How do populations aggregate?

Abstract

Background: Understanding the relationship between populations at different scales plays an importantrole in many demographic analyses. Objective: We show that when a population can be partitioned into subgroups, the death rate forthe entire population can be written as the weighted harmonic mean of the death rates ineach subgroup, where the weights are given by the numbers of deaths in each subgroup.This decomposition can be generalized to other types of occurrence-exposure rates. Usingdifferent weights, the death rate for the entire population can also be expressed as anarithmetic mean of the death rates in each subgroup. Contribution: We use these relationships as a starting point for investigating how demographers can correctly aggregate rates across non-overlapping subgroups. Our analysis reveals conceptual links between classic demographic models and length-biased sampling. To illustrate howthe harmonic mean can suggest new interpretations of demographic relationships, wepresent as an application a new expression for the frailty of the dying, given a standarddemographic frailty model.