Symmetries between life lived and left in finite stationary populations

Abstract

BACKGROUNDThe Brouard-Carey equality describes the symmetries between the distribution of life lived and life left in stationary populations. This result was formally proved for populations of infinite size and continuous time, and a subsequent attempt to prove it for populations of finite size is invalid. OBJECTIVEWe attempt to provide a formal mathematical proof of the Brouard-Carey equality for finite stationary populations. CONCLUSIONSThe symmetries between life lived and life left in finite stationary populations can only be proved if time is explicitly discretized. The proof is more complex than in a continuoustime framework, but it conforms with the kinds of data usually available to researchers. CONTRIBUTIONThe main contribution of this paper is to offer a complete and formal proof of the symmetries between life lived and life left for finite stationary populations in a discrete-time framework. This result is a useful tool for the study of human and non-human populations when the assumption of stationarity is acceptable, especially when subject ages are unknown, but individuals are followed-up until death.

Time-to-death patterns in markers of age and dependency

“…In retrospect (after the death of a cohort), the thanatological age structure of a cohort at a given past point in time is a fixed characteristic. Since a closed birth cohort is akin to a stationary population, 2 one may be tempted to assume that since chronological and thanatological age structures are symmetrical in stationary populations (Brouard 1989, Vaupel 2009, Villavicencio and Riffe 2016) that the patterns of demographic characteristics within cohorts might also demonstrate an analogous kind of symmetry. This is not so; even in the case of stationary populations or extinct cohorts, the age profiles of other demographic characteristics in the population are decidedly different when viewed chronologically versus thanatologically.…”

Life lived and left: Estimating age-specific survival in stable populations with unknown ages

“…The main result in (1*) that describes the relationship between life lived and life left in stable populations has already been proved for the particular case of stationary populations with growth rate r = 0 (Brouard 1989;Vaupel 2009;Villavicencio and Riffe 2016). To our knowledge, Vaupel was the first to generalize it to stable populations in an unpublished manuscript from 2013.…”

“…Vaupel (2009) proved that in stationary populations of infinite size and in continuous time, the probability that a randomly selected individual is age a equals the probability that an individual has exactly a time left until death. Villavicencio and Riffe (2016) suggested an alternative proof for empirical and finite stationary populations in a discrete-time framework.…”

A unified framework of demographic time

“…2 in which individuals dying at different ages but in the same time period are grouped together. To our knowledge, the TPD diagram has only appeared once in the literature, as a didactic aid in a proof of symmetry between chronological and thanatological age structure in discrete stationary populations (Villavicencio and Riffe 2016). TPD diagrams may also be useful to arrange events or durations that are logically aligned (or may only be aligned) by time of termination.…”