PAPP103 - S09: Models of populations with variable growth rates

Application 5: Life tables without a population at risk

If one makes the assumption that mortality is more-or-less unchanging over a relatively short time period, then deaths by age will grow at the same rate as the population by age. Thus, if one has data on deaths by age for two adjoining periods and assumes unchanging mortality, one can calculate a life table directly by adjusting the deaths by their own growth rates.

Deaths at age x in the life table, d_x, equal L_xm_x. Thus, taking the equation for actual deaths in the population,

D_{x} = B_{0} e^{- \sum_{a = 0}^{x - 1} r_{a} - 0.5 r_{x}} \frac{L_{x}}{l_{0}} m_{x}

by rearranging it and setting l₀= B₀, one obtains:

d_{x} = D_{x} e^{\sum_{a = 0}^{x - 1} r_{a} + 0.5 r_{x}}

If one is prepared to assume constant mortality and has death data for two adjoining periods of duration h years, one can calculate the growth rate in each age group from the two sets of data as:

r_{a} = \frac{\ln D_{a} (t + h) - \ln D_{a} (t)}{h}

The average numbers of deaths each year in the two periods is:

D_{x} (\bar{t}) = \frac{\sqrt{D_{x} (t + h) \cdot D_{x} (t)}}{h}

By adjusting these average numbers of deaths using the growth rates of the deaths, one can estimate d_x without knowing anything about the population at risk of dying. Once d_x has been calculated, cumulating the measures downward from the oldest age group yields successive l_x values ending in l₀. From these values one can calculate all the other columns of the life table, including the age-specific death rates and life expectancy at birth.

This application of variable r relationships is particularly useful in historical demography as registers of deaths were often kept before enumerations of the population had begun.