Location–Scale Models in Demography: A Useful Re-parameterization of Mortality Models

Abstract

Several parametric mortality models have been proposed to describe the age pattern of mortality since Gompertz introduced his “law of mortality” almost two centuries ago. However, very few attempts have been made to reconcile most of these models within a single framework. In this article, we show that many mortality models used in the demographic and actuarial literature can be re-parameterized in terms of a general and flexible family of models, the family of location–scale (LS) models. These models are characterized by two parameters that have a direct demographic interpretation: the location and scale parameters, which capture the shifting and compression dynamics of mortality changes, respectively. Re-parameterizing a model in terms of the LS family has several advantages over its classic formulation. In addition to aiding parameter interpretability and comparability, the statistical estimation of the LS parameters is facilitated due to their significantly lower correlation. The latter, in turn, further improves parameter interpretability and reduces estimation bias. We show the advantages of the LS family over the typical parameterization of mortality models with two illustrations using the Human Mortality Database.

Appendices

Appendix A: Weibull and the Log–Location–Scale Family

Here, we provide a similar derivation of Sect. 2.3 to show that the Weibull model belongs to the log–location–scale (LLS) family of mortality models.

Among its various parameterizations, the Weibull model can be expressed in the form:

$$\begin{aligned} \mu (x)=ab \, (ax)^{b-1}, \quad x>0 \,, \end{aligned}$$

(A.1)

where $a>0$ and $b>0$ are parameters (Lawless 2011). From the life table functions introduced in Sect. 2.1, we can then derive the density function f(x) of the Weibull model:

$$\begin{aligned} f(x) = \mu (x)l(x) = \frac{b}{x} \, (ax)^{b} \, {\hbox {exp}} \left[ - (ax)^{b} \right] . \end{aligned}$$

(A.2)

The Weibull model can be re-parameterized in terms of the LLS family. In particular, the LLS Weibull model for the force of mortality is

$$\begin{aligned} \mu (x) = \frac{1}{c\,x} \, \mu _{\mathrm{LLS}} \left( \frac{\ln (x)-u}{c} \right) = \frac{1}{c\,x} \, {\hbox {exp}} \left( \frac{\ln (x)-u}{c} \right) , \end{aligned}$$

(A.3)

where $u \in {\mathbb {R}}$ and $c >0$ are the location and scale parameters, respectively. The corresponding LLS density function of the Weibull model can be expressed as:

$$\begin{aligned} f(x)&= \frac{1}{c\,x} \, f_{\mathrm{LLS}} \left( \frac{\ln (x)-u}{c} \right) \\ &= \frac{1}{c\,x} \, {\hbox {exp}} \left[ \frac{\ln (x)-u}{c} - {\hbox {exp}} \left( \frac{\ln (x)-u}{c} \right) \right] . \end{aligned}$$

(A.4)

Indeed, if we let the location and scale parameters be $u = - \ln (a)$ and $c=\frac{1}{b}$, and we substitute them in Eqs. (A.3) and (A.4), we obtain the classic Weibull formulas in Eqs. (A.1) and (A.2). As such, the Weibull model belongs to the LLS family of mortality models defined in Eqs. (2) and (4).

Appendix B: Location–Scale Functional form of Twelve Parametric Models of Mortality

Table 3 presents the classic, location–scale (LS) and log–location–scale (LLS) functional forms f(x), $f_{\mathrm{LS}}(\cdot )$ and $f_{\mathrm{LLS}}(\cdot )$ of the mortality models presented in Sect. 2.4.

Table 3 Mortality models belonging to the location–scale (LS) and log–location–scale (LLS) families and models closely related to them, together with their parameterization in terms of the classic, LS and LLS functional forms f(x), $f_{\mathrm{LS}}(x)$ and $f_{\mathrm{LLS}}(x)$

Appendix C: Derivation of the Best Fitting Parametric Model

The selection of the best fitting parametric model can generally be made along different metrics and criterion. In this article, the estimation of a model’s parameter is achieved by maximum likelihood (Sect. 2.5); within a Poisson framework, the Bayesian Information Criterion (BIC, Schwarz 1978) is therefore a natural metric to compare different models, as it provides a good trade-off between model parsimony and accuracy.

Specifically, within a Poisson framework, the deviance is often used as a measure of discrepancy between observed and fitted data, and it is defined as:

$$\begin{aligned} \hbox {Dev}= 2 \sum _{y} \sum _{x} \left[ D_{x,y} \, \ln \left( \frac{D_{x,y}}{\hat{D}_{x,y}} \right) - (D_{x,y} - \hat{D}_{x,y} )\right] , \end{aligned}$$

(C.1)

where $D_{x,y}$ and $\hat{D}_{x,y}$ denote the observed and fitted number of deaths at age x and year y, respectively. This is a “badness of fit” measure, as higher values correspond to worse models in terms of goodness of fit.

In the two-dimensional age and time setting, the BIC can then be computed as:

$$\begin{aligned} \hbox {BIC}=\hbox {Dev}+\ln (m n)\,\hbox {ED} \end{aligned}$$

(C.2)

where m and n are the dimensions (length) of age and time, respectively. ED denotes the effective dimension, or total number of parameters, of a model. Lower BIC values are associated with better models, and the trade-off between accuracy and parsimony is accounted for by the two components of the BIC.

Appendix D: Section 3.1: Additional Results

Here, we present some additional results corresponding to the analyses of Sect. 3.1.

Table 4 shows the BIC and rankings of the different LS models. From the table, it emerges that the Minimal Generalized Extreme–Value (MinGEV) model is the best specification for both genders in the four countries.

