A model of the age patterns of births by parity in natural fertility populations

This research develops a convolution model to express the age patterns of fertility at each birth order in natural fertility populations in terms of six parameters, directly representing the proximate determinants of fertility, and a series of parity level indicators. The parity level indicators at each birth order are simply the proportions of women in a cohort who will eventually have births at each birth order it the age‐related fecundity decline is controlled. The Coale‐McNeil nuptiality model is adopted to represent the age pattern of first marriage rates and the natural fertility schedule employed in the Coale‐Trussell fertility model is incorporated to adjust age effects. The fast Fourier transform is used in solving the model numerically. It proves that the model is able to provide excellent fits to fertility for rural Chinese women in the 1950s.     

Software announcement: LIPRO 2.0

We describe a method for fitting the Coale‐Trussell model to fertility rates or to counts of births and exposure by single years of age. The procedure maximizes a quasi‐likelihood function and can easily be implemented using standard software. An extension to handle covariates is discussed.     

Turbulent dynamics in a XVIIth century population

"A reconstruction of the population of the Pays de Caux [France] (1589-1700) yields the time series of a fertility behavior indicator....An attempt is made to explain this general temporal structure by using a simulation model based on the autoregulation model (the so-called European Marriage Pattern), putting into play a choice of the spouse function, a fertility function, modalities of marriage and remarriage, under the environmental forcing of the reconstructed mortality conditions. The correspondence between reconstruction and simulation turns out to be quite good....A second simulation with simulated mortality conditions shows a bifurcation point: as the mean frequency of crisis increases, the state of the system leaves the lower level and concentrates more and more in the higher level." (SUMMARY IN FRE)     

Ansley J. Coale on Increases in Expectation of Life and Population Growth

The rapid decline in mortality after the end of World War II, in combination with a much slower downward adjustment of fertility, resulted in an extraordinary acceleration of world population growth. In a contribution prepared for the 1959 Vienna conference of the International Union for the Scientific Study of Population, Ansley J. Coale présented a concise and spirited exploration of the influence of mortality and fertility on the levels and patterns of growth and on the distribution of the population by age. Using the stable population model as his tool of exploration, Coale présents a comparative analysis of the implications of movements between stable states, making imaginative illustrative assumptions on changes over time and highlighting the often surprising and counterintuitive results of such calculations. The full text of this article, omitting summaries in English and French, is reproduced below from pp. 36–41 in Union Internationale pour I?étude scientifique de la population, Internationaler Bevölkerungskongress, Wien:Im Selbstverlag, 1959. Ansley Coale was one of the most prominent figures in demography in the second half of the twentieth century. He was born in 1917 and was educated at Princeton University. He spent his entire professional career at Princeton, as a member of the economics faculty and in association with the Office of Population Research, of which he was director from 1959 to 1975. In 1967/68 he was president of the Population Association of America, and from 1977 to 1981 he was president of IUSSP. His many scientific works include Population Growth and Economic Development in Low‐Income Countries (1958), coauthored with Edgar M. Hoover, a book that was highly influential in shaping the international population policy agenda from the 1960s on—lately receiving renewed attention as a predictor of the “demographic dividend” benefiting economies as a result of the transition to low fertility. He was initiator and leader of the Princeton project exploring the causes of the decline in marital fertility in Europe, culminating in the 1986 book, coedited with Susan C. Watkins, The Decline of Fertility in Europe. His most lasting contribution to population studies, however, was in the field of formal demography, as both teacher and scholar. His research in this area is exemplified by the 1972 book The Growth and Structure of Human Populations: A Mathematical Investigation, and in the application of demographic models to the estimation and analysis of population data. Ansley Coale died on 5 November 2002, at the age of 84.     

