Life Table Radix And Cohort Counts

Introduction

A life table expresses a survival model in a count scale that is easy to interpret. Its starting count, called the radix, is a reference size for a hypothetical cohort. The entries \(\ell_x\) then represent how many members of that cohort are alive at age \(x\), according to the assumed survival model. The cohort is a mathematical device: it need not correspond to a group of people who were actually observed.

The radix affects the size of the counts, but not the model’s survival probabilities. Understanding that distinction helps prevent a common error: treating a life table count as though it were itself a probability, or assuming that a larger radix implies different mortality.

Definition: The radix is the starting number of lives in a hypothetical cohort, denoted \(\ell_0\) when the cohort begins at age 0. The life table entry \(\ell_x\) is the number from that original cohort who survive to exact age \(x\), under the specified survival model. When the model uses the survival function \(S_0(x)=P(T>x)\), where \(T\) is age at death, the cohort count is \(\ell_x=\ell_0 S_0(x)\).

Counts, Probabilities, and the Radix

The survival function \(S_0(x)\) gives the probability that a life aged 0 survives beyond age \(x\). Multiplying that probability by the starting cohort size gives the corresponding survivor count. For example, if \(S_0(30)=0.85\) and \(\ell_0=10{,}000\), then the model assigns \(8{,}500\) members of the hypothetical cohort to survival beyond age 30.

In a probability model, \(\ell_x\) need not be an integer. It is the expected number of survivors when \(\ell_0\) independent lives share the same survival distribution. A published or constructed table may round entries to whole numbers for convenience. That rounding is a presentation choice; it does not alter the underlying probability model. When doing calculations, use unrounded counts or probabilities when they are supplied, rather than introducing rounding early.

A radix is a scale choice. If the radix is multiplied by a positive constant, every survivor count is multiplied by the same constant. For a cohort beginning at age 0, the ratio of survivors at age \(x\) to the original cohort size is the survival probability to age \(x\):

\frac{\ell_x}{\ell_0} = \frac{\ell_0 S_0(x)}{\ell_0} = S_0(x).

The starting cohort size cancels. Thus a table with radix \(1{,}000\) and one with radix \(100{,}000\) can represent exactly the same mortality assumptions. Their entries have different scales, but the proportion surviving to each age is the same.

Theorem: If two life tables use the same survival model but radices \(r_1\) and \(r_2\), their age-\(x\) entries are \(r_1S_0(x)\) and \(r_2S_0(x)\). Therefore each entry is proportional to its radix, while the proportion surviving, \(\ell_x/\ell_0\), is independent of the radix.

Why the Scaling Works

The result follows directly from the meaning of a survivor count. Consider a cohort of \(\ell_0\) lives, each with probability \(S_0(x)\) of surviving beyond age \(x\). For each life, define an indicator that equals 1 if that life survives beyond \(x\), and 0 otherwise. The total number of survivors is the sum of these indicators. Each indicator has expected value \(S_0(x)\), so the expected total is the sum of \(\ell_0\) equal expected values:

E[\text{number surviving beyond }x] = \underbrace{S_0(x)+\cdots+S_0(x)}_{\ell_0\text{ lives}} = \ell_0S_0(x).

This also explains why changing the radix does not change the survival model. The probability for each life is unchanged; only the number of lives to which that probability is applied changes. The formula concerns an expected count. A realized group of people could have a different number of survivors because of random variation.

Identify the cohort’s starting age and radix.
For a cohort beginning at age 0, the radix is \(\ell_0\). If the cohort begins at another age, use its count at that starting age as the scale.

Find the relevant survival probability.
Use the stated survival model to determine the probability of reaching the target age from the cohort’s starting age.

Multiply to obtain the survivor count.
Apply the survival probability to the starting count. Keep the probability as a proportion, not a percentage number such as 85.

Check the scale and interpretation.
The survivor count cannot exceed the starting cohort count under a model with no entry into the cohort, and its ratio to the starting count should equal the survival probability.

Worked Applications

Worked Example: A Uniform Lifetime Model and Two Survivor Counts

A hypothetical cohort has radix \(\ell_0=12{,}000\). Assume each member’s age at death \(T\) is uniformly distributed from age 0 to age 80, so \(T\sim\operatorname{Uniform}(0,80)\). Find the model survivor counts at ages 20 and 50, and verify the proportions of the radix represented by those counts.

Method. For this uniform distribution, the survival function is \(S_0(x)=(80-x)/80\) for \(0\leq x\leq80\). Multiply each survival probability by the radix. The common mistake to avoid is using the cumulative probability of death by age \(x\), rather than the probability of surviving beyond age \(x\).

At age 20,

S_0(20)=\frac{80-20}{80}=\frac{60}{80}=0.75, \qquad \ell_{20}=12{,}000(0.75)=9{,}000.

At age 50,

S_0(50)=\frac{80-50}{80}=\frac{30}{80}=0.375, \qquad \ell_{50}=12{,}000(0.375)=4{,}500.

The checks are \(9{,}000/12{,}000=0.75\) and \(4{,}500/12{,}000=0.375\), exactly the model survival probabilities. The model counts are 9,000 at age 20 and 4,500 at age 50.

