Which causes account for a difference in life expectancy
Cause contributions · reconstructed HCD short list
What it shows
The piece takes a difference in both-sex period life expectancy, constructed here from the Human Cause-of-death Database (HCD) reconstructed short list, between any two of 795 observed country-years and carries it apart into sixteen cause groups. Each cause’s bar is the sum of its row in a nineteen-band grid.
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)]. Each age’s contribution is then split across the sixteen causes in proportion to the cause-specific difference in death rates at that age. When those cause differences cancel and the all-cause rates agree to machine precision, the split uses the discrete sensitivity of life expectancy to that age’s rate instead of a 0/0 ratio. Every schedule is a reconstructed HCD short-list year; nothing on the page is modelled or projected. Life expectancy itself is built here from the sixteen cause rates and HCD exposures, not from the Human Mortality Database (HMD) table.
How to read it
The cause-total panel is the answer: sixteen signed bars, one for each reconstructed short-list group. Rust means the reference country-year is ahead on that cause; blue means the comparison is. The scale under the bars is in years. The Total column is that row’s Approach-III sum. Once both fronts have finished, those sixteen totals add to the constructed difference: the table’s All causes × Total cell, and both sidebar cards.
Two replacement fronts advance from opposite ends of the age axis. Behind each front the reference schedule’s all-cause mortality has been replaced by the comparison’s and the life table rebuilt, so each sidebar total is that hybrid’s gain in life expectancy. A labelled vertical line marks the band each front has reached. Both hybrids arrive at the comparison life expectancy together.
The age panel is those same sixteen rows, opened across nineteen abridged bands from age 0 to 85 and over. Rust marks cells where the reference country-year is ahead, blue where the comparison is; darker cells are larger contributions. The legend uses that same ahead. Age labels on the grid thin when the window is narrow; the table under the figure keeps every band.
Bars and cells reveal Approach-III contributions only after both fronts have passed an age, so a partial bar is not yet the gap. Where the two directional age rows in the table separate, the directions disagree; Direction under Methods explains why.
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 change over time. Swapping reference and comparison flips every contribution’s sign and leaves the magnitudes untouched.
The visual
Data & sources
| Quantity | Provenance | Source |
|---|---|---|
| Cause-specific deaths, 16 groups | ● observed, reconstructed | HCD reconstructed short list, both sexes (sex = 3), after ill-defined redistribution |
| Exposures | ● observed | HCD exposures, both sexes |
| Cause-specific death rates | computed | Deaths divided by exposures at extract; published HCD rates (per 1,000,000) are a check |
| All-cause death rate | computed | Sum of the sixteen shipped cause rates, not a separately shipped S000 |
| ℓ(x), L(x), T(x), e(x) | computed | Abridged Chiang life table from those rates; HMD columns are not used |
| Contributions by age and cause | computed | Arriaga (1984), Approach-III of Murthy (2005), then the hybrid cause split |
Eighteen countries on every reconstructed year the series holds: 795 country-years, and any two can be decomposed against each other. That is the complete reconstructed collection published with HMD. Russia ends in 2014. Ukraine ends in 2013. England and Wales (GBRTENW) is not the United Kingdom. Germany’s reconstructed series ends in 2020. Moldova and Romania take exposures from country specialists rather than from HMD; they stay in the picker and are excluded from the HMD life-expectancy check.
These are reconstructed schedules 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.
The default pair is the United States 2023 against France 2023, a difference of 4.364 years, and 2023 is the last year both report: the French reconstructed series ends there while the American runs to 2024. Three of the sixteen groups carry three-quarters of that difference, and neoplasms runs against it, so the United States leads on that group by 0.567 years while trailing overall. Widening the range is one click away: Russia 1994 against Spain 2019 is 20.338 years and among the widest the collection holds, though Russia 1994 is the minimum of the post-Soviet mortality crisis rather than a typical year for the country, and that series ends in 2014.
What it is — and isn’t
The decomposition is an exact accounting identity on the life expectancy constructed here. Contributions sum to that constructed difference to machine precision, because consecutive terms in Arriaga’s age formula cancel; the closure error printed under the panel is floating-point noise.
