Estimating Mortality Force with MLE and Exposed-to-Risk Data

Introduction

In our last lesson, we focused on estimating $q_{x}$ , the probability of dying within a one-year period, using the actuarial estimate. This gave us a practical tool for working with life tables. Today, we will shift our perspective from a discrete annual probability to a continuous, instantaneous rate of mortality. This quantity, known as the force of mortality and denoted by $μ_{x}$ , is a cornerstone of modern survival modeling.

For someone with your statistical background, you can think of $μ_{x}$ as the hazard rate from survival analysis. It represents the instantaneous potential for death at a precise age $x$ , given survival to that age. This lesson will show you how to estimate $μ_{x}$ from observed data using a method you are very familiar with: maximum likelihood estimation (MLE). By the end of this lesson, you will be able to estimate the force of mortality using deaths data and the crucial concept of 'exposed to risk'.

The Force of Mortality and Central Exposed to Risk

First, let's formalize the concept of the force of mortality. It is defined as the limit of the probability of dying in a very short interval of time, divided by the length of that interval.

[PDF] BS3b Statistical Lifetime-Models

This section from a University of Oxford lecture note provides a precise mathematical definition of the force of mortality.

Please read the section "Force of mortality". Focus on the definition given in equation (2): the formal definition. This shows how \mu_t is an instantaneous conditional probability rate. Also, note the relationship derived in equation (3), which shows how the survival function can be recovered by integrating the force of mortality.

The key insight is that $μ_{x}$ is a rate, not a probability. It can exceed 1 and provides a more fundamental description of the mortality process than the annual probability $q_{x}$ . In practice, modeling $μ_{x}$ is often simpler and more flexible than modeling $q_{x}$ , especially at very high ages.

To estimate this rate from data, we need two key pieces of information for each age interval (e.g., from age $x$ to $x + 1$ ):

$d_{x}$ : The number of deaths observed in that age interval.
$E_{x}^{c}$ : The total time all individuals were observed to be alive and "at risk" of dying during that age interval. This is known as the central exposed to risk.

This contrasts with the initial exposed to risk ( $E_{x}$ ) from our last lesson, which was a simple count of people at the start of the year. The central exposed to risk is a measure of person-time (e.g., person-years) and is a more precise measure of exposure.

Let's look at a practical example of how $E_{x}^{c}$ is calculated.

A whiteboard illustration showing the calculation of central exposed to risk for five individuals over a one-year observation period (Jan 1, 2021 to Jan 1, 2022). The table tracks their date of birth, entry date into the study, and exit date.

Let's break down the exposure calculation for a few individuals in this example to see how it works. We are calculating the exposure during the year 2021. The "age" is defined as age at last birthday.

Individual A:
- DOB: 01/Mar/2000. On 01/Jan/2021, A is 20 years old. A turns 21 on 01/Mar/2021.
- Enters: 01/Jan/2021. Exits: 01/July/2021.
- Exposure while age 20: From 01/Jan/2021 to 01/Mar/2021 = 2 months.
- Exposure while age 21: From 01/Mar/2021 to 01/July/2021 = 4 months.
- Total exposure for A ( $E_{x}$ ) is $2 + 4 = 6$ months.
Individual B:
- DOB: 01/May/2000. On 01/Jan/2021, B is 20 years old. B turns 21 on 01/May/2021.
- Enters: 01/Jan/2021. No exit date, so B is observed for the whole period.
- Exposure while age 20: From 01/Jan/2021 to 01/May/2021 = 4 months.
- Exposure while age 21: From 01/May/2021 to 01/Jan/2022 = 8 months.
- Total exposure for B ( $E_{x}$ ) is $4 + 8 = 12$ months.

The table in the image calculates the total exposure for each individual and then splits it by age. The columns $E_{20}^{c}$ and $E_{21}^{c}$ show the total exposure time (in months) for all individuals when they were age 20 and 21, respectively. The total central exposed to risk for age 20 is the sum of the $E_{20}^{c}$ column, and similarly for age 21.

Maximum Likelihood Estimation of the Force of Mortality

With the concepts of $μ_{x}$ and $E_{x}^{c}$ established, we can now formulate the estimation problem. Our goal is to find the value of $μ_{x}$ that maximizes the likelihood of observing our data (the $d_{x}$ deaths and $E_{x}^{c}$ exposure).

A standard and powerful approach is to model the number of deaths, $D_{x}$ , using a Poisson distribution. This is natural for your background, as the Poisson distribution models the number of events occurring in a fixed interval of time or space, given a known average rate.

[PDF] A Practitioner's Guide to Statistical Mortality Graduation - SOA.org

This guide provides an excellent statistical framing for mortality estimation, focusing on the Poisson model.

Please read the section "A Statistical Approach to Graduation". Pay close attention to: The Poisson assumption for the number of deaths, D_x. Here, m_x is the central death rate, which we treat as our estimate for \mu_x. The likelihood and log-likelihood functions for the model parameters. This demonstrates how to construct the likelihood from the Poisson probability mass function.

Let's walk through the derivation of the MLE. We make a simplifying assumption that the force of mortality is constant over the year of age $x$ , and we denote this constant rate by $μ$ .

The Model: The number of deaths $D_{x}$ in the age interval is assumed to follow a Poisson distribution with a rate parameter $λ$ . This rate is the product of the constant force of mortality $μ$ and the total time at risk $E_{x}^{c}$ .
$D_{x} \sim Poisson (λ) where λ = μ \cdot E_{x}^{c}$
The Likelihood: The probability of observing $d_{x}$ deaths is given by the Poisson PMF:
$P (D_{x} = d_{x}) = \frac{e ^{- μ E_{x}^{c}} ( μ E _{x}^{c} ) ^{d_{x}}}{d _{x} !}$
This is our likelihood function $L (μ)$ .
The Log-Likelihood: As usual, it's easier to work with the log-likelihood, $l (μ)$ :
$l (μ) = ln (L (μ)) = - μ E_{x}^{c} + d_{x} ln (μ E_{x}^{c}) - ln (d_{x}!)$ $l (μ) = - μ E_{x}^{c} + d_{x} ln (μ) + d_{x} ln (E_{x}^{c}) - ln (d_{x}!)$
Maximization: To find the MLE, we take the derivative with respect to $μ$ and set it to zero.
$\frac{d l}{d μ} = - E_{x}^{c} + \frac{d _{x}}{μ} = 0$
The Estimator: Solving for $μ$ gives us the maximum likelihood estimator, $μ^$ :
$μ^= \frac{d _{x}}{E _{x}^{c}}$

This remarkably simple and intuitive result states that the MLE for the force of mortality is the total number of observed deaths divided by the total time those individuals were exposed to the risk of death. This quantity is also known as the central death rate, $m_{x}$ .

For a deeper dive into the theory, the same estimator can be derived under the more fundamental assumption of a continuous-time, two-state (alive/dead) Markov process, without needing the Poisson approximation.

[PDF] BS3b Statistical Lifetime-Models

This resource derives the MLE from first principles and then confirms it using the Poisson model.

Scan through these sections to see the different ways the same result is reached. In section 3.5, the MLE is derived directly from the likelihood of continuous lifetimes, showing \hat{\mu} = d/\ell, where \ell is the total exposed to risk. In section 6.2, the same estimator is derived using the Poisson model, confirming that \hat{\mu}_{x+1/2} = D_x / E_x^c. In section 10.4.1, this is generalized for multi-state Markov models, showing that any transition rate is estimated as the number of transitions divided by the time spent in the initial state. This connects the concept to your strong background in Markov processes.

Variance of the Estimator

Under the Poisson model, we know that the variance of a Poisson random variable is equal to its mean.

Var (D_{x}) = μ E_{x}^{c}

Using the delta method or asymptotic properties of MLEs, we can find the variance of our estimator $μ^$ .

Var (μ^) = Var (\frac{D _{x}}{E _{x}^{c}}) = \frac{1}{( E _{x}^{c} ) ^{2}} Var (D_{x}) = \frac{μ E _{x}^{c}}{( E _{x}^{c} ) ^{2}} = \frac{μ}{E _{x}^{c}}

In practice, we estimate this variance by plugging in our MLE for $μ$ :

Var (μ^) \approx \frac{μ ^}{E _{x}^{c}} = \frac{d _{x} / E _{x}^{c}}{E _{x}^{c}} = \frac{d _{x}}{( E _{x}^{c} ) ^{2}}

Conclusion

This lesson bridged the gap from the discrete probability of death, $q_{x}$ , to the continuous rate, or force, of mortality, $μ_{x}$ . We established a robust statistical method for its estimation based on the principle of maximum likelihood.

Key Takeaways:

The force of mortality, $μ_{x}$ , is the instantaneous rate of death at age $x$ , equivalent to the hazard rate in survival analysis.
To estimate $μ_{x}$ , we need the number of deaths $d_{x}$ and the central exposed to risk $E_{x}^{c}$ , which is the total person-time of exposure in the age interval.
Assuming a Poisson distribution for the number of deaths, $D_{x} \sim Poisson (μ E_{x}^{c})$ , the maximum likelihood estimator for a constant force of mortality over an age interval is:
$μ^= \frac{d _{x}}{E _{x}^{c}}$
This estimator is also known as the central death rate, $m_{x}$ .
The estimated variance of this estimator is $\frac{d _{x}}{( E _{x}^{c} ) ^{2}}$ , which is crucial for constructing confidence intervals.

Preview of the Next Lesson

The age-specific estimates $μ^_{x}$ we've just calculated are often called "crude" rates. When calculated for each age, they can fluctuate wildly due to random variation, especially at ages with little data. In our next module, we will begin exploring graduation, the process of smoothing these crude rates to produce a model that is both true to the underlying data and sufficiently smooth to be useful for practical actuarial work. We will start by fitting parametric formulas, like the Gompertz and Makeham laws, to these crude rates.

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