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

TL;DR

This paper analytically derives a model linking the Weibull shape parameter to healthy life years lost, providing a new method to estimate health expectancy from extended life tables and validating it against WHO data.

Contribution

It introduces a generalized survival-mortality model incorporating a healthy life years lost parameter, extending classical life tables with an accessible Excel tool.

Findings

The HLYL parameter correlates with Weibull shape parameter changes over time.

The proposed model aligns with WHO's Healthy Life Expectancy estimates.

An Excel tool for practical application is provided.

Abstract

Matsushita et al (1992) have done 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 are illustrated on the diagram 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. The cumulative hazard of this model can express the additive process of applying a force in a material for enough time before cracking. To pass to the human data, the Weibull model and the cumulative hazard can express the additive process which disabilities and diseases cause the human organism during the life span leading to healthy life years lost. In this paper we further analytically derive a more general model of survival-mortality in which we estimate a parameter related to the Healthy Life Years Lost (HLYL) and 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 have proposed provides results similar to those provided by the World Health Organization for the Healthy Life Expectancy (HALE) and the corresponding HLYL estimates. An analytic derivation of the mathematical formulas is presented along with an easy to apply Excel program. This program is an extension of the classical life table including four more columns to estimate the cumulative mortality, the average mortality, the person life years lost and finally the HLYL parameter bx. The latest versions of this program appear in the Demographics2019 website