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.     

Findings and implications of a correlation analysis of the closed and the open birth intervals

A correlation analysis of data on four fertility variables viz. closed birth interval, open birth interval, age and parity, collected in a survey of about 2,000 married women in the reproductive ages in rural India, is carried out in order to study the interrelationships among these variables. Two hypotheses are formulated governing the relationship of the closed and open birth intervals with the fecundability distribution of fertile women, and Parity Progression Ratios which are largely influenced by the proportion of women becoming sterile after each parity. The data lend support to the hypothesis that while the closed intervals are influenced mainly by the distribution of fecundability of women of non-zero fecundability, the open birth intervals are influenced mainly by Parity Progression Ratios or the proportion of women becoming secondarily sterile after each parity. The analysis suggests that we can use the mean open intervals of women classified by parity as indices of fertility, and such an index is comparable to the index of average age of women of given parity. In areas where it is difficult to ascertain the correct age of women this might be of practical value.     

Parity patterns of birth interval distribution

A simplified model is presented for the measurement of inter-live birth interval distributions. Comparative analyses of "effective fecundability," the mean length and variation of postpartum infecundity, and the parity progression ratio based on this measurement model are presented using data from U.S. marriage cohorts and from an Indian sample. Parity differentials in these reproduction parameters are investigated, and the ramifications for fertility estimate and projection discussed.     

Systematic and chance components in fertility measurement

Abstract A Monte Carlo simulation model of fertility is developed to incorporate a decline in fecundability with age of women as well as a variation between women, and a family planning strategy in which couples prefer certain sizes of families and seek to lengthen the last planned birth interval. In the simulation of family planning and non-family planning populations, segmental parity tables are computed. These are tables in which the numbers of women in the columns representing the completed family sizes are sub-divided by rows representing the sizes of the families for shorter durations of marriage. The couples' success in their plans, under various conditions, and the rapidity with which family sizes stabilize, are among the results obtained from the simulation. The application of the tables to the estimation of family size preferences from observed family size distributions is also discussed.     

On some probability distributions for forward birth interval

Statisticians from Varanasi, India have proposed 2 mathematical models for forward birth interval which are more realistic than past models. For example, past models have assumed fecundability to be constant throughout a woman's reproductive life. On the other hand, the proposed models assume fecundability changes with parity. Further the statisticians use all types of marital durations making their models usable even for short marriage durations. The 1st model uses a parity specific model to develop a model for forward birth interval regarless of parity. The statisticians also include moments of the 1st and 2nd models as expressed for the rth order raw movement of these models to estimate some of their parameters. These models can be used to conduct prospective surveys of family planning programs since birth intervals can best evaluate the effect of these programs. These models have not yet been advanced enough to follow women over a fixed period, however. They only observe women until the last woman conceives. The statisticians do not apply the models to parity specific data because they were not available. Yet they do run the models with parities 0 and 1. The models reveal that the mean length of forward birth interval is strongly influenced by the conception rate which the woman has at the time of the survey. Thus, if women with a certain parity in a community begin an effective family planning method on the survey day, which in effect would reduce the prevailing conception rate, that method would significantly postpone the next live birth.     

Some comments on article by K. Srinivasan: 'A probability model applicable to the study of inter-live birth intervals and random segments of the same'

Abstract This paper discusses a probability model of birth intervals, proposed by K. Srinivasan, for the study of data collected in some fertility surveys. New formulae for mean and variance of the "open interval" distribution are given.     

Reflections on Recent Levels and Trends of Fertility and Mortality in Egypt

This paper discusses a probability model of birth intervals, proposed by K. Srinivasan, for the study of data collected in some fertility surveys.

New formulae for mean and variance of the “open interval” distribution are given.     

Fecundability and its relation to age in a sample of Taiwanese women

Abstract In this paper reliable estimates of mean fecundability at marriage for a sample of heterogeneous Taiwanese women are obtained by using a set of data collected retrospectively. The effects of 'truncation bias' and 'memory bias' are estimated by studying the relationships between mean fecundability and the duration of marriage. Then the variations in fecundabilities by age at marriage are studied. The data are taken from an intensive fertility survey of married women between the ages of 20 and 39 conducted in Taichung city of Taiwan towards the end of 1962 prior to a year-long family planning action programme. Fecundability, the probability of conception in the absence of contraception, is estimated by using a Type I geometric model and is estimated from the observed distribution of first pregnancy intervals-the period between the onset of marriage and the beginning of first conception. The estimated fecundability level relatively free from truncation and memory bias is 195±3 per 1.000 women. The fecundability level increases with wife's age at marriage (up to 25) independently of its association with the duration of marriage.     

