A non‐linear model of fecundability,postpartum amenorrhea,and sterility*

This paper proposes a convolution model of fecundability, controling for the effects of postpartum amenorrhea and unobserved heterogeneity in fecundability. Simulation analysis was used to assess the validity and reliability of estimates derived from the model. Analysis showed that the model captured the mean and standard deviation of age at the onset of sterility in simulated populations where sterility followed either a Gompertz, a gamma, or a lognormal distribution. The model performed well when sterility was specified by either a lognormal or a gamma distribution. The model also accurately estimated fecundability and postpartum amenorrhea. Next, the model was found to fit data from 17th and 18th century French Canadian birth histories. In this French Canadian sample the mean age at sterility was found to be 46.3 years using a gamma model. The decline in fecundability was almost linear after age 30. Thus, fecundability at age 40 had declined to about one‐third of that observed at age 30. Variability in individual fecundability was quite high. For example, women with fecundability one standard deviation above the mean had about 2.3 times as high fecundability as women one standard deviation below the mean.     

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.     

Maximum likelihood estimates for the parameters of a continuous time model for first conception

The duration of time between two successive births or between marriage and first birth is an indicator of the level of fertility of a couple. Potter and Parker (1964) and Singh (1961, 1967) have suggested the Type I Geometric as a distribution appropriate for representing the length of interval to first conception leading to a live birth. Potter and Parker estimated the parameters of this distribution with the help of the first two moments. Majumdar and Sheps (1970) pointed out the limitations of these moment estimates and gave a method to obtain maximum likelihood estimates, based on formulas which are too involved for solution without the help of a computer. Singh proposed a continuous probability distribution based on another set of assumptions for the above situation. He outlined a method to obtain best asymptotically normal estimates of the parameters. These estimates are obtained after several iterations starting from any set of consistent estimates. The objective of this paper is to show that it is relatively easier to obtain maximum likelihood estimates of the parameters of the continuous model, which describes the data on duration to first conception as well as does the discrete model. Simple expressions for the moment and maximum likelihood estimates with the corresponding covariance matrices are obtained. Application is made to three sets of data.     

Heterogeneity,dependence among causes of death and Gompertz

Heterogeneity in a population with respect to mortality, or variation in “frailty”; among members of that population, which has been discussed extensively in the literature over the last decade and a half is essential to any realistic model of dependence among causes of death. The main problem then is the development of a mortality model incorporating heterogeneity and cause of death which is both realistic and of manageable proportions.

In a recent paper (J. H. Pollard, 1991), it has been shown that many life table results are remarkably insensitive to the strict shape of the mortality curve, at least for more developed populations, and that accurate approximations can in many cases be obtained knowing only the mortality rates at two representative ages (e.g. 50 and 70). These results and the Gompertz “law”; of mortality can be used to develop manageable approximate formulae for the expectation of life under heterogeneity and correlation among the causes of death. The formulae are confirmed by simulation.

Numerical results indicate, somewhat surprisingly, that the effects of correlation among causes of death, even at quite high levels, on expectation of life and changes on expectation of life when particular causes of death are reduced or eliminated are relatively minor.     

Age patterns of mortality for older women: An analysis using the age‐specific rate of mortality change with age

The age‐specific rate of mortality change with age, defined by k(x) = d Inμ(x)/dx, where μ(x) is the age‐specific death rate at exact age x, is estimated for middle and old ages in ten selected populations that are considered to have relatively accurate age data. For females in each of the study populations, k(x) follows a bell‐shaped curve that usually peaks around age 75. In some of the populations, the age pattern of k(x) for males is confounded with substantial cohort variations, which seem to reflect long‐term impacts of their World War I experiences.

Among the mathematical models proposed by Gompertz, Makeham, Perks and Beard, only the Perks model is consistent with the bell‐shaped pattern of k(x). It is shown that, if the risk of death for every individual follows the Makeham equation and if the individual frailty is gamma‐distributed, then the age‐specific death rate follows the Perks equation.     