Worked Example: Combining Survival Across Two Age Intervals

A hypothetical cohort begins at age 0 with radix \(8{,}000\). Assume a piecewise-constant force of mortality: \(\mu_x=\ln(5/4)/10\) for \(0\leq x<10\), and \(\mu_x=\ln(8/5)/10\) for \(10\leq x<20\). Find \(\ell_{10}\) and \(\ell_{20}\).

Method. Under a constant force \(\mu\) over an interval of length \(t\), the probability of surviving the interval is \(e^{-\mu t}\). Apply the first interval’s survival probability to the radix, then apply the second interval’s probability to the survivors at age 10. The trap is applying the second interval’s probability directly to the original radix and thereby ignoring the first interval.

From age 0 to age 10, the survival probability is

e^{-10\ln(5/4)/10} =e^{-\ln(5/4)} =\frac{4}{5} =0.8.

Therefore,

\ell_{10}=8{,}000(0.8)=6{,}400.

From age 10 to age 20, the conditional survival probability is

e^{-10\ln(8/5)/10} =e^{-\ln(8/5)} =\frac{5}{8} =0.625.

The age-20 count must start from \(\ell_{10}\):

\ell_{20}=6{,}400(0.625)=4{,}000.

As a check, the total survival probability from age 0 to age 20 is \(0.8(0.625)=0.5\), and \(8{,}000(0.5)=4{,}000\). The counts are \(\ell_{10}=6{,}400\) and \(\ell_{20}=4{,}000\).

Worked Example: Changing the Radix Without Changing Mortality

Assume a hypothetical lifetime model has \(S_0(40)=0.60\). From age 40 onward, assume a constant force of mortality \(\mu=\ln(10/9)/5\) per year. A cohort has radix \(25{,}000\). Find the counts at ages 40 and 45, then find the corresponding counts if the same model is represented with radix \(100{,}000\).

Method. Use the given survival function value to reach age 40, and the constant-force model for the next five years. Then rescale each count in proportion to the new radix. The mistake to avoid is treating the larger radix as a change in the survival probability.

For the cohort with radix \(25{,}000\), the age-40 count is

\ell_{40}=25{,}000(0.60)=15{,}000.

The five-year survival probability from age 40 to age 45 is

e^{-5\mu} =e^{-5\ln(10/9)/5} =e^{-\ln(10/9)} =\frac{9}{10} =0.90.

Thus the age-45 count is

\ell_{45}=15{,}000(0.90)=13{,}500.

The new radix is \(100{,}000\), which is four times \(25{,}000\). Therefore its counts are four times the original counts: at age 40, \(4(15{,}000)=60{,}000\); at age 45, \(4(13{,}500)=54{,}000\). The checks are \(60{,}000/100{,}000=0.60\) and \(54{,}000/100{,}000=0.54\). The second proportion also equals \(0.60(0.90)=0.54\). The mortality assumptions have not changed; only the scale of the cohort has changed.

Common Interpretation Errors

Calling \(\ell_x\) a probability. It is a count, or a model expected count. The corresponding proportion is \(\ell_x/\ell_0\) for a cohort beginning at age 0.
Using the radix as if it were fixed by the mortality model. The radix is a chosen scale. A different positive radix produces proportionally different counts under the same survival assumptions.
Confusing survival with death by age \(x\). If a model gives a death probability by age \(x\), it is not the quantity to multiply by the radix to obtain survivors. Use the probability of survival beyond age \(x\).
Restarting with the original radix at an intermediate age. When following a cohort through successive intervals, the next count is based on the survivors at the start of that interval, not on the original cohort size.
Assuming every tabulated count must be an observed integer. The model count may be fractional before rounding. A life table can be a scaled representation of probabilities, not a record of a particular group.

Key takeaway: A life table radix sets the cohort’s scale. Under a survival model, \(\ell_x=\ell_0S_0(x)\); changing \(\ell_0\) scales the counts but leaves the survival probabilities unchanged.

Check Your Understanding

For each problem, distinguish the model probability from the count produced by the chosen radix.

A hypothetical cohort has radix \(5{,}000\), and its survival model gives \(S_0(12)=0.92\). Find \(\ell_{12}\). (Answer: \(4{,}600\))
A cohort has radix \(20{,}000\), and \(T\sim\operatorname{Uniform}(0,100)\). Find \(\ell_{25}\) and \(\ell_{60}\). (Answer: \(\ell_{25}=15{,}000\), \(\ell_{60}=8{,}000\))
A cohort with radix \(10{,}000\) has survival probability \(0.70\) from age 0 to age 30, followed by conditional survival probability \(0.80\) from age 30 to age 40. Find \(\ell_{30}\), \(\ell_{40}\), and the proportion of the radix surviving to age 40. (Answer: \(\ell_{30}=7{,}000\), \(\ell_{40}=5{,}600\), proportion \(=0.56\))
A model gives \(\ell_{18}=6{,}300\) for radix \(9{,}000\). The same model is represented with radix \(30{,}000\). Find the new age-18 count and explain what happens to the survival probability to age 18. (Answer: \(21{,}000\); the survival probability remains \(6{,}300/9{,}000=0.70\))
A hypothetical cohort has radix \(16{,}000\). The force of mortality is constant at \(\mu=\ln 2/10\) per year from age 0 to age 10 and constant at \(\mu=\ln(4/3)/5\) per year from age 10 to age 15. Find \(\ell_{10}\) and \(\ell_{15}\). (Answer: \(\ell_{10}=8{,}000\), \(\ell_{15}=6{,}000\))

What You'll Learn