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Link Functions — μ_xt vs q_xt

Two statistical frameworks

GAPC models split into two groups depending on the link function chosen for the linear predictor η_xt.

Models: LC, APC, RH

\[ D_{xt} \sim \text{Poisson}(E_{xt}\,\mu_{xt}), \qquad \log(\mu_{xt}) = \eta_{xt} \]
Symbol Meaning
\(\mu_{xt}\) Central mortality rate — expected deaths per person-year lived
\(E_{xt}\) Central exposed-to-risk (person-years lived in year \(t\) at age \(x\))
\(D_{xt}\) Observed death counts

The fitted values returned by fit.fitted_rates and fc.rates are central mortality rates \(\mu_{xt}\). They can exceed 1 at very old ages.

Models: CBD, M6, M7, M8

\[ D_{xt} \sim \text{Binomial}(E_{xt},\,q_{xt}), \qquad \text{logit}(q_{xt}) = \eta_{xt} \]
Symbol Meaning
\(q_{xt}\) Probability of dying between exact ages \(x\) and \(x+1\)
\(E_{xt}\) Initial exposed-to-risk (number of lives observed at age \(x\) at start of year \(t\))
\(D_{xt}\) Observed death counts

The fitted values returned by fit.fitted_rates and fc.rates are probabilities of death \(q_{xt} \in (0, 1)\).

Converting between μ and q

Under the Uniform Distribution of Deaths (UDD) assumption within each year of age:

\[ q_{xt} = 1 - e^{-\mu_{xt}} \qquad \Longleftrightarrow \qquad \mu_{xt} = -\log(1 - q_{xt}) \]
import numpy as np
import pystmomo as ps

data = ps.load_ew_male()
lc_fit  = ps.lc().fit(data.deaths, data.exposures,
                      ages=data.ages, years=data.years)
cbd_fit = ps.cbd().fit(data.deaths, data.exposures,
                       ages=data.ages, years=data.years)

# LC gives μ → convert to q
mu  = lc_fit.fitted_rates          # central mortality rates
q   = 1 - np.exp(-mu)             # UDD conversion to probability of death

# CBD gives q directly
q_cbd = cbd_fit.fitted_rates

# Compare forecasts on the same scale
fc_lc  = ps.forecast(lc_fit,  h=20)
fc_cbd = ps.forecast(cbd_fit, h=20)

q_lc_forecast = 1 - np.exp(-fc_lc.rates)   # now comparable to fc_cbd.rates

Exposure type matters for CBD

The Poisson group uses central exposure (person-years lived). The Binomial group ideally uses initial exposure (lives at start of year). The two are related by:

\[ E^o_{xt} \approx E^c_{xt} + \tfrac{1}{2}\,D_{xt} \]

where \(E^o\) is initial and \(E^c\) is central.

# If you only have central exposure, approximate initial exposure for CBD:
Ext_initial = data.exposures + 0.5 * data.deaths

cbd_fit = ps.cbd().fit(data.deaths, Ext_initial,
                       ages=data.ages, years=data.years)

In practice, for adult ages (50+), the difference is small and both exposure types yield similar fitted values.

Summary table

Model Link fitted_rates Exposure type
LC log μ_xt — central rate Central
APC log μ_xt — central rate Central
RH log μ_xt — central rate Central
CBD logit q_xt — prob. of death Initial (ideally)
M6 logit q_xt — prob. of death Initial (ideally)
M7 logit q_xt — prob. of death Initial (ideally)
M8 logit q_xt — prob. of death Initial (ideally)