Populations with constant immigration

We consider a Leslie‐type model of a one‐sex (female) population of natives with constant immigration. The fertility and mortality schedule of the natives may be below or above replacement level. Immigrants retain their fertility and mortality, their children adopt the fertility and mortality of the natives. It is shown how this model may be written in a homogeneous form (without additive term) with a Leslie‐type matrix. Reproductive values of individuals in each age group are discussed in terms of a left eigenvector of this matrix. The homogeneous form of our projection model permits the transformation into a Markov chain with transient and recurrent states. The Markov chain is the basis for the definition of genealogies, which incorporate immigration. It is shown that genealogies describe the life histories of individuals in a population with immigration. We calculate absorption times of the Markov chain and relate them to genealogies. This extends the theory originally designed for closed populations to populations with immigratioa     

Stabilization,birth waves,and the surge in the elderly*

Substantial regularities characterize the transition to stability that follows a shift from one set of vital rates to another. The new vital, rates interact with the population's initial age composition and generate birth waves whose amplitude and attenuation depend on the ratio of ultimate to initial growth and on the new pattern of stable net maternity. A greater change in growth and a later stable net maternity pattern produce larger fluctuations in the number of births. Stabilization begins at the youngest ages and proceeds upward. Sixty years after the shift, the birth waves have largely disappeared and the proportion under age 15 approximates the stable level implied by the new rates. Those patterns are manifest in the stabilization of both observed and Coale‐Demeny model stable populations.

When fertility falls, the new stable population has a larger fraction at all ages above (approximately) 30, with greater changes characterizing the extremes of life. Fifteen years after the fall, there is a trough in the number at ages 0–14. Sixty years after the fall, when the largest pre‐decline cohort is age 60–74 and the smallest post‐decline cohort is age 45–59, there is a surge in the relative size of the elderly population. Thus after two generations, the birth waves produced by a rapid decline in fertility accentuate the effects of population aging.     

Population growth with variable family size

The Sharpe‐Lotka continuous time deterministic model of population growth is developed to take account of some possible forms of mother‐daughter fertility association, characterised here by a bivariable measure, A. This leads to a linear double integral equation for which, subject to certain conditions, a finite time solution can be found by Laplace transform methods and thus also model specific results relating the intergenerational fertility effect to the long term population growth rate and magnitude are established. The quantitative implications of the theory are illustrated by a consideration of a general bilinear form of A and in this context numerical results illustrating the finite time growth and also the long term distribution of fertility levels in the stable female population are obtained. In particular, it is shown that different fertility specific subpopulations can coexist indefinitely.     

The mathematics of sex and marriage,revisited

We analyze the problem of modeling marriages in a two‐sex model of population dynamics. We first deal with the problem of incomplete and inconsistent census data and then use a simulator to compare the performance of a variety of marriage functions in modeling births and couples during the ten‐year period between consecutive U.S. censuses. Unlike most empirical methods for comparing marriage functions based on goodness of fit, the differences in the projections of the various functions in our method are of the same magnitude (or even smaller) than the errors between the projected and real data. We observe that for the population of the United States, the harmonic mean function frequently found and used in the literature is a quite poor performer when compared with many other functions in the family we use.     

Persistent age distributions for an age‐structured two‐sex population model*

In this paper we formulate an age‐structured two‐sex population model which takes into account a monogamous marriage rule and the duration of marriage. We are mainly concerned with the existence of exponential solutions with a persistent age distribution. First we provide a semigroup method to deal with the time‐evolution problem of our two‐sex population model. Next, by constructing a fixed point mapping, we prove the existence of exponential solutions under homogeneity conditions.     

Factors Associated with the Finnish Fertility Decline Since 1776

The analysis of annual age-specific fertility rates in Finland over more than 200 years reveals the existence of a significant early fertility decline at the end of the eighteenth century preceding the secular decline that started around 1910. A reconstruction of age-specific proportions married by a simulation model based on Coale's marriage model indicates that the mean age at marriage increased and the proportion ever-marrying decreased substantially during the period of the early fertility decline. A modification of the index of family limitation applied under certain assumptions to overall fertility rates also indicates that fertility was essentially natural until 1910. Cross-lagged correlation analysis shows that infant mortality does not influence subsequent fertility in the pre-modern period. Finally, a number of socio-economic indicators are related to fertility, and conclusions are drawn from the Finnish case about several hypotheses in the field of demographic transition.     

Population momentum: A wider definition

Summary

Keyfitz has derived an elegant formula for estimating the ultimate size of an initially stable, growing population that abruptly reduces its fertility to replacement level. Reduction of fertility is achieved by the rather unrealistic device of dividing the original age schedule nffertility rates by the net reproduction rate. Only the inertia of the age distribution is thus accounted for, but not that of the fertility schedule. The key idea of an abrupt imposition of a fixed regimen capable in the long run of generating zero population growth may be retained, but the regimen made more realistic. By elaborating the population setting, such disparate ZPG regimens as reduction of marital fertility by contraception, delayed and/or less universal marriage, raised mortality risks, or permanent net out-migration may be formulated. Convergence of the populaton to stationarity becomes a two-phase process: a primary adjustment period of changing fertility rates followed by a period of age adjustment.

