PAPP103 - S09: Models of populations with variable growth rates

Application 3: Estimating life tables from proportions

Using variable growth rate methods, one can study the loss of characteristics that everyone shares on entry to a population using proportional data. For example, everyone is unmarried at birth. Therefore, one can use variable growth rate methods to estimate net nuptiality tables from two sets of age-specific data on the proportions of a population who are single.

This is valuable way of obtaining life tables in situations in which complete enumerations of the population of interest are unavailable, so one cannot calculate its age-specific growth rates, but one does have sample data from repeated surveys of the population. In essence the method entails constructing a life table from two cross-sectional sets of current status data by adjusting for age-specific growth in the prevalence of the characteristic of interest.

The approach is illustrated here using two sets of data on the proportion of children who are orphaned. Ignoring emigration, the population with living mothers P* is subject to two decrements – death, m_x, and orphanhood, π_x:

$$P_x^{*}=B_0^{*}{e}^{-{\displaystyle \sum _{a=0}^{x-1}{r}_a^{*}-0.5r_x^{*}}}\cdot e^{-{\displaystyle \sum _{a=0}^{x-1}{m}_a-0.5m_x}}\cdot e^{-{\displaystyle \sum _{a=0}^{x-1}{\pi }_a-0.5{\pi }_x}}$$

where r_a* is the growth rate at age a of the unorphaned population. For the total population:

$$P_x=B_0{e}^{-{\displaystyle \sum _{a=0}^{x-1}r{}_a-0.5r_x}}\cdot e^{-{\displaystyle \sum _{a=0}^{x-1}{m}_a-0.5m_x}}$$

Every child’s mother is alive when the child is born (to a close approximation in a few sad cases) and therefore B₀ ≡ B*₀. Thus, if one assumes no differential mortality of orphans so that this term cancels out, division of the population with living mothers by the total population produces:

$$\frac{P_x^{*}}{P_x}=B_0^{*}{e}^{-{\displaystyle \sum _{a=0}^{x-1}\left(r_a^{*}-r_a\right)-0.5\left(r_x^{*}-r_x\right)}}\cdot e^{-{\displaystyle \sum _{a=0}^{x-1}{\pi }_a-0.5{\pi }_x}}$$

The LHS term of this equation represents the actual proportion with living mothers at time t, S_x(t). On the RHS of the equation, the difference between the two growth rates equals the growth rate in the proportion not orphaned and the second exponentiated term equals the life table proportion of the population with a living mother for the period between the two surveys:

$$S_x(\tau )=e^{{\displaystyle \sum _{a=0}^{x-1}{\pi }_a-0.5{\pi }_x}}$$

Moving the stationary population term to the LHS, one gets:

$${{\displaystyle S}}_x(\tau )={{\displaystyle S}}_x(\overline{t})e^{{\displaystyle \sum _{a=0}^{x-1}{r}_a^s+0.5r_x^s}}$$

Thus, by taking the growth out of the observed proportions, one can obtain synthetic cohort or life table measures.