Bongaarts 2004 — Long-Range Trends in Adult Mortality Models and Projection Methods

Summary

Bongaarts proposes and validates the shifting logistic model for adult mortality — a logistic force-of-mortality specification in which the slope parameter $\beta$ is empirically constant within populations over time, so that mortality improvement manifests as a rightward shift of the entire senescent mortality schedule rather than a uniform decline in rates. Using this model he then develops a new projection method as an alternative to Lee-Carter, arguing that Lee-Carter's assumption of time-invariant age-specific improvement rates is violated empirically and leads to implausible long-run age patterns.

Key Claims

• The three-parameter logistic model $\mu(x,t) = \alpha(t)e^{\beta x}/[1 + \alpha(t)e^{\beta x}] + \gamma(t)$ fits adult mortality (ages 25–109) extremely well across $14$ Human Mortality Database (HMD) countries, 1950–2000, for both sexes ($R^2 \approx 0.999$).
• The slope parameter $\beta$ is nearly constant within each population over time (coefficient of variation: $2.2\%$ for females, $3.0\%$ for males) — an order of magnitude less variable than the level parameter $\alpha$ or background mortality $\gamma$. This constancy extends back to Swedish data from 1850.
• Because $\beta$ is constant, changes in $\alpha$ imply a pure rightward shift of the senescent mortality schedule by $S(t) = -\ln(\alpha(t)/\alpha(t_0))/\beta$ years. The shift approximates the change in senescent life expectancy: $S(t) \approx \Delta e_s(t)$. For Swedish females the shift was $4$ years from 1850–1950 and $7$ years from 1950–2000.
• Background mortality $\gamma(t)$ declined sharply before 1975 and has since reached a low-level plateau in most high-income countries.
• Lee-Carter assumes the proportional rate of mortality improvement $\rho(x,t)$ is constant over time at each age. Under the shifting model, $\rho_s(x,t) = \dot{e}_s(t) \cdot k_s(x,t)$, which varies with both time and age; this confirms the empirically documented instability of $\rho(x,t)$ across decades.
• Short-run projections (1975→2000 for Swedish females): the new method and Lee-Carter perform nearly identically (projected $e(25)$ of $57.7$ vs. observed $57.5$; Lee-Carter $58.0$).
• Long-run projections (2000→2100): the two methods diverge substantially in age pattern; Lee-Carter produces little improvement at highest ages and implausibly large improvements at ages 60–80 because it freezes different b(x) values per age group.

Concepts Introduced or Extended

• Shifting Logistic Model
• Lee-Carter Model
• Period Mortality
• Linear Shift Model

Entities Mentioned

• John Bongaarts
• John R. Wilmoth
• Griffith Feeney
• James W. Vaupel

Quotes

"Instead of interpreting mortality as rising or falling, the schedule of the force of senescent mortality can be viewed as shifting to higher or lower ages over time."

"The Lee-Carter method may produce implausible results in projections over many decades. Some investigators have attempted to address this limitation by adding complexity and additional parameters to the Lee-Carter model. The alternative approach proposed next provides a simpler solution."

My Take

The $\beta$-constancy finding is the paper's most important empirical contribution and has been independently confirmed (Gavrilov and Gavrilova 1991, Thatcher 1999). The shifting interpretation is conceptually elegant and connects to Bongaarts and Feeney (2002, 2003) and Wilmoth's (2005) linear shift model. The projection method is well-motivated theoretically: projecting $\log[\alpha(t)]$ linearly is the same autoregressive integrated moving average (ARIMA) / random-walk-with-drift as Lee-Carter's $\kappa(t)$, so the two methods share the same underlying extrapolation, but the shifting structure constrains the age pattern in a logistically plausible way.

The main limitation is that the forecasting comparison is purely illustrative — one application (Swedish females), no cross-validated forecast accuracy study across populations or time horizons. The theoretical advantage for long-run projections (plausible age shape preservation) is compelling, but the empirical case for dominance over Lee-Carter awaits the "further research" the conclusion calls for. The background mortality $\gamma(t)$ extrapolation is also underspecified; Bongaarts notes it is unstable and defers it, which matters little now but could accumulate error in very long forecasts.