Risk Modelling and Survival Analysis · Estimation procedures for lifetime distributions

Estimating Mortality Rates from Exposed to Risk

Updated 11 October 2026 · Fact-checked

Exposed to risk is the total time (or number of lives) observed at risk of dying. Central exposed to risk E^c_x is total time lived, giving μ̂ = d ÷ E^c_x. Initial exposed to risk E_x counts lives at the start of the year, giving q̂ = d ÷ E_x. Convert between them using E_x ≈ E^c_x + ½d.

Understand Estimating Mortality Rates from Exposed to Risk

A mortality investigation observes lives for a period and records who dies. To turn deaths into a rate, you must divide by how much risk was faced. That measure is the exposed to risk.

The simplest model is the two-state model: a life is either alive or dead. The only transition is alive to dead, at rate μ_x (the force of mortality). If you assume μ_x is constant over the age range, the number of deaths d depends on the total time lived by all lives, not on how many lives there were.

Central exposed to risk, E^c_x, is the total time spent alive and under observation in the age range, summed over all lives. A life who dies after 0.4 years contributes 0.4. A life who survives the year contributes 1. It matches the rate μ_x, so μ̂_x = d_x ÷ E^c_x. This is the maximum likelihood estimate when μ is constant.

Initial exposed to risk, E_x, is the number of lives at the start of the observation year, with each life who dies during the year counted as if they had been observed for the full year. It matches the probability q_x, so q̂_x = d_x ÷ E_x. Compared with central exposure, it adds back the unexpired part of the year for those who died.

The two are linked. If deaths occur on average mid-year, E_x ≈ E^c_x + ½d_x. Also q_x ≈ μ_x for small rates, and under constant μ, q_x = 1 − e^(−μ). For a large group, D_x (deaths) is modelled as binomial(E_x, q_x) with initial exposure, or approximately Poisson(E^c_x × μ_x) with central exposure. The Poisson model gives mean E^c μ and variance E^c μ. The binomial gives mean E q and variance E q(1 − q). The estimator μ̂ has variance approximately μ ÷ E^c.

When exposure is computed from census data, you count lives at set dates and approximate exposure by averaging, for example using the census method with the trapezium rule. Exact-age data is covered under related topics.

Key rules to remember

Central exposed to risk estimate of force of mortality: μ̂_x = d_x ÷ E^c_x; Valid as the MLE when μ is constant over the age and period. E^c_x is the total time under observation.
Initial exposed to risk estimate of q_x: q̂_x = d_x ÷ E_x; E_x counts each life from the start of the year, including those who die or leave part-way if the deaths are treated as full-year lives.
Link between exposures: E_x ≈ E^c_x + ½ d_x; Assumes deaths occur on average half-way through the year. For a death at fraction t of the year, the exact addition is (1 − t).
Poisson approximation for deaths: D_x ~ Poisson(E^c_x μ_x); Var(μ̂_x) ≈ μ_x ÷ E^c_x; Mean and variance of D_x are both E^c_x μ_x. Good when μ is constant.
Binomial model for deaths: D_x ~ Binomial(E_x, q_x); Var(q̂_x) = q_x(1 − q_x) ÷ E_x; Needs initial exposure and independent lives with the same q_x.
Constant force relationship: q_x = 1 − e^(−μ); Holds for one year when μ is constant over the year. q ≈ μ if μ is small.
Census approximation of exposure: E^c_x ≈ ∫ P_x(t) dt ≈ average of census counts × period length; P_x(t) is the number of lives aged x last birthday at time t. The trapezium rule is the usual approximation.

How to solve Estimating Mortality Rates from Exposed to Risk questions

Use this method for most questions on estimating mortality rates from exposure.

1Identify what is asked: a rate μ̂ or a probability q̂. This decides central or initial exposure.
2Read the data and note the observation period, the ages, and the number of deaths d.
3Build the exposure. For central exposure, add up the time each life was observed. For initial exposure, count lives at the start, with those who died counted for a full year.
4If one exposure is given and the other is needed, convert using E_x ≈ E^c_x + ½d, or use exact death times if given.
5Compute μ̂ = d ÷ E^c or q̂ = d ÷ E, stating the assumption (constant force, or deaths mid-year).
6If asked for variance or a confidence interval, use Var(μ̂) ≈ μ̂ ÷ E^c or q̂(1 − q̂) ÷ E, then use the normal approximation.
7Check that the answer is reasonable: q and μ should be close and below 1 for ordinary ages.
8State the result with units (per year) and any assumptions used.

Quickest way: Match the rate to the exposure

When to use it: Use in MCQs and short written parts where exposure is already given or easy to count.

