10.1 Some mathematics

First we need some strict definitions, so that the correct questions are asked, and the correct answers are obtained.

The cause-specific cumulative hazard functions are \[\begin{equation*} \Gamma_k(t) = \int_0^t \alpha_k(s) ds, \quad t > 0, \quad k = 1, 2, 3. \end{equation*}\] The total mortality is \[\begin{equation*} \begin{split} \lambda(t) &= \sum_{k=1}^3 \alpha_k(t)\quad t > 0 \\ \Lambda(t) &= \sum_{k=1}^3 \Gamma_k(t), \quad t > 0 \end{split} \end{equation*}\] and total survival is \[\begin{equation*} S(t) = \exp\left\{-\Lambda(t)\right\} \end{equation*}\] So far, this is not controversial. But asking for a “cause-specific survivor function” is.