The Modal Age at Death and the Shifting Mortality Hypothesis

Abstract The modal age at death is used to study the shifting mortality scenario experienced by low mortality countries. The relations of the life table functions at the modal age are analyzed using mortality models. In the models the modal age increases over time, but there is an asymptotic approximation towards a constant number of deaths and standard deviation from the mode. The findings are compared to changes observed in populations with historical mortality data. As shown here the shifting mortality scenario is a process that might be expected if the current mortality changes maintain their pace. By focusing on the modal age at death, a new perspective on the analysis of human longevity is revealed. (ProQuest: ... denotes formulae omitted.) 1. Introduction Lexis (1878) considered that the distribution of deaths consisted of three parts: a decrease in the high number of deaths with age after birth to account for infant mortality; deaths centred around the late modal age at death (referred to hereafter as modal age at death), accounting for senescent mortality; and premature deaths that occur infrequently at young ages between the high infant mortality and senescent deaths. Life expectancy, or the mean of the life table distribution of deaths, is the indicator most frequently used to describe this distribution. In a regime with a high level of infant mortality, life expectancy will be within the age range of premature deaths, even when most deaths occur around ages zero and the modal age at death. The early stages of the epidemiological transition (Omran 1971) are characterized by a reduction in infant mortality. These changes in infant mortality have been captured very accurately with the rapid increase in life expectancy over time. However, an alternative perspective is to study the age where most of the deaths are occurring, that is the modal age at death. Currently, mortality is concentrated at older ages in most countries. Life expectancy has slowed down its rapid increase and is now moving at a similar pace as the late modal age at death. Studying the modal age at death provides an opportunity to have a different perspective of the changes in the distribution of deaths and to explain the change in mortality at older ages (Kannisto 2000, Kannisto 2001, Robine 2001, Cheung et al. 2005, Canudas-Romo 2006, Cheung and Robine 2007, Canudas-Romo and Wilmoth 2007). This study describes the important role of the modal age at death in populations experiencing mortality decline. The aim of this article is to study the shifting mortality hypothesis by assessing the changes in the late modal age at death. The shifting mortality hypothesis suggests a shifting force of mortality schedule which retains its shape over time as mortality falls (Bongaarts and Feeney 2002, 2003). Bongaarts (2005) furthers this idea by describing the shifting mortality regime as one where adult mortality is assumed to shift to higher ages over time. These characteristics of the shifting hypothesis can be observed in any of the life table functions: the hazard function, the survival function and the density function describing the distribution of deaths. The interrelations of these three curves imply that changes in one will be carried on to the others (Wilmoth 1997). Under the shifting mortality regime a shift in the density function describing the distribution of deaths implies that the hazard function declines but retains its shape, and that the survival function increases as the curve moves further to the right. In the present study, these three functions are analyzed together with the changes that occur in the modal age at death. The debate on "How long do we live?" initiated by Bongaarts and Feeney has been focused on the study of life expectancy at birth (Barbi et al. 2008, Feeney 2006, Goldstein 2006, Guillot 2006, Horiuchi 2005, Rodriguez 2006, Schoen and Canudas-Romo 2005, Vaupel 2005, Wachter 2005,Wilmoth 2005). …