The Life Table

The life table has been a key tool of actuaries for some 200 years and is the basis for calculating life expectancy.

Consider a large group, or "cohort", of U.S. males, for example, who were born on the same day. If we could follow the cohort from birth until all members died, we could record the number of individuals alive at each birthday -- age x, say -- and the number dying during the following year. The ratio of these is the probability of dying at age x, usually denoted by q(x). It turns out that once the q(x)'s are all known the life table is completely determined.

In practice such "cohort life tables" are rarely used, in part because individuals would have to be followed for up to 100 years, and the resulting life table would reflect historical conditions that may no longer apply. Instead, one generally works with a period, or current, life table. This summarizes the mortality experience of persons of all ages in a short period, typically one year or three years. More precisely, the death probabilities q(x) for every age x are computed for that short period, often using census information gathered at regular intervals (every ten years in the U.S.). These q(x)'s are then applied to a hypothetical cohort of 100,000 people over their life span to produce a life table. An example from the National Center for Health Statistics (1997) is given below.

```
Table 1: Abbreviated decennial life table for U.S. Males.
-----------------------------------------------------------
x     l(x)  d(x)     q(x)     m(x)   L(x)     T(x)   e(x)
-----------------------------------------------------------
0   100000  1039  0.01039  0.01044  99052  7182893   71.8
1    98961    77  0.00078  0.00078  98922  7083841   71.6
2    98883    53  0.00054  0.00054  98857  6984919   70.6
3    98830    41  0.00042  0.00042  98809  6886062   69.7
4    98789    34  0.00035  0.00035  98771  6787252   68.7
5    98754    30  0.00031  0.00031  98739  6688481   67.7
6    98723    27  0.00028  0.00028  98710  6589742   66.7
7    98696    25  0.00026  0.00026  98683  6491032   65.8
8    98670    22  0.00023  0.00023  98659  6392348   64.8
9    98647    19  0.00020  0.00020  98637  6293689   63.8
10    98628    16  0.00017  0.00017  98619  6195051   62.8

20    97855   151  0.00155  0.00155  97779  5211251   53.3

30    96166   197  0.00205  0.00205  96068  4240855   44.1

40    93762   295  0.00315  0.00316  93614  3290379   35.1

50    89867   566  0.00630  0.00632  89584  2370098   26.4
51    89301   615  0.00689  0.00691  88993  2280513   25.5

60    81381  1294  0.01591  0.01604  80733  1508080   18.5

70    64109  2312  0.03607  0.03674  62953   772498   12.0

75    51387  2822  0.05492  0.05649  49976   482656    9.4
76    48565  2886  0.05943  0.06127  47121   432679    8.9

80    36750  3044  0.08283  0.08646  35228   261838    7.1

90     9878  1823  0.18460  0.20408   8966    38380    3.9

100      528   177  0.33505  0.40804    439     1190    2.3
-----------------------------------------------------------```

The columns of the table, from left to right, are:

x: age

l(x), "the survivorship function": the number of persons alive at age x. For example of the original 100,000 U.S. males in the hypothetical cohort, l(50) = 89,867 (or 89.867%) live to age 50. These values are computed recursively from the m(x) values using the formula l(x+1) = l(x)*exp[-m(x)], with l(0), the "radix" of the table, arbitrarily set to 100,000. For example, l(2) = l(1)*exp[-m(1)] = 98961*exp(-0.00078) = 98883.

d(x): number of deaths in the interval (x,x+1) for persons alive at age x, computed as d(x) = l(x) - l(x+1). For example, of the l(50)=89,867 persons alive at age 50, d(50) = l(50)-l(51) = 89867 - 89301 = 566 died prior to age 51.

q(x): probability of dying at age x. Also known as the (age-specific) risk of death. Generally these are derived using the formula q(x) = 1 - exp[-m(x)], under the assumption that the instantaneous mortality rate, or force of mortality, remains constant throughout the age interval from x to x+1. By construction, q(x) is also equal to d(x)/l(x). Thus, for example, q(50) = d(50)/l(50) = 566 / 89,867 = 0.00630.

m(x): the mortality rate at age x. Generally these quantities are estimated from the data, and are the sole input to the life table. That is, all other quantities are determined once the m(x)'s are specified. By construction, m(x) = d(x)/L(x), the number of deaths at age x divided by the number of person-years at risk at age x. Note that the mortality rate, m(x), and the probability of death, q(x), are not identical. For a one year interval they will be close in value, but m(x) will always be larger.

L(x): total number of person-years lived by the cohort from age x to x+1. This is the sum of the years lived by the l(x+1) persons who survive the interval, and the d(x) persons who die during the interval. The former contribute exactly 1 year each, while the latter contribute, on average, approximately half a year, so that L(x) = l(x+1) + 0.5*d(x). This approximation assumes that deaths occur, on average, half way in the age interval x to x+1. Such is satisfactory except at age 0 and the oldest age, where other approximations are often used; for details see the National Center for Health Statistics (1997) or Schoen (1988).

T(x): total number of person-years lived by the cohort from age x until all members of the cohort have died. This is the sum of numbers in the L(x) column from age x to the last row in the table.

e(x): the (remaining) life expectancy of persons alive at age x, computed as e(x) = T(x)/l(x). For example, at age 50, the life expectancy is e(50) = T(50)/l(50) = 2,370,099/89,867 = 26.4.

In the above life table, mortality rates for U.S. males at each age were determined from census data for a short period (3 years). The rates applied to the hypothetical cohort of 100,000 U.S. males in the table throughout the lifetime of the cohort. All the other columns of the life table are derived from m(x) as indicated above. See Schoen (1988), Anderson (2002), or National Center for Health Statistics (1997) for details.

Notes

1. Life expectancy is not the same as median survival time, the latter being the time at which only 50% of a cohort are still alive. For example, of the 100,000 persons alive at age 0, 51,387 are alive at age 75, and 48,565 are alive at age 76. The median survival time at birth (age 0) is thus between 75 and 76 additional years (and can be shown to be 75.5), while the life expectancy at birth is e(0) = 71.8 additional years.

2. The calculation of life expectancy for a person should not be confused with predicting their survival time. While newborn U.S. males have a life expectancy of 71.8 years, any given U.S. male may die tomorrow or live to age 100. One need not predict actual survival times in order to compute life expectancy (the average survival time).

References

Anderson RN (1999). United States life tables, 1997. National vital statistics reports; vol 47 no 28. Hyattsville, Maryland: National Center for Health Statistics.

Anderson TW (2002). Life expectancy in court: A textbook for doctors and lawyers. Vancouver BC: Teviot Press.

National Center For Health Statistics (1997). U.S. decennial life tables for 1989-1991, volume 1, number 1. Hyattsville, Maryland: Author.

Schoen R (1988). Modeling multigroup populations, chapter 1. New York: Plenum Press.