The present paper treats what happens when a fixed ZPG sterilization regimen, defined by a minimum age of sterilization γ and constant continuous risk φ of sterilization among unsterilized wives aged γ to β, is imposed abruptly (or else progressively over an interval T) upon an initially stable, growing population. Additional sources of residual growth are: (1) the nine-month lag in sterilization effect owing to pregnancy: (2) the more youthful pattern of child-bearing under sterilization: (3) the extra adjustment period (of length β-γ-0.75) of changing fertility rates; and (4) any delays in exposing elements of the population to the sterilization regimen.

Two questions are pursued. First, how important are the additional sources of residual growth? Secondly, how do their relative sizes vary as a function of the characteristics of the initial population?     

On the distribution of completed parities when fertility is heritable

Over the last one hundred years, there has been, in many developed countries, a demographic convergence towards the two child family. The possible implications for population growth of such a tendency are considered in this paper in terms of both family limitation and also the intergenerational transmission of fertility. These two effects interact so that as the proportion of two‐child families increases, the possible influence of mother‐daughter fertility associations on population growth decreases, though even now it could override otherwise significant changes in either or both of the birth and death intensities. In particular, it is shown that according as to how fertility is transmitted through generations, it is still possible to have zero growth rates consistently with a widely dispersed stable distribution of family size as well as a typical mortality regime.     

Statistical modeling of selected aspects of the childbearing process with application to world fertility survey countries∗

The childbearing process should be monitored in developing countries experiencing high population growth rates and high levels of maternal and infant mortality. A mathematical model for estimation of certain aspects of the childbearing process, which requires only data on age‐specific fertility rates, is developed. Synthetic maternal childbearing indices, namely, mean ages at first and last birth, length of reproductive life span, inter‐birth spacing, and proportion of childless women, in addition to the well‐known mean age at childbearing, for the WFS countries are obtained using the proposed model. The indices are free from age truncation effects, and, under certain assumptions, provide information about a cohort's completed fertility before the women stop reproducing. The effects of women's residence and education on fertility are also examined.     

Estimation of vital rates by means of monte carlo simulation

Monte Carlo simulation has been used to estimate age-specific fertility and mortality rates for a small population,the French-derived isolate of Northside on St. Thomas, U. S. Virgin Islands. Estimates were based on data collected in a household census and genealogical survey and on birth, death, and marriage records for the years 1916to 1966. During this 50-year period (in which the population size increased from 202 to 657), the numbers of births and deaths were too, small to estimate age-specific rates directly, and in addition, death registration was incomplete. Mortality rates were estimated using a simulation program in which mortality was the only stochastic variable. A model mortality schedule was chosen which most accurately reproduced the growth pattern of the population over the 50-year period. To estimate fertility rates, a more complex simulation model was used in which fertility, nuptiality, and mortality were random variables with probability distributions. Preliminary estimates of fertility were made from the birth records and used as input to this simulation program. Birth probabilities were adjusted empirically from one set of simulation runs to the next, until population growth rates, as well as other demographic characteristics, were similar in the real and simulated populations. The birth rates which produced the best fit to the real population data were taken as the estimated age-specific fertility schedule. To reproduce the real population age structure more closely, secular changes in birth probabilities were applied.     

Estimating the goodman,keyfitz, pullum kinship equations: An alternative procedure∗

As is often the case in demography, Goodman, Keyfitz and Pullum (1974) developed their theory of the interrelationships of fertility, mortality and kinship numbers by means of continuous mathematics [integrals], but resorted to ad hoc finite approximations for calculating results in concrete empirical cases. Their reason: ‘Ordinarily, we cannot evaluate the l(x) and m(x) functions for arbitrary values of x, since the data are usually collected for five‐year age intervals’ [p. 24]. Recent developments in computer software now provide an alternative, two‐step procedure that avoids extensive programming of finite approximation algorithms: 1) using a popular scientific curve‐fitting package, functions are found to represent particular sets of discrete data on fertility and mortality, 2) the resulting functions and parameter estimates are then inserted directly into the kinship equations, and the integrals evaluated numerically using readily available mathematics software. This procedure has potentially wide application in other areas of population mathematics where theory is given by integrals and other continuous expressions, but data are for discrete age groups.     

