Relation of the Weibull Shape Parameter with the Healthy Life Years Lost Estimates: Analytical Derivation and Estimation from an Extended Life Table

Matsushita et al. (J Clin Epidemiol 45(10):1165–1175, 1992) showed an interesting finding. They observed that the shape parameter of the Weibull model presented systematic changes over time and age when applied to mortality data for males and females in Japan. They have also estimated that this parameter was smaller in the 1891–1898 data in Japan compared to the 1980 mortality data, and they presented an illustrative figure for females, where the values of the shape parameter appear close to the corresponding survival curves. However, they have not provided an analytical explanation of this behavior of the shape parameter of the Weibull model. Of course, the Weibull model is ideal to model the fatigue of materials. In particular, the cumulative hazard of this model can express the additive process of applying a force for enough time before cracking. Turning our attention to human data, the Weibull model and the cumulative hazard can express the additive process with regard to what disabilities and diseases affect the human organism during the lifespan leading to healthy life years lost. In this paper, we analytically derive a more general model of survival-mortality in which we estimate a parameter related to the Healthy Life Years Lost (HLYL), leading to the Weibull model and the corresponding shape parameter as a specific case. We have also demonstrated that the results found for the general HLYL parameter we proposed are similar to those provided by the World Health Organization for the Healthy Life Expectancy (HALE) and the corresponding HLYL estimates. An analytical derivation of the mathematical formulas utilized is presented together with an easy to apply program in Excel. This program is an extension of the classical life table which includes four more columns to estimate the cumulative mortality, the average mortality, the person life years lost, and finally the HLYL parameter b x. The latest versions of this program appear in the Demographics2019 website at http://www.asmda.es/demographics2019.html and in the Demographics2020 website at http://www.smtda.net/demographics2020.html .