Life Table

Life Table: Concept Architecture

Follow a hypothetical group through ages or intervals

A life table organizes mortality and survival into a consistent sequence of ages, numbers surviving, deaths, person-years lived, and remaining life expectancy. It translates a set of age-specific death probabilities into answers such as the probability of surviving from age 60 to 70 and the expected remaining years at age 95. Its cohort of survivors is commonly hypothetical and begins with an arbitrary number called the radix; a period life table applies a period's mortality schedule across all ages rather than literally following one birth cohort through time.

Life tables support population health measurement, insurance and pension analysis, and health economic models. Their interpretation depends on time origin, population, age grouping, mortality source, and assumptions about deaths within and beyond the observed intervals. A life table is a structured calculation; it is not by itself a causal estimate of the effect of a treatment or a forecast of any one person's death age.

Read the core columns in order

For an interval from exact age $x$ to $x+n$, conventional life-table symbols include the following. Their values are linked: one cannot alter a death probability and leave later survivor counts unchanged.

Start from $l_x$ and a death probability. The recurrence for a closed age interval is:

$${}_nd_x=l_x{}_nq_x,\qquad l_{x+n}=l_x-{}_nd_x.$$

The death probability is not the same as a central death rate ${}_nm_x$, which divides observed deaths by population person-time exposure in an age band. A life-table method converts rates into probabilities using an assumption about how much of the interval people who die live through. If ${}_na_x$ is their average time lived after entering an interval of width $n$, a standard approximation is:

$${}_nq_x=\frac{n\,{}_nm_x}{1+(n-{}_na_x){}_nm_x}.$$

The choice of ${}_na_x$ must match the age band and data, with special attention to infancy and the oldest ages. Do not replace a probability with a rate merely because both are reported per person or multiplied by 1,000.

Count person-years and accumulate remaining life

For a one-year age interval, if deaths are approximated as occurring halfway through the year, person-years are $L_x\approx(l_x+l_{x+1})/2$. More generally, the closed-interval formula is ${}_nL_x=n l_{x+n}+{}_na_x{}_nd_x$. This counts full intervals for survivors and partial intervals for those who die. An open final age group has no finite $n$ and requires its own tail method; blindly applying the midpoint formula to it is invalid.

Work backward from the oldest interval to accumulate total future person-years, then divide by those alive at each starting age:

$$T_x=\sum_{a\geq x}{}_nL_a,\qquad e_x=\frac{T_x}{l_x}.$$

Here the sum is schematic for a sequence of age intervals, including an appropriately closed or open final interval. $T_x$ has units of person-years and $e_x$ has units of years per person. If the radix changes from 1,000 to 100,000 but the probabilities remain the same, $l_x$, $d_x$, $L_x$, and $T_x$ scale, while $q_x$ and $e_x$ do not.

Build a small table and check each column

Consider an intentionally simple hypothetical group of 1,000 people at age 95. Assume death probabilities of 0.20 during ages 95–96, 0.25 during 96–97, and one-third during 97–98. For these closed one-year intervals assume deaths occur on average halfway through. At age 98, 400 survivors remain; suppose a separately justified open-age-tail model assigns them 400 total future person-years, or one remaining year per survivor. This last assumption is an input, not a fact deduced from the previous three probabilities.

The first row has 200 deaths and 800 survivors at 96. Its 900 person-years equal $(1{,}000+800)/2$. Working backward, the 97-year-old cohort has $T_{97}=500+400=900$ person-years, the 96-year-old cohort has $T_{96}=700+900=1{,}600$, and the 95-year-old cohort has $T_{95}=900+1{,}600=2{,}500$. Thus $e_{95}=2{,}500/1{,}000=2.50$ years. The probability of surviving from 95 to 98 under these assumed probabilities is $l_{98}/l_{95}=400/1{,}000=0.40$. These large late-age probabilities and rounded tail are solely pedagogic.