Effect of aggregation on the estimation of trend in mortality

There are three approaches to analyzing and forecasting age‐specific mortality: (1) analyze age‐specific data directly, (2) analyze each cause‐specific mortality series separately and add the results, (3) analyze cause‐specific mortality series jointly and add the results. We show that if linear models are used for cause‐specific mortality, then the three approaches often give close results even when cause‐specific series are correlated. This result holds for cross‐correlations arising from random misclassification of deaths by cause, and also for certain patterns of systematic misclassification. It need not hold, if one or more causes serve as “leading indicators”; for the remaining causes, or if outside information is incorporated into forecasting either through expert judgment or formal statistical modeling. Under highly nonlinear models or in the presence of modeling error the result may also fail. The results are illustrated with U.S. age‐specific mortality data from 1968–1985. In some cases the aggregate forecasts appear to be the more credible ones.     

Estimating birth stopping and spacing behavior

A decomposition of age-specific marital fertility rates into indices related to spacing and stopping is developed by using Coale and Trussell's indices and the first few parity progression ratios. This approach leads to estimates of the mean birth interval among low-parity births that can be used to address the issue of fertility control early in marriage. In this way the model addresses several of the most serious limitations of Coale and Trussell's approach. The usefulness of the proposed indices is demonstrated by applications to historical data from the United States and Europe.     

Pre‐procreative ages in population stability and cyclicity

Age‐specific models of population renewal (with and without feedback) which imply convergence to a stable state for some levels of fertility or feedback may imply the presence of long‐term cycling around a constant or exponentially changing equilibrium for other levels of fertility or feedback. The switch from one regime to the other is a “bifurcation.”; The conditions for bifurcation involve the roots of an analogue of Lotka's Equation.

Typically bifurcation is induced by raising the strength of feedback or the level of fertility. It has been known since the early 1980s, however, that this is sometimes impossible. It is sometimes impossible even with the linear renewal equation itself and with the most basic of non‐linear models, Lee's cohort feedback model.

Here it is proved that this typical route to bifurcation does not fail for these basic models in the presence of a condition which always holds for realistic applications with higher organisms: the existence of a span of ages before the onset of fertility.

Specifically, a strictly positive lower bound on ages of procreation is proved to be sufficient to guarantee the existence of a rescaling of Lotka's Equation for which the real part of some complex root vanishes. This result holds for absolutely Lebesgue‐integrable (signed) net maternity functions on the positive real line and for absolutely summable (signed) net maternities on the positive integers.

It follows that Coale's rescaling device for the analysis of approach to stability in stable population theory can be implemented for all realistic human net maternity schedules. It also follows that the many special cases of the cohort feedback model throughout population biology will all generate persistent cycling instead of stability if feedback is sufficiently strong.     

A demographic study of the British ducal families

The available records of the ducal families of the British Isles have been studied in order to determine fertility and mortality among the highest social class.

The expectation of life was considerably higher for females than males, but a large part of the difference could be explained by deaths from violence. Mortality fell rather abruptly about the middle of the eighteenth century, and perhaps again in the twentieth century. At other times mortality has fallen gradually.

The mortality of the aristocracy was similar in Britain and the Continent. The differences are rather in favour of Britain, especially for children and old people.

The mean age at marriage rose from 22 to 29 for men, and from 17 to 24 for women, between the fourteenth and the eighteenth centuries. Thereafter it has scarcely varied. Eldest sons have always married at younger ages than did their brothers.

Between about 1760 and 1860, the rate of fertility was remarkbly high. To a large extent, falling mortality accounts for the sudden rise in fertility in the mid-18th century, but it does not explain all the increase. After 1860 or so, fertility fell, as in the general population, and at present ducal families are just failing to reproduce themselves.

In every period, roughly one in six of all marriages of completed fertility were childless. The decline in fertility was thus brought about by a reduction in the proportion of large families.

Especially since 1700, marriages into another peerage family produced more children than did other marriages. There is no evidence that the first child was significantly more often male than were subsequent children.