A comparison of three maximum likelihood models for stage-frequency data

Summary A comparison has been made between the estimates obtained from maximum likelihood estimation of gamma, inverse normal, and normal distribution models for stage-frequency data. Results have been compared for six of sets of test data, and from many sets of simulated data. It is concluded that (1) some estimates may differ substantially between the models, (2) estimates from the correct model have little bias, and estimated standard errors are generally close to theoretical values, (3) there are problems in determining degrees of freedom for chi-squared goodness of fit tests, so that it is best to compare test statistics with simulated distributions, and (4) goodness of fit tests may not discriminate well between the three models.     

Estimators of a type I geometric distribution from observations on conception times

In order to study distributions of fecundability, Potter and Parker fitted a Pearson Type I geometric distribution (with parametersa andb) to data from the Princeton Fertility Study. They, and subsequently other authors, estimateda andb from the observed moments of the month of first conception. A critical analysis of this method has shown that moment estimators ofa andb are moderately reliable only within a specified range of values ofa. Outside this range, either the estimators are extremely inefficient or their variances are not defined at all. Caution should therefore be taken in adopting this procedure. Furthermore, no moment estimate is defined whena is less than 2. It seems preferable to derive maximum likelihood estimates which have certain optimal properties and are defined for all permissible (i.e. positive) values ofa andb. For large samples, we here present: the covariance matrix (where defined) of the moment estimators, methods of obtaining maximum likelihood estimates and their covariance matrix, and the variances of estimates of specified moments of the fecundability of the sample. Results were obtained for three sets of data; in all cases, the maximum likelihood estimates fit the data better than do the moment estimates. Despite a substantial improvement, however, the fit is still poor for the two sets of data from the Princeton Fertility Study. Possible explanations are: a) that the departures from the assumption of constant fecundability for each couple are sufficient to produce the poor fit, b) that the data are inaccurate, or c) that the method of defining the sample of women from whom the data were obtained resulted in an over-representation of short conception times. The relative importance of these factors is difficult to establish.     

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.     

A Gompertz model for birth interval analysis

During recent years birth intervals have been analysed on a life table basis. This method retains both closed and open intervals, and so reflects behaviour that deliberately avoids the next birth entirely. When life tables are prepared separately for each birth order, markedly different patterns of movement toward the next birth can appear from one parity to the next. This is illustrated for Korean survey data, with historical trends given across marriage cohorts.

A Gompertz model is found to fit the family of curves that show the cumulative proportion giving birth within each interval closely. Its three parameters have direct intuitive interpretations, one being equal to the parity progression ratio and the other two controlling the pace of childbearing before and after the point of peak activity within the interval. The model is useful for interpolation and projection, and provides an efficient summary of the otherwise cumbersome detail given in a life table. Testing against additional data sets is suggested.     

Estimates of fecundability from a truncated distribution of conception times

A population consisting of women who have conceived before a time t ignores all women whose conception time exceeds t. Such a population is considered as a truncated population, and the samples are called truncated samples. Under the assumption that fecundability among women varies according to a Beta distribution (with parameters a and b), the distribution of conception times in a truncated population can be considered as truncated Type I geometric. This paper presents an algorithm to obtain the moment and maximum likelihood estimates of a and b from the truncated samples. Large sample properties of the estimators are also studied. Examples using the Hutterite and the Princeton Fertility Survey data are given.     

Mixed estimation of old‐age mortality

The estimation of the mortality of the “oldest old”; is subject to considerable random error, but important prior information exists that can be used to make the estimates more robust. Mixed estimation is a method of incorporating auxiliary information into the statistical estimation of linear models. We extend the method to cover general maximum likelihood estimation, and show that the mixed estimator can be represented approximately as a weighted average of the purely data based estimator and the auxiliary estimator. The methods can be applied to the analysis of the old‐age mortality via logistic and Poisson regression. A major advantage of the mixed estimator is the simplicity with which it can incorporate partial prior information. Moreover, no special software is needed in the fitting. We show how the targeting methods of Coale and Kisker can be represented as mixed estimation in a natural way that is more flexible than the original proposal. We also derive empirical estimates of the target information based on pooled data from several countries with high quality data. We consider the mortality of Finland at ages 80 +, study the reliability of the evidence of mortality crossover, and derive estimates of life expectancy at age 100.     

