A modification of a continuous time model for first conception

The interval between marriage and the first conception leading to a live birth plays an important role in the determination of fertility components. Several probability models (Potter and Parker, 1964; Singh, 1961, 1964, 1967) based on varying sets of assumptions relating to this interval have been propounded in the recent past. All of them are based on the assumption that the females under study are susceptible to conception at the time of marriage. However, in certain situations, where some of the females already pregnant at the time of marriage report to have conceived within a short interval following marriage, this condition is not satisfied and these models become unsuitable. A probability distribution which is an inflated form of the continuous model proposed by Singh, for the time of the first conception leading to a live birth, is presented in this paper. It describes reasonably well the data on first conception times in the context of premarital conceptions. Simple expressions for the maximum likelihood estimators of the parameters involved in the model are obtained and a method for finding the asymptotic covariance matrix of the estimators is outlined. The model is applied to four sets of data.     

Stability and change in temporal distance between the elderly and their children

Drawing on a developmental model of late-life migration, this paper investigates how older people’s health and social characteristics influence stability and change in their temporal distance from their children. Data from the Longitudinal Study of Aging are used to examine both discrete transitions and continuous change in distance over a four-year period. Decline in older parents’ physical health increased the propensity of parents and children to become temporally closer to each other. Among those parent-child pairs who had become closer, the conjunction of declining health and widowhood increased both the degree of non-coresident proximity and the likelihood of transition to coresidence. The findings portray a geographically resilient family that adjusts to the changing needs of its older members.     

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.     

Pre-procreative ages in population stability and cyclicity

"Humans' protracted maturation before childbearing is an extreme example of a general characteristic of higher organisms: a positive lower bound on ages of procreation. The pre-procreative span, much discussed for its evolutionary and social ramifications, has consequences also for the mathematics of population renewal....This paper proves the presence of a pre-procreative span sufficient, in and of itself, to guarantee the existence condition for bifurcation for all models in one important class. The class includes the best-studied examples of age-specific systems with and without homeostatic feedback in purely discrete and in purely continuous formulations."     

Dynamics of Asset Poverty in South Korea

Studies of economic insecurity tend to focus on changes in incomes and transitions in and out of income poverty. Yet family economic conditions are shaped by more than just income. Levels of assets and wealth, and changes in these over time also play a role. To identify which groups of the poor have been structurally trapped in poverty over time, using the Korean Welfare Panel Study, we examined the dynamics of asset poverty from 2005 to 2014. We defined three asset poverty lines by operationalizing assets as resources for either future consumption or development. Findings show that asset poverty experience in a previous year significantly increased the probability to incur asset poverty by 5–12% for all analysis samples. In addition, the probability of incurring asset poverty decreased with home ownership, higher disposable income, and greater diversification of the household portfolio. Our findings suggest that the asset poor are likely to fall into structural and persistent poverty over time, and an asset-building approach is needed to improve the asset poverty status of households in South Korea.

     

Measures of Inequality: Application to Happiness in Nations

What is a good measure for happiness inequality? In the context of this question, we have developed an approach in which individual happiness values in a sample are considered as elements of a set and inequality as a binary relation on that set. The total number of inequality relations, each weighed by the distance on the scale of measurement between the pair partners, has been adopted as an indicator for the inequality of the distribution as a whole. For models in which the happiness occurs as a continuous latent variable, an analogous approach has been developed on the basis of differentials. In principle, this fundamental approach results in a (zero) minimum value, and, more importantly, also in a maximum value. In the case where happiness is measured using a k-points scale, the maximum inequality is obtained if all ½N sample members select the lowest possible rating (Eq. 1) and the other ½N the highest possible one (k). This finding even applies to the truly ordinal case, i.e., if the distances between the successive ratings on the scale are unknown. It is, however, impossible to quantify the inequality of some measured sample distribution, unless all distances of the k categories of the scale of measurement are known or at least estimated, either on an empirical basis or on the basis of assumptions. In general, the numerical application of the method to continuous distributions is very complicated. An exploration on the basis of a relatively simple model with a linear probability density function suggests that the inequality of a beta probability distribution with shape parameters a and b increases as the value of these parameters decreases. A contour plot, obtained by numerical integration, demonstrates this relationship in a quantitative way. This approach is applicable to judge the aptness of common statistics of dispersion, among which the standard deviation and the Gini coefficient. The former is shown to be more appropriate than the latter for measuring inequality of happiness within nations.     

