Do intersections of mortality-rate and survival functions have significance?

Abstract

Common points of intersections have frequently been reported among members of families of linearized mortality-rate and survival functions. A general condition for the existence of such intersections is derived. It is shown that a common point of intersection between straight-line functions exists if and only if the intercepts of the functions are linearly related to their slopes. This slope-intercept condition is applied to a didactic model to illustrate its generality and to three models, the Gompertz-Makeham, the Weibull, and the logistic, which are often used in the analysis of mortality data. The slope-intercept condition for the Gompertz-Makeham mortality-rate model proves to be the well-known Strehler-Mildvan correlation. Families of mortality-rate functions or of the corresponding survival functions but not both may display common points of intersection. Differences between the ages at which survival functions intersect and those at which the associated mortality-rate functions intersect are calculated to be of the order of magnitude of 10 to 20 years. Survival function intersections lie close to the limit of human life span but often arise in consequence of unsupported extrapolations of data obtained at younger ages. These and other results lead to the conclusion that, in themselves, the intersections of survival and mortality-rate functions are not of great importance. To the extent that significance can be attributed to the intersections, it lies in the existence of linear relationships between their slopes and intercepts.