Perturbation Analysis of Longevity Using Matrix Calculus

Hal Caswell, Woods Hole Oceanographic Institution

Perturbation analysis derives the effect of changes in one or more parameters on some dependent variable. The first applications to longevity were due to Keyfitz, Pollard, and Vaupel, who derived the sensitivity of life expectancy to changes in mortality. These results can be extended by describing demography with age- or stage-classified matrix models and applying Markov chain theory and matrix calculus. Individuals are classified by age, maturity, developmental stage, health status or other measures. Many indices of longevity can be calculated from the resulting model. Matrix calculus gives the sensitivity and elasticity of these indices with respect to stage-specific vital rates. I present the sensitivity and elasticity of life expectancy, variance in lifespan, life lost (e-dagger), and the distribution of age or stage at death, to changes in age- or stage-specific mortality. The sensitivities and elasticities are easily computed for arbitrarily complicated life cycles or individual transition structures.