Net delay of next conception by contraception: A highly simplified case

Abstract A probability model, building on the work of Perrin and Sheps, is presented and applied. The model makes it possible to follow a cohort of women from a pregnancy outcome to next conception. Principal simplifying assumptions are homogeneity among women and constant fecundity. Cases treated include no contraception; a single contraceptive method practised; and a second contraceptive method practised following a first one. Expected durations to next conception may be calculated in relation to the following factors: outcome of previous pregnancy, length of anovulation, natural fecundability, time first contraceptive method is initiated, characteristics of first contraceptive (effectiveness and continuation rate), characteristics of second contraceptive, and gap between termination of first and commencing second contraceptive. By suitably pairing runs of the model, one can construct experiments in which the only differentiating factor is use and non-use of a specified contraceptive, and, by differencing the corresponding mean durations to next conception, compute the net delay of next conception produced by that contraceptive.     

The precision of population projections studied by multiple prediction methods

A population projection is a prediction of a random vector variable X _T . which represents the size and age/sex distribution of the population in year T. The population is assumed to be closed and to develop according to fixed and known schedules of birth and death probabilities as a multitype branching process. The precision of the usual projection e _T (= EX _T) is studied by a family of prediction intervals of linear functions of the vector of deviations X _T — e _T , which has a preassigned probability level. This family is obtained by a multi-normal approximation and an argument similar to the one leading to Scheffé's method of multiple comparison. From the family of prediction intervals, an upper limit of the total absolute deviation Σ |X _iT ? e _it | is obtained, and the ratio of this limit to the projected total population is proposed as a measure of the relative precision of the projection. For a numerical study, Norwegian population data is used.     

The impact of infant and child death on subsequent fertility in Ethiopia

This paper uses hazard regression models to assess the impact of experienced infant and child mortality on the risk of subsequent conceptions in Ethiopia. The purpose of this paper is to test for the presence of a fertility response to an infant or child death, net of the effects of truncated breastfeeding on fecundity. Using retrospective birth history data from a national survey in Ethiopia, we find a significantly higher risk of a conception in the months following the death of an index child, even after controlling for postpartum amenorrhoea and breastfeeding status. The fertility response is strongest after the death of the fourth or fifth child, which is when most women in Ethiopia are at or near their desired family size. However, we find no evidence of a fertility response to the death of a nonindex child. We attribute the higher risk of a conception following an index child’s death to the intentional efforts of couples to reduce the waiting time to a next birth and thereby replace the deceased child. However, absent evidence of replacement fertility in response to the death of older nonindex children, we interpret the response to the death of an index child as an emotional response to child loss rather than a conscious strategy to meet a fertility target.

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.     

The effect of marital status at first birth on marital dissolution among adolescent mothers

