Mortality Laws

From the time of De Moivre (1725), suggestions have been made as to the law of mortality as a mathematical formula, of which the most famous is perhaps that of Gompertz (1825). Since John Graunt (1620–1674) life tables had been constructed empirically. These life tables represented mortality over the whole human lifespan, and it was natural to ask if the mathematical functions defined by the life table could be described by simple laws, as had been so successfully achieved in natural philosophy. The choice of function that should be described by a law of mortality (a mathematical formula depending on age) has varied, as different authors have considered μx , qx , or mx (see International Actuarial Notation) (or something else) to be most suitable; in modern terminology we would model the parameter that is most natural to estimate, given the underlying probabilistic model for the data. It should be noted that, since qx and mx are all μ x+ 2 to first order, it makes little difference at younger ages whether the formula is used to represent the functions μx , qx , or mx (see below for advanced ages). Since laws of mortality are attempts to summarize empirical observations, they have been intimately linked with the statistical techniques of analyzing mortality data, and, nowadays, would be described as parametric forms for quantities appearing in statistical models. In some cases, these models may have a physiological basis and may attempt to model the ageing process. A typical modern pattern of mortality can be divided into three periods. The first period is the mortality of infants, namely, a rapid decrease of mortality during the first few years of life. The second period contains the so-called ‘accident hump’ where the deaths are mainly due to accidents, for example, in cars or motor bicycles. The third period is the almost geometric increase of mortality with age (the rate of increase slowing down after age 80) – senescent mortality. We will now describe the most important laws of mortality that have been proposed and their features. De Moivre (1725) [7] used a 1-parameter formula: