# 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 $$x$$, $$\mu(x)$$, has the exponential form $\mu(x) = \alpha e^{\beta x}$

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 $$x$$, $$l(x)$$, $\mu(x) = -\frac{d}{dx} \log l(x)$ the expression for $$l(x)$$ is $l(x) = \exp\left(-\frac{\alpha}{\beta}\left(\exp(\beta x) - 1\right)\right)$ It then follows that the density of deaths at age $$x$$, $$d(x)$$ is $d(x) = \mu(x) l(x) = \alpha \exp(\beta x) \exp\left(-\frac{\alpha}{\beta}\left(\exp(\beta x) - 1\right)\right)$ which probably looks worse than it is. $$d(x)$$ tells us about the distribution of deaths by age. It is a density, so $\int d(x) = 1$ Say we observe death counts by age, $$y(x)$$, which implies a total number of deaths of $$D$$. If we multiply the total number of deaths $$D$$ by $$d(x)$$, 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 $$d(x)$$ which best describes the data we observe, $$y(x)$$.

## 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 $$\beta$$:

$\mu(x) = \beta \exp\left(\beta (x - M)\right)$ 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 $$d(x)$$ 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:

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

### End notes

This is part of an ongoing project with Josh Goldstein on modeling mortality rates for a dataset of censored death observations. Thanks to Robert Pickett who told me about the Tissov et al. paper and generally has interesting things to say about demography.

1. A good reference for this is Essential Demographic Methods, Chapter 3.