What the arithmetic assigns to a cause is an accounting share, and it says nothing about what would follow if that cause were eliminated. The two country-years differ in every other respect as well, and the decomposition holds none of those other differences constant. A temporal comparison carries the same caution: one country in two years decomposes two period schedules (the cause-specific mortality prevailing in 1960 against that prevailing in 2019) and describes no one who lived through the interval.
Methods
Life tables
HCD publishes reconstructed deaths and rates, not life expectancy. Two inputs are taken from that file, and only two: the sixteen cause-specific rates and the exposures used to form them. Every other quantity on this page (ℓ(x), d(x), L(x), T(x), e(x), the all-cause rate, and therefore every contribution) is computed here.
The table is abridged on nineteen bands that every file fills: age 0, 1–4, five-year groups through 80–84, and 85 and over. The all-cause rate at each age is the sum of the sixteen shipped cause rates. Chiang’s conversion gives the probability of dying in a closed interval, \(n q_x = n \cdot {}_n M_x / (1 + (n - {}_n a_x) \cdot {}_n M_x)\), with \(q = 1\) in the open interval.
Mean age at death in the interval uses Andreev–Kingkade (HMD Methods Protocol v6, Table 3) at age 0, evaluated on the both-sex infant rate and then averaged across the male and female formulae. That is not HMD’s death-weighted combination of the two sexes, which this file does not have. Age 1–4 takes \({}_4 a_1 = 1.5\) (the HMD convention). Five-year bands take \(n/2\). The open interval takes \(a = 1/m\) and \(q = 1\).
Survivorship follows \(\ell(x+n) = \ell(x)[1 - q(x)]\) from a radix of 100 000. Person-years in a closed interval are \(n\cdot\ell(x+n) + {}_n a_x\cdot d_x\). The single-year person-years formula omits the width \(n\) and would collapse \(e_0\) to about twenty years on these bands. The open interval is \(L = \ell\cdot a\). \(T\) accumulates \(L\) from the top, and life expectancy is \(T(0)/\ell(0)\).
HMD’s published \(\ell\) and \(e\) are not read: reading them would give two observed tables and nothing else. Rebuilding from rates is what makes a hybrid table possible, because a hybrid takes the comparison’s all-cause rate on bands the front has passed and the reference’s on the rest, and has no published columns of its own.
Opening the last band at 85 rather than 110 is the largest expected gap against HMD’s both-sex \(e_0\). That residual is a check.
Decomposition
Superscript 1 marks the reference country-year and 2 the comparison; n is the width of the age band (1, 4, or 5; the open interval has no finite n); 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 band, so adding them across the nineteen bands 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.
Each age’s total is then split across the sixteen causes. When the all-cause rates differ by at least \(\varepsilon = 10^{-12}\),
\[\Delta(x,i) = \Delta^{\mathrm{III}}(x) \cdot \frac{m^2_{x,i} - m^1_{x,i}}{m^2_x - m^1_x}.\]
When they do not — the case where causes move in opposite directions and cancel — the piece uses the discrete sensitivity of \(e_0\) to a one-band perturbation of size \(\delta = 10^{-8}\) on the midpoint schedule, \(\Delta(x,i) = s_x \cdot (m^2_{x,i} - m^1_{x,i})\). The cause totals on the bar panel are \(\Delta(i) = \sum_x \Delta(x,i)\).
Andreev, Shkolnikov and Begun (2002) is cited as literature, not as the estimator drawn here.
Direction
The two fronts are not a second method. Replacing all-cause mortality one band at a time and rebuilding the table reproduces Arriaga exactly: run from 85 and over down and the age row 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. Each sidebar readout is that hybrid’s gain in life expectancy, not a running sum of the cause cells.
Direction never changes the total. It changes only which age is credited with the interaction. On these nineteen bands the disagreement is usually largest in the open interval, and it grows with the size of the difference: 0.080 years at 85 and over for the default pair, the United States and France in 2023, against 1.544 years at 85 and over for a far wider pair, Russia 1994 and Spain 2019. The United States and Japan in 2019 sit between them at 0.256 years. It shrinks further as two populations converge: 0.003 years at age 0 and 0.016 years at 85 and over for Estonia and Sweden in 2024. The cause cells stay on Approach-III; there is no unique cause-split of either single direction.