Dynamics of Mixed Coalitions Under Social Cohesion Constraints

ABSTRACT

Parameters for the birth and death diffusion life table model subject to downward jumps randomly occurring at a constant rate are estimated. The jump magnitudes have a beta distribution with support [0, l_x ], where l_x is the total number of survivors prior to the jump. The estimation method is maximum likelihood. The Cramer–Rao Lower bound and the asymptotic distribution for the MLE are derived. The model is applied to the U.S. men′s population from 1900 to 1999.     

Period Parity Progression Ratios and Birth Intervals in England and Wales, 1941–1971: A Synthetic Life Table Analysis

A method is presented for analysing maternity history data to provide period estimates of parity progression ratios, birth intervals and related indices. This is applied to a sample of the marriage and maternity histories from the Census of England and Wales of 1971 and shows: (a) a general increase through the 1950s and into the 1960s in period estimates of marriage and parity progression ratios, especially in the progression from first to second birth; (b) a general acceleration of fertility with, again, the second birth interval becoming particularly short and compact; and (c) very steep declines in third and fourth birth progression ratios from the mid-1960s. Birth interval distributions altered during the period examined. Decomposition of a progression-based total fertility index shows change in the ratios for lower birth orders to have dominated the fertility upswing and declines in ratios for higher birth orders to have initiated the subsequent decline.     

Multistate life‐tables and regression models

A survey is given of the use of modern statistical techniques in event history analysis, and in particular in the study of multi‐state life‐tables in demography. Emphasis is placed on the interplay between partial likelihood and nonparametric maximum likelihood based methods, a) when analysing semi‐Markov models or models with repeated spells, and b) in frailty models for inobservable heterogeneity.     

Determining the birth function for an age structured population

This paper deals with an inverse problem in age‐structured population dynamics, namely the recovery of the unknown birth function from the additional or overposed data consisting of the total population over a time interval equal to the maximum life span of the species. Conditions on the data are given to guarantee the existence and uniqueness of a solution, and the question of continuous dependence of the birth function on the data is addressed. Some numerical simulations are presented to indicate that one can in fact use the methods of the paper to reconstruct the birth function.     

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.     

A Gompertz fit that fits: Applications to canadian fertility patterns

In this paper an attempt is made to refine the method of fitting the Gompertz function to the cumulative fertility rates by using iterative techniques. The method is tested with the historical data series for the Canadian population. The demographic implication of the parameters of the Gompertz function as fitted to the fertility distribution is examined, and the usefulness of the method in projecting future fertility trends is studied. The Makeham function is also fitted to the fertility distribution by the same iterative technique, and the relative efficiency of this function is compared with that of the Gompertz.     

A model of numbers of births in three countries,with persistent forty‐year cycles

A non‐linear model of fertility is described, which was derived from data for more than a century from England and Wales, New Zealand, and the U.S.A. The demographic transition is modelled with a logistic function, and age‐specific fertility rates are estimated using lognormal distributions. The stepwise inclusion of a partner availability estimate in the model showed that it accounts for twenty‐nine percent of otherwise unexplained variance. Projected future births stabilise in sustained or limit cycles with periods a little longer than 40 years, and amplitudes at least 7% of the mean. The necessary conditions for cycle persistence are outlined on a graph of maximum and minimum fertility parameters.     

The multistate stable population model with immigration

This paper outlines the discrete‐time and continuous‐time formulations of the stable population model with immigration, showing their commonality. It then illustrates how the model can be extended to include multiple interacting populations, and goes on to consider a multistate version of reproductive value that further illuminates the evolutionary dynamics of an “open”; model of multistate population growth and redistribution. Attention is restricted to results arising from a fertility regime that is below replacement level.