DEMOGRAPHY© Volume 23, Number 3 August 1986

AGE INTERVALS AND TIME INTERVALS: REPLY TO KIM

Samuel H. PrestonPopulation Studies Center, University of Pennsylvania, 3718 locust Walk, Philadelphia, Pennsylva­nia 19104

Ansley J. CoaleOffice of Population Research, 21 Prospect Avenue, Princeton University, Princeton, New Jersey08540

Professor Kim has performed a valuable service in deriving certain discreteanalogs to many of the equations presented in our paper (Preston and Coale 1982).She is correct that, when the time interval separating two data sets that are arrayedby age or by duration in a state is equal to the age or duration interval in which thedata are presented, there is no value for demographic estimation in resorting toequation (1) of her paper. Standard procedures for intercensal survival analysisprovide much the same answers, and with less effort. In this circumstance, usingequation (1) is like counting sheep by adding up the number offeet and dividing byfour.

We and our collaborators have stressed in a series of papers (e.g., Bennett andHoriuchi 1981; Preston and Bennett 1983) that equation (1) and its offshoots arevaluable precisely because they can be used when age intervals are not equal to timeintervals, which is probably the most common circumstance facing the analyst.Consider the problem of constructing a life table from two single-year age distribu­tions separated by an arbitrary time interval, say 8 years. If the age distributions areaccepted as accurate (and if the population is a closed one), then the single-yearpopulation at age a in the first census minus the single-year population at age a + 8 inthe second census equals the number of deaths experienced by this cohort over the8-year interval. By intra-cohort interpolation it is possible to estimate the number ofpersons crossing each exact age from a + 1 to a + 8, to estimate the number ofdeaths experienced by the cohort within each single-year age interval, and toestimate the number of person-years lived in each single-year age interval from a toa + 8. By combining such estimates it is possible to construct a life table. Forexample, the deaths at each age divided by the person-years lived at each age yield aschedule of age-specific mortality rates that can be converted into the probability ofdying in each age interval and then into a qx function.

Intra-cohort interpolation was used by Coale (1984) to estimate the number ofpersons crossing each birthday in intercensal periods in China from 1953 to 1964 andfrom 1964 to 1982. Interpolation factors were obtained by two procedures, aniterative process that begins with linear interpolation, and model interpolationfactors. The total number attaining each exact age in the intercensal period was thenconverted into a life table by a formula using the calculated age-specific growth ratein each single-year age interval. A life table could equally well have been calculatedby an analogous interpolation procedure establishing the number of deaths and thenumber of person-years lived in each age interval. Kim's examples, using as they doa single-year age distribution at a time interval of only one year, are not analogous tothe example just described. Her identities apply only if the data from the twocensuses are grouped into age intervals equal to the duration of the intercensalperiod. For censuses in 1953 and 1964, her procedure permits the calculation only oflILll/llLO' lIL22/lILlI' etc.

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464 DEMOGRAPHY, volume 23, number 3, August 1986

This advantage for equation (1) is principally one of convenience; it seems certainthat some variant of intercensal survival analysis could be adapted to any combina­tion of age-time intervals that would yield the same results as any particularoperationalization of equation (1). We are presently working on such elaborations(e.g., Coale 1985). It also seems clear that equation (1) will retain heuristicadvantages by presenting a simple and unified framework applicable to all datacircumstances.

A second advantage of equation (1) is analytic; it forms the basis for newexpressions that are useful in demographic estimation and that could not have beenderived in its absence. One such instance is an "integrated procedure" described byPreston (1983a), designed specifically to deal with defective data by combiningequation (1) with a logit model life table system. Obviously, such procedures wouldnot have been developed if we believed, as Kim seems to think we do, that equation(1) solves all problems of defective data. The relative analytic advantage of equations(1) and (2) is particularly salient in view of the fact that Kim's equations contain noexpression for the birth rate or the infant mortality rate. These important demo­graphic indicators are instead combined by Kim into a single expression (h') for thepopulation size below age one. The continuous equations permit an exact represen­tation of these quantities, and the main product of the integrated procedure is, infact, an estimate of the birth rate.

Kim terms her equations "exact accounting identities." Exact accounting identi­ties in discrete age and discrete time intervals based on equation (1) were alsopresented in Appendix 2 of Preston and Coale (1982). Our discrete-time discrete-ageidentities, however, are not identical to Kim's. We have attempted to follow throughthe logic of period life table construction by relying on events occurring within theage-time rectangles shown in Kim's figure. Kim instead follows cohorts up thediagonal and chains together cohort survival experience. Each of her cohortsoccupies only half of any rectangle in the lexis diagram. There is no guarantee thather procedure will produce the conventional period life table, one that is based uponthe force of mortality function observed within an age-time rectangle. We canconstruct an example where her procedure would produce quite misleading results(e.g., if a cohort is very small compared to adjacent ones and has very unusualmortality). But we would expect, in general, differences to be very small. Preston(1983b) has shown that the two procedures will necessarily produce identical resultsfor n-year wide age and time intervals when geometric means of the two observa­tions on .N; (the number of persons between exact ages x and x + n) are used in thediscrete versions of equations (1) and (2) in Kim's paper.

In short, Kim has helped to demonstrate the unity of demographic accountingsystems and the foundations of the new variable growth rate procedures. But we seelittle virtue in saying that one way of looking at cohort and period relations is correctand that all others are inaccurate, roundabout, or mistaken.

REFERENCES

Bennett, N. G., and S. Horiuchi. 1981. Estimating the completeness of death registration in a closedpopulation. Population Index 47:207-221.

Coale, A. 1984. Life table construction on the basis of two enumerations of a closed population.Population Index 50:193-213.

--. 1985. An extension and simplification of a new synthesis of age structure and growth. Asian andPacific Census Forum 12:5-7.

Preston, S. H. 1983a. An integrated system fur demographic estimation from two age distributions.Demography 20:213-226.

Age Intervals and Time Intervals: Reply to Kim 465

---. 1983b. Estimation of Certain Measures in Family Demography Based Upon Generalized StablePopulation Relations. Presented to IUSSP Conference on Family Demography. Population Council.New York, December 12-14, 1983. Forthcoming, Oxford University Press.

Preston, S. H., and N. Bennett. 1983. A census-based method for estimating adult mortality. PopulationStudies 37:91-104.

Preston, S. H., and A. Coale. 1982. Age structure, growth, attrition and accession: a new synthesis.Population Index 48:217-259.