Notes on Life Table Construction
I worked this all out when in Yangon, but had no time to write it down, and on needing it here I had forgotten and had to work it out over again last night.
I want to make a life table from age-specific death rates using the formula
nqx = n´nmx/[1+(n-nax)nmx]
where nmx denotes the age-specific death rate for the age interval x to x+n, nax is the average age at death for persons who die in this age interval, and nqx is standard life table notation for the conditional probability of dying in the interval x to x+n given survival to age x.
Two questions then arise. First, what is the derivation of the formula? Second, how do we calculated the nax values for extreme age groups? For intermediate age groups, nax may be taken simple as n/2.
The neat thing about calculating nax is that the formula turns out to be identical to that for the singulate mean age at marriage. Recall that the latter is, verbally, person years lived by single women during the specified marriage ages, a1 to a2, say, (generally taken to be 15 and 50, respectively), less the length of this range a2-a1 times the proportion of women remaining single at age a2 (this is generally estimated by averaging the proportions single at the age groups just above and below and dividing by the total number of ages in these groups), this difference being divided, finally, by one minus the proportion of persons remaining single at age a2. The general formula would divide by the difference between the proportions ever married at ages a1 and a2, that is, by the proportion of women marrying in the specified age range, but the proportion ever married at age a1 is generaly taken to be one. The formula for this, using standard life table notation, is
[nLx - nlx+n]/[lx - lx+n]
[notation problem here, but I'm thinking how to do this with ascii characters only, and nLx, e.g., works fine, only when subscripts are more complicated does ambiguity arise.]
The formula for nax is by precise analogy
nax = [nLx - nlx]/[lx - lx+n]
which is readily calculated given a suitable life table. Since we are constructing a life table, we won't have lx and nLx values, but the obvious expedient is to compute nax from a suitable proxy, e.g., a model table. Having constructed an initial life table in this way, we could use it to recalculate nax values and compute a second life table from the given data, and one suspects strongly that continuing this process would yield a convergent set of nax values and corresponding life tables. This is reminiscent of Keyfitz's old JASA paper `A Life Table that Iterates to the Data,' though that construction assumes a stable population, whereas this one assumes a stationarty population.
Now on to the derivation of the basic formula. We begin with a Lexis diagram with a vertical line representing the mid-point of an n-year period and horizontal lines at ages x and x+n. Let nPx denote the population at this time, also average person years lived during any year, sinc the population is assumed stationary, and nDx the number of death between ages x and x+n during a single year. The age-specific death rate is then nDx/nPx.
From Vietnam Consultancy, 1995