Transforming gompertz's function for fertility analysis: The development of a standard for the relational gompertz function

Summary The relational Gompertz function improves upon the Gompertz for fertility analysis by achieving a better fit in the tails of the distribution. This is obtained by a transformation of the age scale corresponding to an empirical standard. This standard is developed from Coale and Trussell's model and is appropriate for use with populations of high fertility. The model is tested on two sets of data and is shown to produce good estimates of completed fertility even for data truncated at quite early ages. Good results are also obtained for declining fertility.     

A Relational Gompertz Model of Male Fertility: Development and Assessment

Brass has proposed a relational Gompertz model of female fertility which, in combination with the standard fertility distribution developed by Booth, has proved useful in a range of applications, such as indirect estimation, demographic modelling, and population projections. This paper develops a standard distribution of male fertility for use in conjunction with the relational Gompertz model. The derivation of the standard takes advantage of the similarity between the shape of male and female fertility distributions. It entails ‘stretching’ the female standard, so that it extends to age 80, and then transforming it, using the Gompertz model into a pattern which is more typical of male fertility distributions in the developing world. An assessment of this new standard by fitting the relational Gompertz model based on it to a series of male fertility distributions from diverse populations, suggests that it performs very well.     

On the use of the truncated Gompertz distribution and other models to represent the parity progression functions of high fertility populations

The Gompertz distribution, developed from the mortality “law”; long used by actuaries and demographers promises to be a useful distribution for many other demographic purposes as well. The continuous distribution can also be adapted to represent discrete data commonly encountered in demographic work, and maximum likelihood estimates of the two parameters are easily calculated using formulae developed in this paper, whether those data be continuous or discrete, truncated below or provided with observations in a final open‐ended interval.

The distribution is unimodel. The use of the truncated form of the distribution, however, allows the researcher to fit it to a wider range of observed distributions, including many for which the density function is monotonic decreasing.

Empirical studies using parity progression data of two high fertility populations indicate that the truncated Gompertz distribution in its discrete form provides a good overall picture of the parity distribution. Interestingly, the simple method of partial sums, commonly employed to fit the Gompertz function, appears to provide parameter estimates which are close to those estimated by maximum likelihood.     

Some demographic consequences of changing cohort fertility patterns: An investigation using the Gompertz function

Abstract In this paper are formulated some convenient summary measures of fertility patterns. These measures, which are based on the Gompertz function, are total lifetime fertility, median age of mothers at childbirth, and inter-quartile range of age of mothers at childbirth. Estimates of the parameters of Gompertz function, based on Canadian data, are used to derive, for each of the summary measures, values which reflect historical fertility experience, and thus give an impression about the range of realistic values for these measures. A simple model of demographic activity which includes the Gompertz function is also considered, and this model is used in computer simulation experiments to determine the macro-demographic effects of changes in each of the three summary measures.     

On the pattern of cohort fertility

Abstract An examination of the patterns of cohort fertility rates in a number of populations revealed the existence of a common pattern showing how mean family sizes of the different age-at-marriage groups of a given marriage cohort are built up over the reproductive span. Standard schedules of fertility patterns are generated by the use of the Gompertz function which defines the distribution of cumulative fertility rates of a given marriage cohort by marriage duration by means of only three parameters. The generated system is tested with the historical series for England and Wales and Sweden and is found adequate to describe widely different childbearing patterns. Some of the demographic implications of the existence of common fertility patterns are examined and the usefulness of the generated system in projecting future fertility trends is briefly discussed.     

Experiments in modelling recent Danish fertility curves

This paper reports the outcome of an exercise in curve fitting to annual sets of Danish age-specific fertility rates for the years 1962 to 1971 by means of least squares. Functions fitted were a cubic spline, the Hadwiger and Coale-Trussell functions, the gamma and beta densities, two versions of a polynomial, and two of Brass's relational procedures, as well as the Gompertz curve. The spline function fitted all curves far better than any of the others. The Coale-Trussell procedure and gamma density were about equal, followed by the Hadwiger function. All of these functions fit the data well. One of the polynomials fit reasonably well, but the rest of the functions were less accurate.     

What is natural fertility? The remodeling of a concept

This paper applies three log-linear models to Louis Henry's original 1961 natural fertility data in order to test various assumptions leading to ways of obtaining a standard natural fertility schedule through explicit modeling. "The models specify that births follow an independent Poisson distribution for each age interval of each population. All parameters are estimated through an iterative maximum-likelihood procedure." The author suggests that the model selected provides better estimates of the standard natural fertility function than previous models.     

A dynamic model of the reproductive process

Summary A dynamic deterministic model of the reproductive process is presented. The model describes and analyses the effect of intermediate fertility variables on fertility. The intermediate fertility variables which are the inputs to the model, include the duration of post-partum amenorrhea, fecundability, incidence of spontaneous and induced abortion, contraceptive use and effectiveness, the distribution of age at first marriage and the age specific risks of marital disruption and remarriage. To test the validity of this model, it is fitted to data obtained from reproductive histories of 512 marriages occurring during the first half of the eighteenth century in Canada.     

