Estimating age-specific mortality using calibrated splines

Figures & data

Figure 1 Mean loss by age (females and males) for five-year abridged schedules derived from the test set, $\mathrm{H}\mathrm{M}{\mathrm{D}}_t$, using six different methods

Note: Mean loss is calculated using equation (17)(17) $\begin{array}{rl}{ϵ}_l(x)& \\ & :=50\times 100\times {ϵ}_d(x)+25\times 10\times {ϵ}_a(x)\\ & +25\times {ϵ}_r(x),\end{array}$(17) . Source: Based on HMD (2016) data.

Figure 2 Death rates for Italian males in 1919: six expansion methods, five-year abridgement

Note: The death rate scale is logarithmic. Source: As for Figure 1.

Figure 3 Mean loss by age (females and males) for 10-year abridged schedules derived from the test set, $\mathrm{H}\mathrm{M}{\mathrm{D}}_t$, using six different methods

Note: Mean loss is calculated using equation (17)(17) $\begin{array}{rl}{ϵ}_l(x)& \\ & :=50\times 100\times {ϵ}_d(x)+25\times 10\times {ϵ}_a(x)\\ & +25\times {ϵ}_r(x),\end{array}$(17) . Source: As for Figure 1.

Figure 4 Death rates for Italian males in 1919: six expansion methods, 10-year abridgement

Note: The death rate scale is logarithmic. Source: As for Figure 1.

Figure 5 Death rates for females and males, Iceland, 1868: five expansion methods and abridged CS rates compared with observed abridged rates

Notes: Grey area refers to 95 per cent confidence interval for abridged rates (which are shown by horizontal lines). Coale–Demeny West model life tables with $e_0=38$ for females and $e_0=30$ for males were used as standards for the Brass relational method. The death rate scale is logarithmic. Source: As for Figure 1.

Figure 6 Death rates for females and males, Iceland, 2001: five expansion methods and abridged CS rates compared with observed abridged rates

Notes: Grey area refers to 95 per cent confidence interval for abridged rates (which are shown by horizontal lines). Coale–Demeny North model life tables with $e_0=82$, for females and $e_0=78$ for males were used as standards for the Brass relational method. The death rate scale is logarithmic. Source: As for Figure 1.

Table 1 Summary statistics for four expansion methods applied to five-year abridged death rates for females and males in Iceland, 1868 and 2001

Year	Sex	$\mathrm{d}\mathrm{e}{\mathrm{v}}_B$	$\mathrm{d}\mathrm{e}{\mathrm{v}}_{HP}$	$\mathrm{d}\mathrm{e}{\mathrm{v}}_{PCLM}$	$\mathrm{d}\mathrm{e}{\mathrm{v}}_{CS}$
1868	Females	38	12	4	13
1868	Males	137	27	7	25
2001	Females	79	27	12	19
2001	Males	319	24	6	24

Note: Goodness-of-fit measure $\mathrm{d}\mathrm{e}\mathrm{v}$ is the Poisson deviance given by equation (20)(20) $\mathrm{d}\mathrm{e}\mathrm{v}=2\sum _i{N}_i\left[{\tilde{m}}_i\mathrm{l}\mathrm{o}\mathrm{g}\left({\tilde{m}}_i/m_i\right)-\left({\tilde{m}}_i-m_i\right)\right].$(20) . $B$ = Brass; $HP$ = Heligman–Pollard; $PCLM$ = Penalized Composite Link Model; $CS$ = calibrated splines.

Source: Based on HMD (2016) data.

Figure 7 Death rates for females and males, Burkina Faso 1996–2006: five expansion methods compared with observed abridged rates

Notes: Kostaki uses United Nations General model life table ($e_0=54$ for females, $e_0=49$ for males) as standard. Grey area refers to age range without reliable death rates. Abridged rates are shown by horizontal lines. The death rate scale is logarithmic. Source: Based on data from IGME (2021) and Ouedraogo (2020).

Figure 8 Death rates for females and males, Norway 1859: five expansion methods compared with censored abridged rates and complete rates

Note: Grey area refers to age range where death rates have been censored. Abridged rates are shown by horizontal lines. The death rate scale is logarithmic. Source: As for Figure 1.

Figure 9 Death rates for female and male overseas-born Australian residents, 2019: five expansion methods, abridged CS rates, and CS fitted to Australian-born residents, compared with observed abridged rates

Notes: CS-AUS refers to CS fit to data for Australian-born residents. Grey area refers to 95 per cent confidence interval for abridged rates (which are shown by horizontal lines). Brass uses the CS fit to Australian-born death rates as standard. The death rate scale is logarithmic. Source: Based on data from ABS (2019a, 2019b).

Table 2 Summary statistics for four expansion methods applied to 10-year abridged death rates for female and male overseas-born Australian residents in 2019

Sex	Intervals	$\mathrm{d}\mathrm{e}{\mathrm{v}}_B$	$\mathrm{d}\mathrm{e}{\mathrm{v}}_{HP}$	$\mathrm{d}\mathrm{e}{\mathrm{v}}_{PCLM}$	$\mathrm{d}\mathrm{e}{\mathrm{v}}_{CS}$
Females	All	1,839	21	174	30
Males	All	906	54	123	16
Females	Closed	15	18	10	11
Males	Closed	905	41	5	8

Note: Goodness-of-fit measure $\mathrm{d}\mathrm{e}\mathrm{v}$ is the Poisson deviance given by equation (20)(20) $\mathrm{d}\mathrm{e}\mathrm{v}=2\sum _i{N}_i\left[{\tilde{m}}_i\mathrm{l}\mathrm{o}\mathrm{g}\left({\tilde{m}}_i/m_i\right)-\left({\tilde{m}}_i-m_i\right)\right].$(20) . $B$ = Brass; $HP$ = Heligman–Pollard; $PCLM$ = Penalized Composite Link Model; $CS$ = calibrated splines.

Source: Based on data from ABS (2019a, 2019b).

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