Increments to life and mortality tempo

This paper introduces and develops the idea of “increments to life.” Increments to life are roughly analogous to forces of mortality: they are quantities specified for each age and time by a mathematical function of two variables that may be used to describe, analyze and model changing length of life in populations. The rationale is three-fold. First, I wanted a general mathematical representation of Bongaart’s “life extension” pill (Bongaarts and Feeney 2003) allowing for continuous variation in age and time. This is accomplished in sections 3-5, to which sections 1-2 are preliminaries. It turned out to be a good deal more difficult than I expected, partly on account of the mathematics, but mostly because it requires thinking in very unaccustomed ways. Second, I wanted a means of assessing the robustness of the Bongaarts-Feeney mortality tempo adjustment formula (Bongaarts and Feeney 2003) against variations in increments to life by age. Section 6 shows how the increments to life mathematics accomplishes this with an application to the Swedish data used in Bongaarts and Feeney (2003). In this application, at least, the Bongaarts-Feeney adjustment is robust. Third, I hoped by formulating age-variable increments to life to avoid the slight awkwardness of working with conditional rather than unconditional survival functions. This third aim has not been accomplished, but this appears to be because it was unreasonable to begin with. While it is possible to conceptualize length of life as completely described by an age-varying increments to life function, this is not consistent with the Bongaarts-Feeney mortality tempo adjustment. What seems to be needed, rather, is a model that incorporates two fundamentally different kinds of changes in mortality and length of life, one based on the familiar force of mortality function, the other based on the increments to life function. Section 7 considers heuristically what such models might look like.