Life Expectancy Decomposition

Definition

A class of demographic methods for attributing the difference between two period life expectancies — across time, populations, or subgroups — to the contributions of specific age groups (and optionally specific causes of death). The core idea is that $\Delta e_0 = e_0(t_2) - e_0(t_1)$ can be expressed as a sum (or integral) of age-specific components $c_x$ , each measuring how much of the overall change is attributable to mortality change at age $x$ .

Key Ideas

Kitagawa (1955) discrete framework: The classical rate-decomposition formula expressing a difference in summary rates as a sum of composition and rate effects; the foundation for all demographic decomposition.
Arriaga (1982) and Andreev (1982) discrete decompositions: Direct discrete formulas for $\Delta e_0$ in terms of age-specific survival and remaining life expectancy quantities. Andreev et al. (2002) extend this to a general algorithmic framework applicable to life expectancy, Gini coefficients, health expectancies, and parity-progression ratios.
Pollard (1982/1988) continuous decomposition: Expresses $\Delta e_0$ as a continuous integral over age: $\Delta e_0 = \int_0^\infty [\mu_x(t_1) - \mu_x(t_2)] T_x\, dx$ , where $T_x = [l_x(t_2)e_x(t_1) + l_x(t_1)e_x(t_2)]/2$ is a weight capturing how many future person-years are affected by the mortality differential at age $x$ . Empirically, $\mu_x$ is replaced by single-year central death rates.
Vaupel and Canudas-Romo (2003): Continuous decomposition "bouquet of formulas" extending the Pollard approach; includes period-cohort, cross-sectional age, and change-over-time formulations.
Age-specific components $c_x$ : Positive $c_x$ means mortality improved at age $x$ (contributing to $e_0$ gain); negative $c_x$ signals mortality retrogression (contributing a loss). The sum $\int c_x\, dx = \Delta e_0$ .
Cause-of-death extension: Decomposition can be applied to cause-deleted life tables to attribute $\Delta e_0$ to specific causes (see Remund et al. 2018 — A Cause-of-Death Decomposition of Young Adult Excess Mortality for an application to young adult excess mortality).

How It Works

Given two life tables at times $t_1$ and $t_2$ , the Pollard (1988) formula computes for each age $x$ :

The mortality differential: $\Delta\mu_x = \mu_x(t_1) - \mu_x(t_2)$ (positive if mortality fell)
The weight $T_x = [l_x(t_2)e_x(t_1) + l_x(t_1)e_x(t_2)]/2$ (average person-years at risk beyond $x$ )
The component: $c_x = \Delta\mu_x \times T_x$

Summing (integrating) $c_x$ over all ages recovers $\Delta e_0$ exactly. The decomposition can be applied separately by sex and for each sub-period to track how the age profile of contributions evolves over time.

Why It Matters

Identifying progress and retrogression: Decomposition makes visible which ages are driving $e_0$ gains and losses in a given period. This is not apparent from overall $e_0$ trends alone.
Spain 1971–2001: Gómez-Redondo and Boe (2005) found negative $c_x$ at young adult male ages (15–29) in the 1980s — driven by human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) and traffic accidents — which would have been invisible in aggregate $e_0$ trends showing steady overall increase.
Young adult excess mortality: Remund et al. (2018) apply cause-of-death decomposition to explain the male excess in mortality at ages 15–35, identifying transport accidents and self-harm as the main drivers.
Policy legibility: By translating mortality rate changes into years-of-life gained or lost, decomposition makes cause-specific interventions (road safety policy, HIV prevention) directly comparable in terms of demographic impact.

Open Questions

How sensitive are decomposition results to the choice of discrete vs. continuous method? (Generally small for well-behaved life tables; can differ at ages with large mortality changes.)
Can decomposition be extended to probabilistic (stochastic) life table projections?