Explaining the differences in life expectancies
Age contributions · observed HMD period life tables
What it shows
The piece takes a difference in life expectancy between any two of 394 observed both-sex tables and carries it apart into the contributions of single years of age. Replacing each table’s mortality with the other’s, from both ends of the age axis at once, produces two profiles that bracket the contribution between them.
The method
The piece runs Arriaga’s (1984) decomposition of a life-expectancy difference in the symmetric Approach-III form of Murthy (2005): the mean of the decomposition taken in each direction, ½[Δ(1→2) − Δ(2→1)]. Every population is an observed Human Mortality Database period life table; nothing on the page is modelled or projected.
How to read it
The upper panel is a survivorship plane: age runs left to right from 0 to 110, and the vertical axis counts survivors out of 100 000 born. The rust curve is the reference population, the blue curve the comparison. Because life expectancy is the area under a survivorship curve, the shaded region between them is the difference being explained.
Two replacement fronts advance through that region, one from age 0 upward and one from age 110 downward. Behind each front, the reference population’s mortality has been replaced by the comparison’s and the life table rebuilt, so each front drags a hybrid curve away from rust and toward blue. A labelled vertical line marks the age each front has reached, crossing both panels. Both hybrids arrive at the comparison curve together.
The lower panel is the contribution profile: what each single year of age adds to the difference, in years, with the age scale labelled beneath it. Two thin lines trace what each front alone credits an age: solid for the front from above, dashed for the front from below. Each bar is their midpoint, drawn once both fronts have passed that age, so bars fill in from the middle outward. Where the two lines separate, the directions disagree; Direction under Methods explains why.
Blue marks ages where the comparison population does better, rust ages where the reference does. Both are named at the right of the lower panel, comparison above and reference below, so the direction never has to be remembered. A profile that crosses zero marks a pair where each population leads at some ages.
Either population can be anything in the list. Choose the same year in two countries and the differential is geographic; choose the same country in two years and it is temporal: the same arithmetic, now decomposing a country’s own improvement over time. Swapping reference and comparison flips every contribution’s sign and leaves the magnitudes untouched.
The visual
Data & sources
| Quantity | Provenance | Source |
|---|---|---|
| q(x), probability of dying in the interval | ● observed | HMD period life tables, both sexes, 1×1 |
| a(x), fraction of the interval lived by those who die in it | ● observed | Same tables (the only other column read) |
| ℓ(x), L(x), T(x), e(x) | computed | Rebuilt here from q(x) and a(x); HMD’s own columns are not used |
| Contributions by age | computed | Arriaga (1984), Approach-III of Murthy (2005) |
Forty countries on sixteen years — 1751, 1850, 1900, 1913, 1921, 1938, 1947, 1950, 1960, 1990, 2000, 2001, 2003, 2010, 2019, 2023 — taken wherever the database holds them. 2019 and 2023 fall on either side of the COVID-19 pandemic. Coverage is ragged: one country at 1751, eight at 1850, eleven at 1900, twelve at 1913, fourteen at 1921, fifteen at 1938, nineteen at 1947, twenty-four at 1950, thirty-two at 1960, thirty-seven at 1990, thirty-eight at 2000, thirty-nine at 2001, forty at 2003, forty at 2010, thirty-seven at 2019, twenty-seven at 2023. 394 tables, and any two can be decomposed against each other. Every country in the piece is present at 2003 and at 2010. Russia, Belarus, and Ukraine appear from 1960 through 2010 and not at 2019. Germany appears from 1990 through 2019: the reunified series starts in 1990. Croatia starts in 2001 and the Republic of Korea in 2003. Australia has no 2023.
A single year is a single year. In small or early populations one epidemic moves a whole life table, so an early differential can record a bad year rather than a lasting gap: Finland’s 1900 life expectancy sits 4.7 years below its own 1895–1905 median and Iceland’s 3.2 below. Iceland 1913 sits 5.4 years above its 1908–1918 median and Italy 1913 3.7 above; Spain 1938 sits 3.4 years below its 1933–1943 median. Japan 1947 sits 6.8 years below the median of its first six published years, Bulgaria 1947 4.6 below, and Slovakia 1950 4.2 below. These are observed tables and are shown unsmoothed, since the decomposition is exact for whatever pair it is given and averaging would trade that for a steadier-looking line.
What it is — and isn’t
The decomposition is an exact accounting identity. Contributions sum to the difference in life expectancy to machine precision, because consecutive terms in Arriaga’s formula cancel; the closure error printed under the panel is floating-point noise.
What the arithmetic assigns to an age is an accounting share, and it says nothing about what would follow if mortality at that age changed. The two populations differ in every other respect as well, and the decomposition holds none of it constant. A temporal comparison carries the same caution: one country in two years decomposes two period tables (the mortality prevailing in 1950 against that prevailing in 2019) and describes no one who lived through the interval.
Methods
Life tables
Two columns are read from HMD, and only two: the probability of dying q(x) and the fraction of the interval lived by those who die in it, a(x). Every other quantity on this page (ℓ(x), d(x), L(x), T(x), e(x), m(x), and therefore every contribution) is recomputed here from that pair.
