Demographic Research Laboratory
Life Tables
If you already have age-specific mortality rates and want to generate a life table, here is a Hypercard script which will do the job for you. Or you may want to look at some examples of life tables generated with this program.
In its simplest form, it can be generated from a set of age-specific death rates. Table 1 is an example of a set of such rates for United States males in 1988 (taken from the Statistical Abstract of the U.S., 1990; you can find similar sets of rates in almost any almanac).
Typically, such tables show a somewhat high rate for the first year of life (due to infant mortality), then very low rates rising steadily through the older ages.
To construct a life table from such data we first convert the death rates per thousand to rates per capita, by dividing each ASDR by 1000 (move the decimal point three places) as in Table 2. This table also shows n, the width of each age interval except for the last.
Next we compute the probability, at age x, of dying in the interval x to x+n (n is the width of the interval). This probability, q, is computed from m and n using the formula
2mn
q = ------
2 + mn
The formula cannot be used on the last line of Table 2 since the value
of n is indeterminate. But the probability of dying in the open-ended
interval must be 1.0 since death is a certainty (ignoring mythical
immortals, of course).
These results are shown in Table 3
The probability of dying is really all we need to construct the life table. Henceforth, we will drop the ASDR, m and n columns (Table 3a). Table 4 shows what would happen if the mortality schedule represented by q were applied to a hypothetical group of 100,000 newborns (this hypothetical group is called the "radix").
If 100,000 people are subject to a probability of dying of 0.01116, then the number of deaths will be 100,000 times .01116. 1,116 deaths will occur. Subtracting these from the 100,000 leaves 98,884 living to enter the second age interval, aged 1. These 98,884 have a probability of dying before age 5 of .00224, so .00224 * 98,884 = 222 deaths will occur in the second interval. These are subtracted, and the table is completed when 21,305 enter the last interval and die there.
Now comes the tricky part. We must calculate how many person-years are lived in each age interval by this imaginary group of 100,000. If no one died in a given interval, then the number of years lived in the interval would be l, the number living at age x, times n, the width of the interval. Thus, if the 98,357 15-year-olds in Table 5 all made it to age 25, they would have lived a total of 98,357 * 10 = 983,570 person-years.
But that didn't happen: 1,503 of them died in the interval. The problem is where in the interval? Did they make it most of the way through? Did most of them die near the beginning of the interval? We just don't know. To reflect our ignorance, we are going to assume that they each made it half-way through the interval. From the total years which would have been lived had they all made it, we subtract half the interval width for the number who died in it. This gives us
n*(l - d/2) = 10*(98,357 - 1,503/2) = 976,055
for years-lived-in-the-interval. This formula (In which "l" is the letter
"ell", not numeral "one"), works for all age groups except the first and
the last.
In the case of the first age interval, age less than one, we know a little bit more about the distribution of deaths within the interval. Infant deaths tend to occur soon after birth. So in this case we are justified in subtacting more than half-a-year for each death. How much more is a matter of experience (and a very important matter since it strongly affects the rest of the table). For our purposes we can substitute a multiplier of .9 for what, in effect, has been our multiplier of .5 (division by 2).
Years lived in the last, open-ended interval is computed from the number living at entry (l) divided by the mortality rate in the interval (m = the ASDR divided by 1000). The reasoning for this is that the number dying in the interval (d) must equal the number entering it (l). The number of deaths (d) should also equal the death rate (m) times the number of years lived (L). Since d = mL and d = l, L = l/m.
The rest is easy. We need total years remaining at each age (T), which we obtain by cumulating the L column backwards, from bottom to top, as in Table 6. At birth the radix of 100,000 people have 7,145,206 years ahead of them. At age 85 the remaining 21,305 people together have 115,318 years remaining. Life expectancies at age x are computed by dividing T by l as shown in Table 7.
The column headings in Table 7 are usually printed with subscripts. All column headings are followed by sub-x (you can, for example, refer to T-sub-15, the total years remaining at age 15). The q, d and L columns are also subscripted before the name of the column with its width, n (thus, sub-10-L-sub-15 refers to the number of years lived in the 10-year interval beginning at age 15). The "e" for life expectancy is usually shown with a tiny little "o" above it, perhaps to distinguish it from the mathematical constant (e = 2.718...).