leq P(T_1 > t) = S_1(t) $$

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  • $\begingroup$ Thank you. Is this proving the opposite of what it said in the paper, though? $\endgroup$
    – bob
    Oct 22, 2020 at 18:31
  • $\begingroup$ @bob I mixed up the sign of the inequality, sorry. The event $\min (T_1, \ldots, T_d) > t$ is included inside every event $T_1 > t$. $\endgroup$ Oct 23, 2020 at 15:42

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