# Gompertz mortality models: the relationship between alpha, beta and mode age

## 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.