Shifting Logistic Model

Definition

The shifting logistic model is a parsimonious model for the force of adult mortality in which the logistic slope parameter $\beta$ is treated as constant within a population over time. Under this constraint, improvements in senescent mortality manifest not as age-uniform rate declines but as a rightward shift of the entire senescent mortality schedule to higher ages. The model was proposed and empirically validated by Bongaarts (2004).

Key Ideas

The standard logistic model. The force of mortality is decomposed into a senescent component and an age-invariant background component: $\mu(x,t) = \frac{\alpha(t)e^{\beta x}}{1 + \alpha(t)e^{\beta x}} + \gamma(t)$ where $\alpha(t)$ is a time-varying level parameter, $\beta$ is the slope (rate of increase with age), and $\gamma(t)$ is background mortality. The logistic functional form captures deceleration of mortality increase at oldest ages (unlike Gompertz), fitting ages $25$ – $109$ with $R^2 \approx 0.999$ across 14 Human Mortality Database (HMD) countries.
Empirical constancy of $\beta$ . Fitted to all Human Mortality Database countries outside Eastern Europe, $1950$ – $2000$ , the coefficient of variation of $\beta$ averages only $2.2\%$ (females) and $3.0\%$ (males) — an order of magnitude smaller than for $\alpha$ or $\gamma$ . For Swedish females, $\beta \approx 0.117$ is stable from $1850$ to $2000$ . This finding confirms earlier results by Gavrilov and Gavrilova (1991) and Thatcher (1999).
The shifting interpretation. When $\beta$ is constant, a change in $\alpha(t)$ is equivalent to a horizontal shift of the senescent mortality schedule: $\mu_s(x,t) = \mu_s(x - S(t), t_0)$ , where $S(t) = -\ln(\alpha(t)/\alpha(t_0))/\beta$ . The senescent mortality schedule at year $t$ is identical in shape to the baseline schedule but shifted $S(t)$ years toward higher ages. This reframes mortality improvement: "death is delayed" rather than "rates fell."
Senescent life expectancy $e_s(t)$ . The shift $S(t) \approx e_s(t) - e_s(t_0)$ , the change in "senescent life expectancy" — a hypothetical life expectancy (LE) computed from senescent mortality alone with no background or non-senescent causes. For Swedish females: shift of $4$ years ( $1850$ – $1950$ ) and $7$ years ( $1950$ – $2000$ ).
Background mortality $\gamma(t)$ . Background mortality declined sharply before 1975 in high-income countries and has since plateaued at low levels. It is modeled separately from the senescent component and extrapolated independently.
Rate of improvement decomposition. Under the shifting assumption, the senescent component of the rate of mortality improvement is $\rho_s(x,t) = \dot{e}_s(t) \cdot k_s(x,t)$ , where $k_s$ is the life-table aging rate for senescent mortality. This is age-varying and time-varying — contradicting Lee-Carter's assumption that $\rho(x,t)$ is constant over time.
Projection procedure. The four-step projection: (1) fit the logistic model to past data, obtaining time series $\alpha(t), \beta(t), \gamma(t)$ ; (2) fix $\beta$ at its average, refit two-parameter model; (3) extrapolate $\log[\alpha(t)]$ with a random walk with drift (the same autoregressive integrated moving average (ARIMA) used for Lee-Carter's $\kappa(t)$ ) and $\gamma(t)$ separately; (4) construct future mortality schedules as logistic curves. Two variants allow non-logistic age patterns: Variant 2 preserves the observed mortality schedule shape and shifts it forward using $S(t)$ .
Critique of Lee-Carter. Lee-Carter extrapolates each age group's death rate at its own historical exponential rate. Differences in $b(x)$ values across age groups eventually cause adjacent age groups' projections to diverge implausibly (either crossing or becoming negative). The shifting logistic model avoids this because the age structure of senescent mortality remains logistically constrained regardless of projection horizon. See Lee-Carter Model.
Relation to linear shift model. The shifting logistic model is a specific parameterization of the broader class of linear shift models (Kannisto et al. 1996) in which mortality improvement = rightward schedule shift. Wilmoth (2005) uses a similar linear shift framework to derive exact period-to-cohort life expectancy conversion rates. See Linear Shift Model.

How It Works

The mathematical equivalence between level change and schedule shift follows from the logistic form. Let $p(t) = \alpha(t)/\alpha(t_0)$ . Substituting into the senescent component:

$\mu_s(x,t) = \frac{p(t)\alpha(t_0)e^{\beta x}}{1 + p(t)\alpha(t_0)e^{\beta x}} = \frac{\alpha(t_0)e^{\beta(x + \ln(p(t))/\beta)}}{1 + \alpha(t_0)e^{\beta(x + \ln(p(t))/\beta)}}$

Defining $S(t) = -\ln(p(t))/\beta = -\ln(\alpha(t)/\alpha(t_0))/\beta$ gives $\mu_s(x,t) = \mu_s(x - S(t), t_0)$ . The proportional change in $\alpha$ corresponds exactly to a shift of $S(t)$ years. When $\alpha$ declines (mortality improves), $S(t) > 0$ and the schedule shifts right. Linear extrapolation of $\log[\alpha(t)]$ implies linear extrapolation of $S(t)$ , which in turn implies linear extrapolation of $e_s(t)$ — i.e., a linear trend in senescent life expectancy.

Why It Matters

Mortality forecasting. The model provides a structurally grounded alternative to Lee-Carter that (a) has the same extrapolation cost (one random walk for $\alpha$ , one for $\gamma$ ), (b) preserves a plausible age shape over long horizons, and (c) resolves the empirical instability in age-specific improvement rates by attributing it to the shifting + background interaction.
Conceptual reframing. The shift interpretation connects biological aging research (the Gompertz slope as a fundamental aging parameter) with demographic forecasting. If $\beta$ reflects the pace of biological aging and is truly stable, the only policy-sensitive dimension of adult mortality change is the timing of onset (captured by $\alpha$ and $S$ ).
SSA and actuarial applications. Lee-Carter is the benchmark used by Social Security Administration (SSA) and the Census Bureau. A shifting logistic alternative that matches Lee-Carter in short-run accuracy but avoids long-run age-pattern distortions is directly policy-relevant for 75-year Old-Age, Survivors, and Disability Insurance (OASDI) projections. See SSA Mortality Forecasting.

Open Questions

No formal out-of-sample forecasting comparison across populations has been conducted as of the paper. The single illustrative case (Swedish females, 1975 base, 2000 horizon) is encouraging but insufficient.
$\beta$ is assumed to be constant across time but is allowed to differ across populations. Whether $\beta$ is converging across populations (as general mortality levels converge) has not been examined.
The shifting property applies only to senescent mortality (ages 25+); the model offers no improvement for child and young adult mortality projection.
The background mortality $\gamma(t)$ extrapolation is underspecified. At current near-zero levels in developed countries the error is small, but over a 100-year horizon even small $\gamma$ misspecification could accumulate.

Sources

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