If the question says rate, force or μ, divide deaths by central exposure.
If the question says probability or q, divide deaths by initial exposure.
Need the other exposure? Add or subtract half the deaths: E = E^c + ½d.
For variance, use μ̂ ÷ E^c for the Poisson case, or q̂(1 − q̂) ÷ E for the binomial case.
Sanity check: q̂ slightly below μ̂ for small rates.

Common mistakes in Estimating Mortality Rates from Exposed to Risk

Using initial exposure to estimate μ, or central exposure to estimate q.
Both are just 'exposure' and the labels look alike.
Fix: Link rate with central and probability with initial. Say it aloud before dividing.
Adding ½d the wrong way round: E^c = E + ½d.
Students forget that initial exposure counts deaths for the full year, so it is the larger.
Fix: E_x is larger than E^c_x because the dead are counted for the full year. So E = E^c + ½d.
Counting a life who died part-way as a full year in central exposure.
Mixing up the two definitions.
Fix: In central exposure, a life who dies at time t contributes only t. In initial exposure, they contribute 1.
Forgetting to include lives who enter or leave mid-year at their actual observation times.
Students count heads at the start instead of tracking time.
Fix: Record entry and exit times, and sum the time each life is actually observed in the age range.
Using the variance of q̂ for μ̂, or vice versa.
Both look like d divided by an exposure.
Fix: Poisson with central exposure gives Var(μ̂) ≈ μ̂ ÷ E^c. Binomial gives q̂(1 − q̂) ÷ E.
Not stating the assumption that μ is constant or deaths occur mid-year.
Students focus on the arithmetic only.
Fix: Write the assumption in one line. Examiners award marks for it.

Worked examples

Example 1

In a one-year investigation, 2,000 lives aged 60 were observed from the start of the year. During the year, 30 died and the remaining lives survived the year. Assume deaths occur on average half-way through the year. Estimate (a) q_60 and (b) μ_60 using central exposure.

Show the solution

(a) Initial exposure E = 2,000 (all lives observed from the start; deaths counted for the full year).
q̂ = d ÷ E = 30 ÷ 2,000 = 0.015.
(b) Central exposure E^c = E − ½d = 2,000 − 15 = 1,985 years.
μ̂ = d ÷ E^c = 30 ÷ 1,985 = 0.015113.

Answer: q̂_60 = 0.015 and μ̂_60 ≈ 0.01511.

Example 2

For a group of lives aged 50, the central exposed to risk was 4,000 years and 20 deaths were observed. Assume a constant force of mortality. (a) Estimate μ. (b) Give the approximate standard error of μ̂. (c) Estimate q_50 using q = 1 − e^(−μ).

Show the solution

(a) μ̂ = d ÷ E^c = 20 ÷ 4,000 = 0.005.
(b) Var(μ̂) ≈ μ̂ ÷ E^c = 0.005 ÷ 4,000 = 0.00000125.
Standard error = √0.00000125 = 0.001118.
(c) q̂ = 1 − e^(−0.005) = 1 − 0.995012 = 0.004988.

Answer: μ̂ = 0.005, standard error ≈ 0.00112, and q̂_50 ≈ 0.004988.

Exam tips

Write the definition of the exposure you use. It earns easy marks and avoids confusion.
In MCQs, the tell-tale word is 'rate' for central exposure and 'probability' for initial exposure.
Always state the assumption: constant μ for the Poisson/MLE result, or deaths mid-year for E = E^c + ½d.
If death times are given, use exact times rather than the mid-year approximation.
For the computer-based paper, show the formula d ÷ E in code or Excel and label the units.

Practice questions from Estimation procedures for lifetime distributions

Estimating Mortality Rates from Exposed to Risk: frequently asked questions

What is the difference between central and initial exposed to risk?

Central exposed to risk is the total time lived by all lives while under observation. Initial exposed to risk is the number of lives at the start of the period, with deaths counted as if observed for the full period. Central goes with rates, initial goes with probabilities.

How do I convert initial exposure to central exposure?

If deaths occur on average half-way through the period, E^c ≈ E − ½d. Equivalently, E ≈ E^c + ½d. If exact death times are known, subtract the unexpired fraction for each death instead.

Why does the Poisson model use central exposure?

Under a constant force of mortality, the number of deaths depends on the total time at risk. The mean is E^c × μ. This makes central exposure the natural measure for the Poisson model.

Is μ̂ equal to q̂?

Not exactly. They are close when rates are small, with q = 1 − e^(−μ) under a constant force. At older ages, the difference becomes more visible.