Central Death Rates Under Constant Force

Why the Conversion Depends on the Mortality Assumption

A central death rate and an annual death probability describe mortality over the same age interval, but they use different denominators. The annual probability is the proportion of people alive at age \(x\) who die before age \(x+1\). The central death rate divides deaths during that interval by the total time lived in the interval by those exposed to death. To convert one measure into the other, we need to know how mortality is distributed within the year.

Under a constant force of mortality, the force is the same at every duration during the year. That assumption leads to a particularly direct conversion: the central death rate equals the constant force, and the annual probability follows from the exponential survival model. This is different from the conversion under a uniform distribution of deaths, which was used in the earlier material on converting central death rates to annual probabilities.

Definition: For the age interval from \(x\) to \(x+1\), the central death rate \(m_x\) is the number of deaths during the interval divided by the total exposure to risk during the interval. In a life table, if \(d_x=l_x-l_{x+1}\) and \(L_x\) is the total person-years lived between ages \(x\) and \(x+1\), then \(m_x=d_x/L_x\). The annual death probability is \(q_x={}_1q_x\).

The life-table expression uses exposure \(L_x\) in person-years, not simply the number of people alive at the start of the year. In a model, the expected deaths and expected exposure replace their observed counterparts. The distinction matters: an observed central rate is calculated from data, while treating it as a constant force is an additional mortality assumption.

Deriving the Constant-Force Conversion

Assume that, conditional on being alive at exact age \(x\), the force of mortality is a constant \(\mu\) throughout the following year. The probability of surviving \(t\) years from age \(x\) is then \(e^{-\mu t}\), for \(0\leq t\leq1\). Consequently, the probability of dying during the year is \(q_x=1-e^{-\mu}\).

To connect this probability to the central rate, let \(l_x\) be the number alive at age \(x\), and consider the expected exposure in the year. At duration \(t\), the expected number alive is \(l_x e^{-\mu t}\); integrating this number over the year gives the exposure. The expected number of deaths is \(l_xq_x\). Therefore:

L_x=l_x\int_0^1 e^{-\mu t}\,dt =l_x\frac{1-e^{-\mu}}{\mu}, \qquad d_x=l_x(1-e^{-\mu}).

Taking the ratio cancels both the starting population and the probability of death:

m_x=\frac{d_x}{L_x} =\frac{l_x(1-e^{-\mu})}{l_x(1-e^{-\mu})/\mu} =\mu.

This is why the conversion is not obtained by setting \(m_x=q_x\): the central rate divides by person-years of exposure, while \(q_x\) divides by the number alive at the start. Under constant force, the exposure calculation makes the central rate equal to the force, not the death probability.

Theorem: If the force of mortality is constant at \(\mu\) throughout the age interval \([x,x+1)\), then \(m_x=\mu\), \(q_x=1-e^{-m_x}\), and \(m_x=-\ln(1-q_x)\). These formulas presume a constant force across the interval; a central rate calculated from data does not, by itself, establish that assumption.

If \(\mu=0\), nobody dies and \(q_x=m_x=0\). The exposure integral in the derivation then has value \(l_x\), as expected; the displayed quotient involving division by \(\mu\) is interpreted by its limit as \(\mu\) approaches zero.

Contrast With the Uniform-Deaths Conversion

Under the uniform distribution of deaths assumption (UDD) for a year, the number alive declines linearly between \(l_x\) and \(l_{x+1}\). Since \(l_{x+1}=l_x(1-q_x)\), the number alive at duration \(t\) is \(l_x(1-tq_x)\). Its integral over the year is \(L_x=l_x(1-q_x/2)\), while deaths are \(l_xq_x\). Thus the UDD central rate and its inverse conversion are:

m_x=\frac{q_x}{1-q_x/2}, \qquad q_x=\frac{m_x}{1+m_x/2}.

The two methods use different within-year survival patterns. Constant force gives exponential decline in the number alive; UDD gives linear decline. Their conversions are consequently not interchangeable. For small central rates their results are close, but a precise calculation should use the assumption specified in the problem.

Assumption within the year	Central rate from annual probability	Annual probability from central rate
Constant force	\(m_x=-\ln(1-q_x)\)	\(q_x=1-e^{-m_x}\)
Uniform distribution of deaths	\(m_x=q_x/(1-q_x/2)\)	\(q_x=m_x/(1+m_x/2)\)

Worked Conversions and Life-Table Checks

In each example, calculations retain full precision until the final reported values. Probabilities and rates are rounded to four decimal places; expected life-table counts and exposures are shown to two decimal places where appropriate.

Worked Example: Convert a Central Rate Under Constant Force

A hypothetical life table has central death rate \(m_x=0.25\) for an age interval. Assume a constant force of mortality throughout the year. Find the annual death probability and compare it with the UDD conversion.

Method. Under constant force, the force is \(\mu=m_x=0.25\). The annual probability is therefore:

q_x=1-e^{-m_x} =1-e^{-0.25} =1-0.7788007831 =0.2211992169.

Rounded to four decimal places, the constant-force result is \(q_x=0.2212\). For comparison, applying UDD to the same central rate gives:

q_x^{\text{UDD}} =\frac{0.25}{1+0.25/2} =\frac{0.25}{1.125} =0.2222222222 \approx 0.2222.

The two answers differ because the within-year assumptions differ. The common mistake is to use the UDD inverse conversion merely because a central rate is given. The stated constant-force assumption selects the exponential conversion, \(1-e^{-m_x}\), not \(m_x/(1+m_x/2)\).