Survivorship follows ℓ(x+1) = ℓ(x)[1 − q(x)] from a radix of 100 000, with L(x) = ℓ(x+1) + a(x)d(x) in closed intervals and L(ω) = ℓ(ω)a(ω) in the open one; T(x) accumulates L from the top down, and life expectancy is T(0)/ℓ(0). HMD publishes ℓ(x) and e(x) too, and they are not used: reading them would give the two observed tables and nothing else. Rebuilding from q(x) and a(x) is what makes a hybrid table possible, because a hybrid takes q and a from one population below the front and the other above it, and has no published columns of its own.
Decomposition
Superscript 1 marks the reference population and 2 the comparison; n is the width of the age interval, one year throughout; ℓ₀ is the radix.
Arriaga’s (1984) original decomposition — Murthy’s Approach-I — splits the contribution of the interval [x, x+n) into a direct effect,
\[DE(x) = \frac{\ell^1(x)}{\ell_0} \left[ \frac{L^2(x)}{\ell^2(x)} - \frac{L^1(x)}{\ell^1(x)} \right]\]
and an indirect-plus-interaction effect,
\[IE(x) = \frac{T^2(x+n)}{\ell_0} \left[ \frac{\ell^1(x)}{\ell^2(x)} - \frac{\ell^1(x+n)}{\ell^2(x+n)} \right]\]
with the open interval carrying the direct term only, T in place of L. That pair is directional: its total for each age equals Chandra Sekar’s effect-interaction forwarded, which assigns each interaction to the youngest age group involved in producing it. Approach-II defers the same interaction to the oldest.
The piece does not draw that. It draws Approach-III, the symmetric average of Arriaga’s original and its mirror, ½[Δ(1→2) − Δ(2→1)]. For e₀, Murthy (2005) gives it in closed form; its per-age total, his equation (1.5), is
\[\Delta(x) = \tfrac{1}{2}(e^2_x - e^1_x)(\ell^2_x + \ell^1_x) - \tfrac{1}{2}(e^2_{x+n} - e^1_{x+n})(\ell^2_{x+n} + \ell^1_{x+n})\]
Here ℓ carries a radix of 1 (the life tables above use 100 000). Each age’s contribution is one term minus the term for the next age, so adding them across the age range cancels every term but the first, and that first term is the difference in life expectancy itself. The identity is exact by construction. Murthy shows it equals the United Nations (1985) and Andreev, Pressat and Pollard (1982) results, while driving the interaction term to nearly zero. Murthy states Arriaga’s formulae in ℓ and e terms; the L and T rendering above is equivalent, and is the form implemented here.
Direction
The two fronts are not a second method. Replacing mortality one age at a time and rebuilding the table reproduces Arriaga exactly: run from the oldest age down and the trace is Arriaga’s original; run from age 0 up and it is the mirror, negated. The fronts are the formula made visible, and their midpoint is Approach-III.
Direction never changes the total. It changes only which age is credited with the interaction, and the disagreement collapses as two populations converge: 0.607 years at age 0 for Portugal and Sweden in 1950, 0.020 years at age 0 for the United States and Japan in 2019, 0.001 years at age 85 by 2019 for Portugal and Sweden.
Measures
Life expectancy at birth counts years lived at every age, and it is the quantity the panel decomposes. It is a level, not a spread. Life disparity e† weights each death by the decedent’s own remaining life expectancy and so measures dispersion in the age at death; the two answer different questions, and a contribution profile for one is not a profile for the other.
Controls
Reference population and Comparison population choose the two life tables. The decomposition always describes comparison minus reference, so Swap reverses the sign of every contribution and leaves the magnitudes untouched. Changing either population rebuilds the tables and restarts the sweep.
Pause and Restart drive the two replacement fronts. Front speed sets how fast they advance, from 4 to 60 ages per second, and changes no quantity: the contributions are computed in full before the animation begins, so the ending is identical at every setting. Reset returns the populations and the speed to their defaults. ↓ PNG exports the current frame with its provenance caption. A reader whose system asks for reduced motion gets the completed sweep on load.
There is no seed and no random draw anywhere in the piece: the same selection always produces the same numbers.
Assumptions
Both tables share an age grid (single years, 0 to 110 with the last open) and a radix. The hybrid tables assume mortality at each age can be exchanged independently: the decomposition’s foundational assumption. Both sexes: HMD’s own combined life table, built from male and female deaths and exposures pooled. Averaging the two single-sex tables would give a different quantity. Figures are therefore not comparable to female- or male-only life expectancies, which run above and below them.
Simplifications
Contributions are single years of age, so the profile carries the full weight of infant mortality in one bar: on the pair the page opens with, Sweden in 1751 against Sweden in 2019, 12.635 of the 44.622-year difference sits at age 0 alone, more than four times the largest contribution at any other age. The panel scales to the largest contribution above age 0 and labels the infant bar with its true value instead of compressing everything else.
No uncertainty is shown. HMD tables are treated as exact, which is defensible for these countries and would not be for data requiring completeness correction.
References
Arriaga, E. E. (1984). Measuring and explaining the change in life expectancies. Demography, 21(1), 83–96.
Murthy, P. K. (2005). A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth. Demographic Research, 12(7), 141–172. https://doi.org/10.4054/DemRes.2005.12.7
Pollard, J. H. (1982). The expectation of life and its relationship to mortality. Journal of the Institute of Actuaries, 109(2), 225–240.
Preston, S. H., Heuveline, P., & Guillot, M. (2001). Demography: Measuring and Modeling Population Processes. Blackwell.
Human Mortality Database. Max Planck Institute for Demographic Research (Germany), University of California, Berkeley (USA), and French Institute for Demographic Studies (France). www.mortality.org
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