Probabilistic Forecasting Using Stochastic Diffusion Models, With Applications to Cohort Processes of Marriage and Fertility

In this article, we show how stochastic diffusion models can be used to forecast demographic cohort processes using the Hernes, Gompertz, and logistic models. Such models have been used deterministically in the past, but both behavioral theory and forecast utility are improved by introducing randomness and uncertainty into the standard differential equations governing population processes. Our approach is to add time-series stochasticity to linearized versions of each process. We derive both Monte Carlo and analytic methods for estimating forecast uncertainty. We apply our methods to several examples of marriage and fertility, extending them to simultaneous forecasting of multiple cohorts and to processes restricted by factors such as declining fecundity.     

Modelling Marital Fertility by Age and Duration: An Empirical Appraisal of the Page Model

Page's model of marital fertility by age and duration is fitted by maximum likelihood techniques to data from 38 of the 41 developing countries that participated in the World Fertility Survey. The results indicate that the model does an excellent job of capturing variations in fertility patterns, with only two parameters. Moreover, national-level estimates of the parameter representing the degree of control of marital fertility correlate reasonably well with the proportion using contraception. On the other hand, estimates of the parameter representing the level of natural fertility correlate well with the duration of breastfeeding and with a measure of contraceptive use for spacing, but also show substantial regional variation. The paper closes with comments on several extensions and applications of the model.     

Empirical analysis of the contribution of age composition to population growth

The method of decomposition is applied to rates of natural increase in order to elucidate the role played by age composition in the growth of populations. A population’s age distribution and fertility schedule are contrasted to those in a "stationary" population having the same mortality rates and having a fertility schedule equal to that of the observed population divided by its net reproduction rate. In this manner it is shown that about one-quarter to one-third of the growth of most current high-growth populations can be attributed to non-stationarity of their age distributions. This fraction will rise, as it has in most industrialized countries, if fertility is reduced and age distributions become middle-heavy. In projections of the 1963 Venezuelan female population with fertility rates declining by 20/0 and 1% annually, more than half of the growth (in numbers) that occurs prior to zero-growth attainment is contributed by non-stationarity of its intervening age distributions.

     

A method of estimating the time of marital fertility decline and associated parameters

Significant research attention has been given to historical patterns of marital fertility transitions in currently industrialized countries. Specifically, studies consider the time of the onset of fertility decline and its distribution among populations or population subgroups; the distribution of pre-decline parital fertility levels; and/or the rate of marital fertility decline. Analyses of pre-decline fertility level and its rate of decline, however, depend upon the procedure used to estimate the time of fertility decline. The Princeton European Fertility Project is the most prominent historical fertility study ever conducted. The procedure employed to estimate the timing parameter in these Princeton studies is described. An alternative statistical procedure is then proposed for detecting the onset of the transition from high to low marital fertility; the method may also be used to find the termination point of decline where the sequence of fertility variables is long enough. Both methods produce maximum likelihood least squares estimates, but the form proposed in the text has conceptual advantages.     

Cohort Parity Analysis and Fertility Transition Dynamics: Reconstructing Historical Trends in Fertility Control from a Single Census

The Cohort Parity Analysis (CPA) method for the indirect measurement of the extent of marital fertility control makes use of parity distribution data of the kind reported, for specified marriage duration and marriage-age (or current age) groups of women, by a fertility census. Corresponding to every such distribution, CPA yields efficient upper and lower bounds on the proportion of controllers in the specified cohort, as well as bounds for the parity distribution of the controllers. As the women who belong to different marriage cohorts are observed at different durations of marriage in a census, it is possible to infer inter-cohort trends in the extent of fertility control (at some specified marriage duration(s)) if one knows the time pattern of the adoption of control among the members of each cohort. It is shown in this paper that robust estimates of the intracohort diffusion pattern can be obtained from the census cross-section itself, by supplementing the basic assumptions of the CPA framework with a plausible assumption about the behaviour of the ‘representative’ controller in a given cohort. The estimation technique is illustrated by applications to data for married women in urban Eire from the Census of Ireland in 1911, and to native white married women in the U.S. South from the U.S. Census of 1910.     

Fertility levels and differentials in Beirut during wartime: an indirect estimation based on maternity registers

In this paper, total fertility estimates for Greater Beirut in the mid-eighties and early nineties are presented, and changes in socio-religious differentials of fertility across time are explored. The baseline information was recorded from registration details for all maternities in Beirut and its inner suburbs in 1984 and 1991: age of mother, number of children ever-born, hospital class, and religion of newborn. An indirect method was used to estimate total fertility from the joint distribution of mothers by age and parity, and, using hospital class as a proxy for social class, differentials in fertility were investigated by Poisson regression. The estimates of total fertility for Beirut shifted from 2.60 in 1984 to 2.52 in 1991, and were higher for Muslims than for Christians in the two periods. The regression analysis showed that: (1) the difference between the two religious groups persisted after control for social class, and in fact applied to the lower social class; (2) fertility dropped between the two dates in the lower social class, and more so for Muslims than for Christians. In comparison with other countries of the region, the decline in Beirut was found to be relatively modest. If the trends assessed in this study were to continue, the religious-based fertility differentials would taper off progressively in the capital city of Lebanon.     