Table 4 BIC values (divided by 100) of nine LS models for adult females and males in Denmark, Japan, Sweden and the USA, 1960–2016

Figure 9 shows the estimated shape $\xi$ parameters of the MinGEV model for the four countries by sex during 1960–2016.

Fig. 9

Estimated shape $\xi$ parameters of the Minimal Generalized Extreme–Value model for female (left) and male (right) adults aged 30–110+ in four high-longevity countries during 1960–2016. (Color figure online)

Figure 10 shows the estimated MinGEV age-at-death distributions for the four countries by sex in 2016. From the figure, it is possible to observe that the share of premature deaths for the USA females and males is higher than for the other three countries. In addition, the smaller compression of the USA distribution of deaths compared to the other countries clearly emerges from the two graphs.

Fig. 10

Age-at-death distributions in 2016 for female (left) and male (right) adults aged 30–110+ in four high-longevity countries corresponding to the Minimal Generalized Extreme–Value model estimates

Figure 11 shows the location u and scale c estimates for six models of the LS family fitted to Swedish adult female and male mortality during 1960–2016. The parameters have been rescaled for comparability, and while here we focus on Sweden, the results are the same for the other countries.

Fig. 11

Location u and scale c rescaled estimates of six LS models for female and male adults aged 30–110+ in Sweden during 1960–2016

The figure shows that the location and scale estimates are very consistent across models: the former are always extremely close to each other, as well as the latter which are characterized by a slightly higher volatility. As such, the very same patterns of shifting and compression dynamics emerge from employing different LS models due to the similarity of the models’ estimates.

Appendix E: Decomposition of Mortality Changes into Location and Scale Effects

Here, we decompose changes in life expectancy at age 30 (${\dot{e}}_{30,t}$) into two components:

$$\begin{aligned} {\dot{e}}_{30,t} = \Delta u + \Delta c, \end{aligned}$$

(E.1)

where $\Delta u$ and $\Delta c$ are the gains in life expectancy resulting from the changes in the location (shift) and scale (compression) parameters, respectively.

Taking advantage of the findings reported in Fig. 11, namely the consistency and comparability of the location–scale parameters across different specification of the LS family, we focus on the decomposition of the Gompertz model. Specifically, we extend the methodology presented by Bergeron-Boucher et al. (2015) to the LS-like parameterization of the Gompertz model.

Equation (7) introduced the LS-like parameterization of the Gompertz model. Here, we make explicit the time dependency of the model by letting the location and scale parameters be a function of time t:

$$\begin{aligned} \mu _{x,t} = \frac{1}{c_t} \, \hbox {e}^{\frac{x-u_t}{c_t}}. \end{aligned}$$

(E.2)

Let a dot on top of a variable denote its derivative with respect to time (Vaupel and Canudas-Romo 2003). The change over time in the force of mortality (${\dot{\mu}}_{x,t}$) can be decomposed into respective components of change for the location (${\dot{u}}_{t}$) and scale (${\dot{c}}_{t}$) parameters:

$$\begin{aligned} {\dot{\mu}}_{x,t}&= {\dot{u}}_{t} \left[ - \frac{\mu _{x,t}}{c_t} \right] + {\dot{c}}_{t} \left[ - \frac{\mu _{x,t}}{c_t} \left( 1 + \frac{x-u_t}{c_t}\right) \right] \\ &= {\dot{u}}_{t} \, f_u(\mu _{x,t}) + {\dot{c}}_{t} \, f_c(\mu _{x,t}), \end{aligned}$$

(E.3)

where $f_u(\mu _{x,t})$ and $f_c(\mu _{x,t})$ are weighting function of the hazard rate for the location and scale parameters, respectively.

Similarly to the force of mortality, we can derive the time change of life expectancy. Specifically, life expectancy at age 30 can be expressed as:

$$\begin{aligned} e_{30,t} = \int _{30}^{\omega} l_{a,t} \, d_a, \end{aligned}$$

(E.4)

where $l_{a,t}$ is the survival function at age a and time t. Changes in life expectancy at age 30 (${\dot{e}}_{30,t}$) can thus be written as:

$$\begin{aligned} {\dot{e}}_{30,t} = \int _{30}^{\omega} {\dot{l}}_{a,t} \, d_a = - \int _{30}^{\omega} l_{a,t} \int _{30}^{a} {\dot{\mu}}_{x,t} \, d_x \, d_a, \end{aligned}$$

(E.5)

where ${\dot{l}}_{a,t}$ is the time derivative of the survival function. If we substitute Eq. (E.3) into Eq. (E.5), we can decompose the changes in life expectancy at age 30 (${\dot{e}}_{30,t}$) into changes due to the location and scale parameters as:

$$\begin{aligned} {\dot{e}}_{30,t} = \underbrace{{\dot{u}}_{t} \int _{30}^{\omega} l_{a,t} \int _{30}^{a} f_u(\mu _{x,t}) \, d_x \, d_a}_{\Delta u}+ \underbrace{{\dot{c}}_{t} \int _{30}^{\omega} l_{a,t} \int _{30}^{a} f_c(\mu _{x,t}) \, d_x \, d_a}_{\Delta c}. \end{aligned}$$

(E.6)

The first term in Eq. (E.6) represents the gain in life expectancy resulting from a change in location ($\Delta u$), corresponding to a shifting pattern, while the second term is the gain in life expectancy produced by a change in variability ($\Delta c$), indicating a compression pattern. These are the equivalent terms of Eq. (E.1) in the Gompertz model. Specifically, we employ discrete approximations to estimate derivatives such as those in Eq. (E.6) (see Bergeron-Boucher et al. 2015, Appendix B).