The sensitivity analysis of population projections

Hal Caswell, Woods Hole Oceanographic Institution
Nora Sánchez Gassen, University of Southampton

Any cohort-component projection can be written as an inhomogeneous (i.e., time-varying) matrix operator: n(t+1) = A[t] n(t) + b(t) where n(t) is the population vector, A[t] the population projection matrix, and b(t) the migration vector, at time t. The results of a projection depend on the time series of mortality, fertility, and immigration used to generate A[t] and b(t). Methods to extrapolate or forecast those vital rates, and to analyze the resulting projection, have become very sophisticated. However, methods to analyze the sensitivity of the results to changes in parameters are still in their infancy. Here, we present a complete sensitivity analysis for projections, using matrix calculus. This novel approach for the first time allows us to systematically analyse the sensitivity and elasticity of any projection output (e.g. population size, age distribution, dependency ratios, short-term growth rates) to changes in age-specific mortality, fertility, and migration, or to any parameters determining those schedules, and to do so for perturbations in any projection year. Sensitivity analysis provides valuable information on the effects of modifications of forecast scenarios, and on the consequences of uncertainty about the values of parameters. The effect of changes in policies and laws (e.g. introduction of immigration quotas) on population size or structure may also be estimated. We apply our methods to a projection of the population of Spain from 2012 to 2052, and identify the ages at which perturbations of mortality, fertility, and immigration will have the largest effects on population size, structure, and dependency ratios.