The construction of the fertility model of variable parity from family planning practice]

Understanding the changing patterns of age specific fertility under the planning system is essential for building a fertility model which reflects birth control policy implementation in China. In building a Parity Variable Fertility Model, 4 basic elements are to be considered: 1) psychosocial, and physiological variables, 2) patterns of the total fertility rate and age-specific fertility rate, 3) socioeconomic development, and 4) distribution of parity-specific fertility. THe natural fertility of women is 17, calculated from a 309-years childbearing period, with 17% of non-susceptible time. In China, about 86% of natural fertility is suppressed by various factors. In this model, the following variables are included: 1) The first marriage ratio, which is the proportion of women in each age group which enters into a first marriage. The range and spread of this ratio is closely associated with the first birth. 2) The first birth ratio, which is the proportion of a marriage cohort to have a first birth each year. 3) the birth interval, which determines the distribution of second births. 4) Regulation coefficient B, which represents birth control regulations which approximately determine the number of second-parity or higher order births. The difference between the fertility level generated from the Parity Variable Fertility Model and reality depends on the implementation of birth control program, the assumptions on regulation coefficients, and changes in social and cultural factors. The model is easy to use, especially for areas where the marriage and fertility records of women of child-bearing are well kept.     

Lifetime probability of additional births by age and parity for American women, 1935–1968: A new measurement of period fertility

A new fertility measurement, probability that a woman of specified parity and age will bear a given number of births in her remaining lifetime, is proposed in this article. The measurement is a summary index of a set of age-parity-specific birth probabilities and in a particular case it is conceptually analogous to the total fertility by birth order but free from the influence of the parity distribution. Fertility of American women for the period 1935-1968 was studied by use of such lifetime probabilities. It appears that the trend of fertility of high birth orders has been parallel with that of low orders. Initiation of the recent decline in fertility depends on parity and age of woman.     

Aboriginal fertility: Trends and prospects

Estimates of Aboriginal fertility compiled from an analysis of 1981 and 1986 Census data on children ever borne by Aboriginal women reveal age-specific fertility rates slightly higher than those of other Australian women at ages above 25, but very much higher rates for younger women. The result is a total fertility ratio more than 50 per cent higher than in the total Australian population, with no more than slight variation between States and Territories. A differential analysis using standardized indices shows considerable differences in levels of fertility of categories of young Aboriginal women classified by education, labour force status and income, and also differences between urban and rural areas. Analysis of prospects for Aboriginal fertility levels confirms the likelihood of continuation in the downward drift in levels of fertility that has been established during the past decade. Comparison of the estimates with another recent set of estimates obtained using the own-children method shows broad conformity in levels of total fertility ratios over time, except in the most recent period, the mid-1980s. Nevertheless, the own-children estimates distort the recent trend and also the age distribution of Aboriginal fertility.     

Immigration and the stable population model

This paper reports on work aimed at extending stable population theory to include immigration. Its central finding is that, as long as fertility is below replacement, a constant number and age distribution of immigrants (with fixed fertility and mortality schedules) lead to a stationary population. Neither the level of the net reproduction rate nor the size of the annual immigration affects this conclusion; a stationary population eventually emerges. How this stationary population is created is studied, as is the generational distribution of the constant annual stream of births and of the total population. It is also shown that immigrants and their early descendants may have fertility well above replacement (as long as later generations adopt and maintain fertility below replacement), and the outcome will still be a long-run stationary population.     

Estimating infant and childhood mortality under conditions of changing mortality

Summary It is well known that estimates of infant mortality obtained using Brass's technique are very accurate. Biases are introduced, however, when one or more of the assumptions on which it relies are violated. Departures from the assumption of constant fertility may be handled by using a variant of the technique which depends on information on the age distribution of surviving children, rather than on indexes of the fertility function. Violations of the assumption of constant mortality - an increasingly common situation in most developing societies - produce upward biases in the estimates. The amount of bias is a function of the speed of mortality decline, the characteristics of the fertility pattern and, finally, of the age of the mother. This paper presents a simple technique which corrects these biases, and in addition, generates estimates of the parameters of the mortality trend. It differs from others in that it uses a cohort definition of mortality decline and relies on knowledge of the age structure of surviving children rather than on indexes of the fertility pattern.     

A probability mixture model of completed parity

This research was designed to find a probability model that would adequately describe completed parity for women in populations characterized by high fertility combined with high zero parity. A negative binomial mixture distribution was adapted for this purpose. The form of this distribution suggests the hypothesis that human populations consist of two subgroups of women with respect to completed parity, defined according to whether they do or do not produce viable offspring. Results of the present research further suggest that the proposed distribution for completed parity has general applicability across human populations, whether they are of a high or low fertility type, and whether they have a high or low zero parity.