Check. Under the constant-force result, \(1-q_x=0.7788007831=e^{-0.25}\), so \(-\ln(1-q_x)=0.25=m_x\), as required.

Worked Example: Recover the Central Rate From an Annual Probability

For a hypothetical age interval, the annual death probability is \(q_x=0.30\). Calculate the central death rate under constant force, then calculate the rate that would be implied by UDD.

Method under constant force. The survival probability across the year is \(1-q_x=0.70\). Inverting \(q_x=1-e^{-m_x}\) gives:

m_x=-\ln(1-q_x) =-\ln(0.70) =0.3566749439 \approx 0.3567.

UDD comparison. Under UDD, the central rate is:

m_x^{\text{UDD}} =\frac{q_x}{1-q_x/2} =\frac{0.30}{1-0.30/2} =\frac{0.30}{0.85} =0.3529411765 \approx 0.3529.

The inputs \(q_x=0.30\) are the same, but the inferred central rates are not. The error to avoid is taking the negative logarithm when UDD is specified, or applying the UDD formula when constant force is specified. The model assumption determines the inverse.

Check. The constant-force rate gives \(e^{-0.3566749439}=0.70\), so the one-year death probability is \(1-0.70=0.30\).

Worked Example: Complete a One-Year Life-Table Step

At exact age \(x\), a hypothetical group has \(l_x=800\) survivors. Its central death rate for the next year is \(m_x=0.15\), and mortality follows a constant force during that year. Find \(q_x\), the expected number of deaths, the expected number surviving to \(x+1\), and the exposure \(L_x\).

Step 1: Convert the rate. The constant force is \(\mu=0.15\), so:

q_x=1-e^{-0.15} =1-0.8607079764 =0.1392920236.

Thus \(q_x\approx0.1393\) to four decimal places.

Step 2: Calculate deaths and survivors. The expected deaths are \(d_x=l_xq_x\):

d_x=800(0.1392920236)=111.4336189 \approx111.43, \qquad l_{x+1}=800-111.4336189=688.5663811 \approx688.57.

Fractional expected counts are appropriate in a model calculation; they do not assert that a realized group must contain a fractional person.

Step 3: Calculate and check exposure. Under constant force, \(L_x=l_xq_x/m_x\), so:

L_x=\frac{111.4336189}{0.15} =742.8907927 \approx742.89\text{ person-years}.

Using unrounded values, \(d_x/L_x=111.4336189/742.8907927=0.15=m_x\). This check confirms that the deaths and exposure are consistent with the assumed central rate.

Common mistake avoided. Do not multiply \(800\) by \(0.15\) to estimate deaths. The central rate is deaths per exposure, not a probability. First convert it to \(q_x\), then multiply the starting lives by \(q_x\).

Using the Assumption Correctly

The constant-force conversion is appropriate when the problem explicitly specifies constant force over the age interval, or when the model being used imposes that assumption. It is also useful for checking a life-table construction: if \(m_x\) is specified as a constant force, the one-year survival factor must be \(e^{-m_x}\), and the annual death probability must be \(1-e^{-m_x}\).

An empirical central rate alone does not tell us whether deaths were uniform, whether the force was constant, or whether some other pattern held within the year. It is computed from observed deaths and exposure. If a question asks for a conversion but does not identify an assumption, the intended method must be established from the stated model or context rather than silently treating the rate as a probability.

A useful way to organize a calculation is to identify the given quantity, name the within-year assumption, apply the matching conversion, and then check the result against survival or exposure. In particular, the constant-force identity \(m_x=\mu\) is a model result; it is not a universal identity between a central rate and a hazard at every age or duration.

Identify the measure given.
Determine whether the input is \(m_x\), \(q_x\), an exposure, or a number of deaths.

State the mortality assumption.
For constant force use \(q_x=1-e^{-m_x}\); for UDD use \(q_x=m_x/(1+m_x/2)\).

Check the result.
Under constant force verify \(1-q_x=e^{-m_x}\), or verify \(m_x=d_x/L_x\) using unrounded values.

Key takeaway: Under a constant force of mortality over one year, the central death rate equals that force, and the annual death probability is \(1-e^{-m_x}\). The UDD conversion is a different model result and should not be substituted without its assumption.

Check Your Understanding

Use the assumption stated in each question, retain full precision during calculations, and round final probabilities and rates to four decimal places.

A hypothetical age interval has \(m_x=0.50\). Assuming constant force, calculate \(q_x\), and calculate the UDD value for comparison. (Answer: constant force \(q_x=0.3935\); UDD \(q_x=0.4000\))
A one-year death probability is \(q_x=0.20\). Find the central death rate under constant force and under UDD. (Answer: constant force \(m_x=0.2231\); UDD \(m_x=0.2222\))
At age \(x\), \(l_x=500\) and \(m_x=0.12\) under constant force. Find \(q_x\), expected deaths, and expected survivors at age \(x+1\). (Answer: \(q_x=0.1131\); deaths \(=56.54\); survivors \(=443.46\))
A constant force applies for two consecutive years and the one-year survival probability is \(0.60\) in each year. Find the force, the two-year survival probability, and the probability of death within two years. (Answer: force \(=0.5108\); two-year survival \(=0.3600\); two-year death probability \(=0.6400\))
A hypothetical interval has 48 deaths and 300 person-years of exposure. Calculate the central rate, then the annual death probability under constant force and under UDD. (Answer: \(m_x=0.1600\); constant-force \(q_x=0.1479\); UDD \(q_x=0.1481\))

What You'll Learn