Changing the open-age assumption changes $T_x$ and $e_x$ at every earlier age in this short example, though it does not change the displayed $q_x$ or $l_x$ before 98. In a real life table, estimate the open interval from appropriate mortality data or a documented model, and report the boundary and sensitivity to tail assumptions.

Know what kind of life table you are using

A period table combines age-specific mortality in a particular calendar period into a hypothetical lifetime under that mortality schedule. It does not predict that today's infant will face those exact rates at every future age. A cohort table follows a birth cohort's experienced mortality and, for living cohorts, necessarily projects rates at future ages. A complete table often uses single years of age; an abridged table uses wider age groups and must account for interval width and within-interval survival. All-cause tables count death from any cause; multiple-decrement tables separate mutually exclusive causes or exits under a carefully specified framework.

The time origin can also be diagnosis, treatment, or entry to a model rather than birth. In that case the rows may describe durations since origin, not chronological ages, and the death or transition probabilities should match that cohort's eligibility and time scale. An actuarial interval table constructed from a clinical study with withdrawals needs censoring assumptions and a defined risk set; it is not automatically the same as an official demographic period life table. Kaplan–Meier estimation handles observed event times directly, while grouped tables aggregate time into intervals.

Check assumptions and data quality

Age-specific deaths, births, population exposures, migration, cause coding, delayed reporting, and age misstatement can affect estimated rates. When age groups contain few deaths, random variation can make life-table quantities unstable; smoothing or pooling may be justified but should be disclosed. Choices for ${}_na_x$, the terminal age, and the open-age model affect person-years and expectancy. Comparisons between geographic or social groups need compatible periods, age definitions, mortality ascertainment, and attention to uncertainty. A life table summarizes the mortality conditions used to construct it; differences between tables need not be caused by the labels applied to those populations.

Conventional single-decrement survival assumes people leave the alive state only by all-cause death. For a specific cause, treating deaths from other causes as though people simply disappeared without consequence can misrepresent real-world cause-specific cumulative incidence. A competing-risk or multi-state model may be needed for that estimand. If estimating mortality for a treated cohort from an external table, avoid using the same death events twice through both background and disease-specific hazards.

Move from the table to economic outcomes

For a model cohort, $L_x$ or an analogous cycle-level person-time calculation supports expected life-years, while health-state utilities can weight appropriate time to produce QALYs. Treatment effects alter event or survival probabilities and should therefore propagate through later survivor counts, costs, and outcomes. A national table can provide background mortality, but clinical trial or registry survival may require disease-specific adjustments, extrapolation, and validation. Matching calendar period, population, sex, age, and model horizon matters more than importing a familiar published $e_x$ without its assumptions.

Life expectancy is a summary output of a table, whereas the table contains the probabilities and person-time path used to compute it. Expected age at death for someone alive at age $x$ under a given distribution is $x+e_x$; $e_x$ alone is remaining years. When comparing strategies, build internally consistent tables under each strategy and compare expected life-years, QALYs, costs, and uncertainty rather than subtracting two isolated life-expectancy values without aligning eligibility and timing.

Report source deaths and exposures, population and period or cohort, radix, interval widths, $q_x$ conversion and ${}_na_x$, open-age method, uncertainty, and any changes made for a modeled patient cohort. Provide enough rows or a reproducible rule to let readers verify that deaths plus survivors, person-years, and backward sums agree.

Sources and further reading

• US National Center for Health Statistics: Method for Constructing Complete Annual U.S. Life Tables. Primary technical method for converting mortality inputs to complete life tables.
• World Health Organization: Methods and data sources for life tables, 2000–2021. Primary demographic methodology for abridged age intervals and mortality estimation.
• UK Office for National Statistics: Period and cohort life expectancy explained. Official explanation of period and cohort table interpretation.
• US National Center for Health Statistics: Life expectancy and life tables. Official overview of life-table uses and age-specific expectancy.