Reconstruction
The short list is the reconstructed series, not the original classification. HCD maps each country’s earlier revisions onto the International Classification of Diseases, 10th Revision (ICD-10) 2016 list and redistributes ill-defined causes (S017) before the sixteen groups are shipped (Pechholdová, Camarda, Meslé and Vallin 2017; Barbieri, Meslé, Poniakina and colleagues 2026). This page never draws S017 as a bar.
Reconstruction is a statistical assignment, not a new medical opinion on each death. A country-year whose original certification was heavy in ill-defined codes has more of its mortality moved by that step.
Cause list
The sixteen groups are HCD’s reconstructed short list (S001–S016). The canvas keeps every group as a row: a total bar, rust or blue for who is ahead, and the nineteen age cells that compose it. The table is the same grid.
| Code | Family | Label on the page | What it holds, briefly |
|---|---|---|---|
| S001 | Infectious | Infectious diseases | Certain infectious diseases |
| S002 | Neoplasms | Neoplasms | All neoplasms |
| S003 | Other | Blood and immune | Blood and blood-forming organs |
| S004 | Other | Endocrine and metabolic | Endocrine, nutritional and metabolic diseases |
| S005 | Other | Mental and behavioural | Mental and behavioural disorders |
| S006 | Other | Nervous system | Nervous system and sense organs |
| S007 | Circulatory | Heart diseases | Heart diseases |
| S008 | Circulatory | Cerebrovascular | Cerebrovascular diseases |
| S009 | Circulatory | Other circulatory | Other and unspecified circulatory disorders |
| S010 | Respiratory | Acute respiratory | Acute respiratory diseases, including influenza, pneumonia, and COVID-19 |
| S011 | Respiratory | Other respiratory | Other respiratory diseases |
| S012 | Other | Digestive | Diseases of the digestive system |
| S013 | Other | Skin and musculoskeletal | Skin, subcutaneous, musculoskeletal and connective tissue |
| S014 | Other | Genitourinary and maternal | Genitourinary diseases and complications of pregnancy, childbirth and the puerperium |
| S015 | Other | Perinatal and congenital | Perinatal conditions, congenital anomalies, and sudden infant death |
| S016 | External | External causes | External causes |
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 country-years. 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 1 to 12 age bands 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 pair to the United States 2023 and France 2023 and restores the default speed. ↓ 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 schedules share the same nineteen-band grid and the same radix. The hybrid tables assume mortality at each age — and, given the proportional split, the cause composition at that age — can be exchanged independently: the decomposition’s foundational assumption.
Both sexes: HCD’s published combined series (sex = 3). 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. The infant \(a_0\) averages the Andreev–Kingkade male and female formulae on the same both-sex \(m_0\), rather than a death-weighted mix of the two sexes.
Simplifications
The age drawing is a signed heatmap of the same sixteen-by-nineteen cells as the table, not a second bar chart. Colour marks direction and magnitude at each age; cause identity lives in the row. Cause totals use the same rust and blue as the heatmap, with the years printed beside them rather than on the mark.
The open interval begins at 85. HMD’s both-sex table continues to 110 with published \(a(x)\). The HCD–HMD residual in \(e_0\) is therefore expected. Among 686 overlapping comparisons it is largest at Japan 2020: 0.580 years (HCD 85.290, HMD 84.710). The piece does not close that gap by borrowing HMD \(q\) or \(a\).
No uncertainty is shown. Reconstructed HCD counts are treated as exact, which would not be defensible for data that still required 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
Pechholdová, M., Camarda, C. G., Meslé, F., & Vallin, J. (2017). Reconstructing long-term coherent cause-of-death series, a necessary step for analyzing trends. European Journal of Population, 33(5), 629–650. https://doi.org/10.1007/s10680-017-9453-1
Barbieri, M., Meslé, F., Poniakina, S., et al. (2026). The Human Cause-of-death Database. Scientific Data. https://doi.org/10.1038/s41597-026-07683-5
Andreev, E. M., Shkolnikov, V. M., & Begun, A. Z. (2002). Algorithm for decomposition of differences between aggregate demographic measures and its application to life expectancies, healthy life expectancies, parity-progression ratios and total fertility rates. Demographic Research, 7(14), 499–522.
Human Cause-of-death Database. Max Planck Institute for Demographic Research (Germany), University of California, Berkeley (USA), and French Institute for Demographic Studies (France). www.mortality.org/Data/HCD
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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