Notation & key assumptions (cont.)

If the force of mortality in an age interval is constant, the age structure in the interval does not matter, and the following equality will hold: * _nm_xⁱ = _nm_xⁱ

Provided that the mortality hazard is constant with respect to age, the associated single decrement probability of surviving the interval can be estimated as:

$${}_n{}^{*}p{}_x^i=e^{-n.{}_{n}M{}_x^i}$$

Note that this formula bears great similarity to the exponential population growth model with a constant growth rate. In this case we are not modeling the growth (or decline) of the population over a time interval, but the decline (mortality can only diminish the population) of the population between two exact ages.

Under the assumption that the cause-specific death rates are constant within age intervals, we do not to need additional estimates for * _na_xⁱ. Instead, we can estimate the person-years lived directly from the life table deaths and death rates. We have previously used the same approach for closing the single decrement life table :

$${}_n{}^{*}L{}_x^i=\frac{{}_n{}^{*}d{}_x^i}{{}_{n}M{}_x^i}$$

As was the case in the development of the single decrement life table, we will have to make special provisions to account for the highly skewed distribution of deaths in the first (two) age group(s).