Produces population and cohort life tables.

Routine ACTBL computes population (current) or cohort life tables based upon the observed population sizes at the middle (IMTH = 0) or the beginning (IMTH = 1) of some userspecified age intervals. The number of deaths in each of these intervals must also be observed.

The probability of dying prior to the middle of the interval, given that death occurs somewhere in the interval, may also be specified. Often, however, this probability is taken to be 0.5. For a discussion of the probability models underlying the life table here, see the references.

Let ti, for i = 0, 1, …, tn denote the time grid defining the n age intervals, and note that the length of the age intervals may vary. Following Gross and Clark (1975, page 24), let di denote the number of individuals dying in age interval i, where age interval i ends at time ti. If IMTH = 0, the death rate at the middle of the interval is given by ri = di/(Mihi), where Mi is the number of individuals alive at the middle of the interval, and hi = ti ‑ ti‑1, t0 = 0. The number of individuals alive at the beginning of the interval may be estimated by Pi = Mi + (1 ‑ ai)di where ai is the probability that an individual dying in the interval dies prior to the interval midpoint. When IMTH = 1, Pi is input directly while the death rate in the interval, ri, is not needed.

The probability that an individual dies during the age interval from ti−1 to ti is given by qi = di/Pi. It is assumed that all individuals alive at the beginning of the last interval die during the last interval. Thus, qn = 1.0. The asymptotic variance of qi can be estimated by

When IMTH = 0, the number of individuals alive in the middle of the time interval (input in IPOP(I)) must be adjusted to correspond to the number of deaths observed in the interval. Routine ACTBL assumes that the number of deaths observed in interval hi occur over a time period equal to hi. If di is measured over a period ui, where ui ≠ di, then IPOP(I) must be adjusted to correspond to di by multiplication by ui/hi, i.e., the value Mi input into ACTBL as IPOP(I) is computed as

Let Si denote the number of survivors at time ti from a hypothetical (IMTH = 0) or observed (IMTH = 1) population. Then, S0 = NPOP when IMTH = 0, and S0 = IPOP(1) for IMTH = 1, and Si is given by Si = Si−1 ‑ δi−1 where δi = Siqi is the number of individuals who die in the i‑th interval. The proportion of survivors in the interval is given by Vi = Si/S0 while the asymptotic variance of Vi can be estimated as follows.

The expected lifetime at the beginning of the interval is calculated as the total lifetime remaining for all survivors alive at the beginning of the interval divided by the number of survivors at the beginning of the interval. If ei denotes this average expected lifetime, then the variance of ei can be estimated as (see Chiang 1968)

where var(en) = 0.0.

Finally, the total number of time units lived by all survivors in the time interval can be estimated as:

The following example is taken from Chiang (1968). The cohort life table has thirteen equally spaced intervals, so AGE(1) is set to ‑5.0. Similarly, the probabilities of death prior to the middle of the interval are all taken to be 0.5, so A(1) is set to ‑1.0. Since IPRINT = 1, the life table is printed by ACTBL.