Gompertz mortality models

Introduction

The Gompertz model is one of the most well-known mortality models. It does remarkably well at explaining mortality rates at adult ages across a wide range of populations with just two parameters. This post briefly reviews the Gompertz model, highlighting the relationship between the two Gompertz parameters, $\alpha$ and $\beta$, and the implied mode age at death. I focus on the situation where we only observe death counts by age (rather than mortality rates), so estimation of the Gompertz model requires choosing $\alpha$ and $\beta$ to maximize the (log) density of deaths.

Gompertz mortality

Here are a few important equations related to the Gompertz model.¹ The Gompertz hazard (or force of mortality) at age $x$, $\mu \left(x\right)$, has the exponential form $$\mu \left(x\right)=\alpha e^{\beta x}$$

The $\alpha$ parameter captures some starting level of mortality and the $\beta$ gives the rate of mortality increase over age. Note here that $x$ refers to the starting age of analysis and not necessarily age = 0. Indeed, Gompertz models don’t do a very good job at younger ages (roughly $<40$ years).

Given the relationship between hazard rates and survivorship at age $x$, $l\left(x\right)$, $$\mu \left(x\right)=-\frac{d}{dx}\log l\left(x\right)$$ the expression for $l\left(x\right)$ is $$l\left(x\right)=\exp \left(-\frac{\alpha }{\beta }\left(\exp \left(\beta x\right)-1\right)\right)$$ It then follows that the density of deaths at age $x$, $d\left(x\right)$ is $$d\left(x\right)=\mu \left(x\right)l\left(x\right)=\alpha \exp \left(\beta x\right)\exp \left(-\frac{\alpha }{\beta }\left(\exp \left(\beta x\right)-1\right)\right)$$ which probably looks worse than it is. $d\left(x\right)$ tells us about the distribution of deaths by age. It is a density, so $$\int d\left(x\right)=1$$ Say we observe death counts by age, $y\left(x\right)$, which implies a total number of deaths of $D$. If we multiply the total number of deaths $D$ by $d\left(x\right)$, then that gives the number of deaths at age $x$. In terms of fitting a model, we want to find values for $\alpha$ and $\beta$ that correspond to the density $d\left(x\right)$ which best describes the data we observe, $y\left(x\right)$.

Parameterization in terms of the mode age

Under a Gompertz model, the mode age at death, $M$ is

$$M=\frac{1}{\beta }\log \left(\frac{\beta }{\alpha }\right)$$ Given a set of plausible mode ages, we can work out the relevant combinations of $\alpha$ and $\beta$ based on the equation above. For example, the chart belows shows all combinations of $\alpha$ and $\beta$ that result in a mode age between 60 and 90.

This chart suggests that plausible values of $\alpha$ and $\beta$ for human populations are pretty restricted. In addition, it shows the strong correlation between these two parameters: in general, the smaller the value of $\beta$, the larger the value of $\alpha$. This sort of correlation between parameters can cause issues with estimation. However, given we know the relationship between $\alpha$ and $\beta$ and the mode age, the Gompertz model can be reparameterized in terms of $M$ and $\beta$:

$$\mu \left(x\right)=\beta \exp \left(\beta (x-M)\right)$$ As this paper notes, $M$ and $\beta$ are much less correlated than $\alpha$ and $\beta$. In addition, the modal age has a much more intuitive interpretation than $\alpha$.

Implications for fitting

Given the reparameterization, we now want to find estimates for $M$ and $\beta$ such that the resulting deaths density $d\left(x\right)$ best reflects the data. If we assume that the number of deaths observed at a particular age, $y_x$, are Poisson distributed, and the total number of deaths observed is $D$, then we get the following hierarchical set up:

$$y\left(x\right)\sim \text{Poisson}\left(\lambda \right(x\left)\right)\lambda \left(x\right)=D\cdot d\left(x\right)d\left(x\right)=\mu \left(x\right)\cdot l\left(x\right)\mu \left(x\right)=\beta \exp \left(\beta (x-M)\right)l\left(x\right)=\exp \left(-\exp \left(-\beta M\right)\left(\exp \left(\beta x\right)-1\right)\right)$$ This can be fit in a Bayesian framework, with relevant priors put on $\beta$ and $M$.