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 is given below. Table 1: Abbreviated decennial life table for U.S. Males. From: National Center for Health Statistics (1997). ------------------------------------------------------- x l(x) d(x) q(x) m(x) L(x) T(x) e(x) ------------------------------------------------------- 0 100000 1039 0.0104 0.0104 99052 7182893 71.8 1 98961 77 0.0008 0.0008 98922 7083841 71.6 2 98883 53 0.0005 0.0005 98857 6984919 70.6 3 98830 41 0.0004 0.0004 98809 6886062 69.7 4 98789 34 0.0004 0.0004 98771 6787252 68.7 5 98754 30 0.0003 0.0003 98739 6688481 67.7 6 98723 27 0.0003 0.0003 98710 6589742 66.7 7 98696 25 0.0003 0.0003 98683 6491032 65.8 8 98670 22 0.0002 0.0002 98659 6392348 64.8 9 98647 19 0.0002 0.0002 98637 6293689 63.8 10 98628 16 0.0002 0.0002 98619 6195051 62.8 20 97855 151 0.0016 0.0016 97779 5211251 53.3 30 96166 197 0.0021 0.0021 96068 4240855 44.1 40 93762 295 0.0032 0.0032 93614 3290379 35.1 50 89867 566 0.0063 0.0063 89584 2370098 26.4 51 89301 615 0.0069 0.0069 88993 2280513 25.5 60 81381 1294 0.0159 0.0160 80733 1508080 18.5 70 64109 2312 0.0361 0.0367 62953 772498 12.0 75 51387 2822 0.0549 0.0565 49976 482656 9.4 76 48565 2886 0.0594 0.0613 47121 432679 8.9 80 36750 3044 0.0828 0.0865 35228 261838 7.1 90 9878 1823 0.1846 0.2041 8966 38380 3.9 100 528 177 0.3351 0.4080 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. d(x): number of deaths in the interval (x,x+1) for persons alive at age x. Thus of the l(50)=89,867 persons alive at age 50, d(50) = 566 died prior to age 51. q(x): probability of dying at age x. Also known as the (age-specific) risk of death. Note that q(x) = d(x)/l(x), so, for example, q(50) = 566 / 89,867 = 0.00630. m(x): the age-specific mortality rate. Computed as 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. [At age 0 and at the oldest age, other methods are used; for details see the National Center for Health Statistics (1997) or Schoen (1988). Note: m(x) = d(x)/L(x).] 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. 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). Construction of a life table 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 below. See Schoen (1988) or National Center for Health Statistics (1997) for details. q(x) = 1 - exp[-m(x)]. This assumes that the instantaneous mortality rate, or force of mortality, remains constant throughout the age interval from x to x+1. d(x) = l(x)q(x) l(x+1) = l(x) - d(x) is determined recursively, with l(0) arbitrarily set to 100,000. L(x) = l(x) + 0.5d(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 used. T(x) = sum of the L(x) column, from age x to the last row of the table. e(x) = T(x)/l(x) Life tables for persons with risk factors To construct a life table for a group of individuals with a common risk factor (or factors), one may either (1) compute age-specific mortality rates from a suitable database, or (2) adjust the mortality rates in the basic life table by adding excess death rates (EDRs) or multiplying by relative risks (RRs), either of which may be available from published studies.