Concept Note - Abridged Life Tables Years of Life Lost and the Counterfactual Path

Summary

Concept note, no raw source. An abridged life table turns age-specific death rates $m_x$ into survivors $l_x$, person-years $L_x$ and life expectancy $e_x$. Years of life lost (YLL) from an event such as the COVID-19 pandemic is the excess in deaths times remaining life expectancy, or the gap between observed life expectancy and the expectancy on a counterfactual “path we were on” extrapolated from pre-event trends (e.g. Lee-Carter). Citations are to the standard literature; numbers below are a toy computed here.

Overview

Motivating PyMC example: E111 “The path we were on: years of life Europe lost to COVID, and whether it came back” (Eurostat deaths, population and life expectancy by age, 2000-2024; abridged life table checked against Eurostat; census breaks measured by age; Lee-Carter by SVD vs a Bayesian negative-binomial version; pooled sensitivities with zero-sum rows; an unidentified level shown failing) - see the gallery. This note gives the formal background; the forecasting model is in Girosi King 2007 - Lee-Carter Model Estimation and Forecasting.

Main Content

Abridged life-table functions (Chiang 1984; Preston et al. 2001)

For age intervals $[x,x+n)$: death rate $m_x=D_x/P_x$ (deaths over person-years); fraction of the interval lived by those dying $f_x$ (so $a_x=n{f}_x$); death probability $q_x=\frac{n{m}_x}{1+n(1-f_x)m_x}$ (open interval $q=1$); survivors $l_{x+n}=l_x(1-q_x)$, deaths $d_x=l_x{q}_x$; person-years $L_x=n(l_{x+n}+f_x{d}_x)$ and, for the open interval, $L_{\omega }=l_{\omega }/m_{\omega }$; $T_x={\sum }_{y\ge x}{L}_y$; life expectancy $e_x=T_x/l_x$.

Years of life lost and excess mortality

Excess deaths $=D_x^{\text{obs}}-D_x^{\text{exp}}$ against an expected (counterfactual) count. YLL $={\sum }_x(D_x^{\text{obs}}-D_x^{\text{exp}})e_x^{\ast }$ with $e_x^{\ast }$ remaining life expectancy at the age of death (period or reference table). Alternative summary: life-expectancy gap $e_0^{\text{cf}}-e_0^{\text{obs}}$, decomposed by age with Arriaga’s (1984) method. The counterfactual defines what excess means: a pre-event average, a linear extrapolation of ${\gamma }_t$ in Lee-Carter (random walk with drift), or a Bayesian forecast with predictive uncertainty.

Why the path matters. Mortality has fallen for decades, so comparing 2020 with a 2015-2019 mean understates the excess. A Lee-Carter or RWD counterfactual extrapolates the decline, but Girosi and King (2007) warn that linear forecasts make age profiles fan out (Girosi King 2007 - Universal Increasing Nonsmoothness of Age Profiles) and that Lee-Carter is biased when the real covariance is not aligned with the drift (Girosi King 2007 - Lee-Carter as a Random Walk with Drift); short horizons (a few years) are far less affected. Practical pitfalls (E111): census breaks in population denominators, open-interval treatment, level identification in the Bayesian model, and whether mortality “came back” to the path (a recovery question, not only a loss).

Examples

Toy abridged table and a one-year shock

Setup: five age groups $[0,1),[1,5),[5,50),[50,80),80+$ with $f=(0.1,0.4,0.5,0.5,\cdot )$ and $m=(0.004,0.0004,0.0015,0.012,0.09)$. Baseline life table: $q=(0.00399,0.0016,0.0653,0.3051,1)$, $l=(100000,99601,99442,92949,64592)$, $e_0=79.07$, $e_{50}=33.15$, $e_{80}=11.11$. Shock: death rates of the last two groups rise 20%. New $e_0=76.75$, a life-expectancy loss of $2.33$ years. With toy populations $(10000,40000,450000,300000,100000)$, excess deaths are $720$ (50-79) and $1800$ (80+), total $2520$. A crude YLL using remaining expectancy at the start of the group is $720\cdot 33.15+1800\cdot 11.11\approx 43,900$ years; using expectancy at the age of death (nearer 65 than 50 in the 50-79 group) would be smaller, so this is an upper bound for that group.

import numpy as np
def life_table(m, n=np.array([1,4,45,30,np.inf]), f=np.array([0.1,0.4,0.5,0.5,0.0]), l0=1e5):
    q = np.where(np.isinf(n), 1.0, n*m/(1+n*(1-f)*m)); l = np.zeros(len(m)); l[0] = l0
    for i in range(len(m)-1): l[i+1] = l[i]*(1-q[i])
    d = l*q; L = np.zeros(len(m))
    for i in range(len(m)-1): L[i] = n[i]*(l[i+1] + f[i]*d[i])
    L[-1] = l[-1]/m[-1]; return l, d, np.cumsum(L[::-1])[::-1]/l      # e_x

PyMC-style sketch of the counterfactual (untested).

with pm.Model():
    # Lee-Carter negative binomial on pre-event years 2000-2019, forecast 2020-2024 as RWD
    mu = exposure * pm.math.exp(alpha[:, None] + beta[:, None] * gamma[None, :])
    pm.NegativeBinomial("deaths", mu=mu, alpha=phi, observed=deaths_pre)
    gamma_future = pm.Deterministic("gamma_f", gamma[-1] + drift*steps + pm.math.cumsum(pm.Normal("xi", 0, sigma_rw, shape=H)))
# excess_draw = observed_deaths - exposure*exp(alpha + beta*gamma_future); YLL per draw with a life_table(...)

Connections

Survival Analysis (life tables are discrete-time hazards), Forecast Evaluation and Backtesting, Hierarchical Models (pooling countries), Factor Analysis and PPCA, Girosi King 2007 - Second-Stage Reestimation and Wilmoth Estimators (Poisson/NB likelihood on counts).

Relevance to marketing measurement

“Excess relative to the path we were on” is the same logic as counterfactual lift in a time-series causal model: the quality of the pre-period forecast and its uncertainty determine whether an estimated effect is distinguishable from drift.

See Also

Girosi King 2007 - Understanding the Lee-Carter Method - Overview