The sequencing of marriage and first birth was expected to play an important role in the stability of marriage among adolescent mothers. We hypothesized that adolescent women who married prior to conception would have the lowest rates of marital disruption, followed by those who married between conception and birth. Adolescent women who gave birth prior to marriage were expected to suffer the highest rates of marital dissolution. The results provide partial support for our hypotheses. There is little difference in the probability of separation between adolescent mothers who had a postmarital conception and those who had a premarital conception but married before the birth. Having a premarital birth, however, significantly increases the probability of marital dissolution. We also hypothesized that marital status at first birth would have less effect on the probability of marital dissolution for blacks than for whites. This, too, is generally supported by our findings. Among black females, those with a premarital birth are the first to suffer a marital disruption, but by the end of ten years there is little difference in the probability of separation among the three marital status groups. In contrast, among white females, those with a premarital birth are the first to experience a disruption, and this differential persists over all subsequent marriage duration intervals. Thus, the sequencing of marriage relative to birth has similar short term effects for whites and blacks, but the effect for blacks is evident only in the short term. Ten years after the marriage, black adolescent mothers have similar rates of marital stability regardless of the sequencing of marriage. This is consistent with the findings of previous research and with our hypothesis; with the black family pattern of lower rates of marriage, higher rates of illegitimacy and higher divorce rates, the sequencing of marriage has no long lasting consequences on marital stability. Finally, our predicted decline in the effect of marital status at first birth over historical time also finds partial support. For white females there has been a change in the effect of marriage-first birth sequencing on separation over time. In the period encompassed by the women in our study, white adolescent mothers who married subsequent to the birth have been the most likely to experience a separation at all marriage duration intervals, but this differential narrows as age at interview declines. Among black females there has been no change in the effect of a premarital birth over time.(ABSTRACT TRUNCATED AT 400 WORDS)     

The solution of time‐dependent population models

We analyze the dynamics of age‐structured population renewal when vital rates make a transition in a finite time interval from arbitrary initial values to any specified final values. The general solution to the renewal equation in such cases is obtained. This solution describes the birth sequence explicitly, and also leads to a general formula for population momentum. We show that the duration of the transition determines the complexity of the solution for the birth sequence. For transitions that are completed in a time smaller than the maximum age of reproduction, we show that the classical Lotka solution found in every textbook also applies, with a small modification, to the time‐dependent case. Our results substantially extend previous work that has often focused on instantaneous transitions or on slow and infinitely persistent change in vital rates.     

A computer model of family building based on expected values

A computerized probability model of family building is described. Called FERMOD, the model is designed to follow the changing distribution of children ever born and birth intervals of a large homogeneous population as it moves through the reproductive period. Use of the model presupposes assumptions about such factors as risks of fetal wastage, lengths of pregnancy, and postdelivery amenorrhea conditional to outcome of pregnancy, length of reproductive period, fecundability, desired family size, preferred birth spacing!, and effectiveness of contraception. Use of FERMOD is illustrated in a limited application to United States fertility. Relations to other models are briefly discussed.     

Large Deviations Analysis of Extinction in Branching Models

Cramer's classical theorem is applied to obtain large deviations in branching processes. This is a new avenue for analysis of models in discrete and continuous time. For the Galton-Watson process a new formula for the rate function in terms of the Legendre transform of its offspring distribution is derived. Further analysis of the approximate path to extinction produces a new interesting formula.     

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.     

Incorporating heterogeneous intercourse records into time to pregnancy models

Information on the timing of intercourse relative to ovulation can be incorporated into time to pregnancy models to improve the power to detect covariate effects, to estimate the day-specific conception probabilities, and to distinguish between biological and behavioral effects on fecundability, and therefore the probability of conception in a menstrual cycle. In this paper, Bayesian methods are proposed for joint modeling of intercourse behavior and biologic fecundability. The model accommodates a sterile subpopulation of couples, general covariate effects, and heterogeneity among fecund couples in menstrual cycle viability and in frequency of unprotected intercourse. Methods are described for incorporating cycles with varying amounts of intercourse information into a single analysis. A Markov chain Monte Carlo algorithm is outlined for estimation of the posterior distributions of the unknowns. Themethods are applied to data from a North Carolina study of couples attempting pregnancy.     

Meet the Parents? Family Size and the Geographic Proximity Between Adult Children and Older Mothers in Sweden

The aim of this study is to estimate the causal effect of family size on the proximity between older mothers and adult children by using a large administrative data set from Sweden. Our main results show that adult children in Sweden are not constrained by sibship size in choosing where to live: for families with more than one child, sibship size does not affect child-mother proximity. For aging parents, however, having fewer children reduces the probability of having at least one child living nearby, which is likely to have consequences for the intensity of intergenerational contact and eldercare.