PAPP104 - S03: Multiple decrement and current status life tables

Multiple decrement versus associated single decrement probabilities

When developing the multiple decrement life table, we have argued that cause-specific probabilities of dying (_nq_xⁱ) must add up to the all cause probability of dying in an age interval. Imagine a world with two causes (i and –i). Everyone can only die from one of those causes and that cause is in principle observable:

{}_{n}q_{x} = {}_{n}q_{x}^{i} + {}_{n}q_{x}^{- i}

The relationship between associated single decrement probabilities and all-cause probabilities is a bit more complex. This is illustrated by the venn diagrams below. In the absence of other causes, cause I would be responsible for the deaths represented by the venn diagram on the left, i.e. it would claim casualties that in reality have succumbed to other causes (the shaded area):

Therefore:

\begin{array}{c} {}_{n}q_{x} \neq {}_{n}^{*}q_{x}^{i} + {}_{n}^{*}q_{x}^{- i} \\ {}_{n}q_{x} = {}_{x}^{*}q_{x}^{i} + {}_{x}^{*}q_{x}^{- i} - ({}_{x}^{*}q_{x}^{i}) ({}_{x}^{*}q_{x}^{- i}) \end{array}