Perturbation analysis of indices of lifespan variability

A number of indices have been used in recent years to calculate lifespan variation, each with different underlying properties. Although these indices are assumed to be interchangeable, little research has been conducted to show under which conditions this assumption is appropriate, or how to compare their responses to the underlying mortality schedule. We compare seven indices of lifespan variation: life disparity, the Gini coefficient, the standard deviation, the variance, Theil's index, the mean logarithmic deviation, and the inter-quartile range. We derive the sensitivity and elasticity of each index by applying Markov chain theory and matrix calculus. Using empirical French and Russian male data we compare the underlying sensitivities to mortality change under different mortality regimes in order to test under which conditions the indices might differ in their conclusions about the magnitude of lifespan variation. Finally we demonstrate how integrating these sensitivities can be used as a method of age decomposition. The result is an easily computable method for calculating the properties of this important class of longevity indices.