A Multistate model of fecundability and sterility

This paper develops a multistate hazards model for estimating fecundability and sterility from data on waiting times to conception. Important features of the model include separate sterile and nonsterile states, a distinction between preexisting sterility and sterility that begins after initiation of exposure, and log-normally distributed fecundability among nonsterile couples. Application of the model to data on first birth intervals from Taiwan, Sri Lanka, and the Amish shows that heterogeneity in fecundability is statistically significant at most ages, but that preexisting sterility and new sterility are unimportant before age 40. These results suggest that sterility may not be an important determinant of natural fertility until later reproductive ages.     

Fertility and replacement: Some alternative stochastic models and results for Brazil

The observed joint distribution of births and child deaths for a cohort of women at a given point in time depends on the number of children that would have been born had the family experienced no deaths, the number of child deaths experienced, and the proportion of these deaths that are replaced by a subsequent birth. In this paper we estimate the parameters of the assumed distributions of these three events using a minimum distance estimation model and data from the 1970 Brazilian census. The parameter estimates are shown to be similar to those obtained previously using a maximum likelihood estimation model. When the data are subdivided according to women's years of schooling, estimates of probability of a child death and mean and variance of children born if no deaths decrease while estimates of probability of replacement of a dead child increase as years of schooling increase.     

A model for estimating fecundability of the currently married woman from the data on her susceptibility status—A cohort approach

A probability model to estimate fecundability of a married woman has been proposed under some mild assumptions. It utilises the knowledge on the susceptibility status of the married women (including menstruation, menopause, pregnancy and amenorrhea) and therefore sets another approach for estimating fecundability. In addition, it is capable of predicting the parity, proportion of foetal losses, fecundability and incidence of secondary sterility. The problem of finding out the consistent estimates of the parameters in the distribution is discussed in section 4. For illustra-tion, the model is applied to a set of simulated data after simplifying many assumptions of the model     

Distribution of Closed Birth Intervals in a Heterogeneous Population

The probability distribution of parity-specific closed birth intervals in a heterogeneous population of women is obtained from Biswas' (1980) concept of gradually decreasing fecundabilities. The distribution of parity-specific last closed birth intervals and its moments of various orders are derived for a heterogeneous female population. Simulations are used to compare with models for a homogeneous female population with constant fecundability. The consideration of heterogeneity allows a better fit for empirical closed birth intervals.     

The Structure of Pregnancy Intervals by Planning Status

Measures of pregnancy intervals are reported for data from the 1975 National Fertility Study in the United States. The structure of such intervals is aggregated into its components by planning status. A measurement model is devised to infer from a survival function some dimensions of the initial distribution of respondents by propensity to experience the event in question. It is adaptable to the study of reproductive intention as well as fecundability and contraceptive effectiveness. The model is applied to the study of the dependence of such behaviour in a given interval on the planning status of the previous pregnancy. Planning status is sequentially well-ordered. Those previously successful with contraceptive use are less likely to fail than those who had previously failed; those who had previously used contraception, whether successful or not, are less likely to fail than those who had not previously used. The credibility of the findings is prejudiced less by the form of model than by the reliability of the basic data.     

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.     

A vital statistics-based procedure for estimating conception rates

A method for estimating conception rates, using vital statistics data, is developed and applied to data on five-year age groups of California women for 1971. The approach is deterministic and allocates total exposure time to the known pregnancy outcomes of live birth, spontaneous abortion, and induced abortion. The population at risk is defined to exclude women who are known to be sterile or sexually inactive. Early fetal loss, premarital conception, and contraceptive use are taken into account. Estimates are made of the fecundability which would obtain ifno contraception were used.     

Intermediate fertility variables and marital fertility rates

Summary Using relatively simple mathematical techniques, an analysis is made of a comprehensive reproductive model that describes the relationships between a set of intermediate fertility variables and the marital fertility rate. Two types of intermediate fertility variables are distinguished: (1) biological parameters and (2) control variables. A homogeneous model is outlined first. Next, this version is extended to include heterogeneity with respect to fecundability and coital rates. Tests of the model with data from two historical populations (i.e. Crulai, 1674-1742, and Tourouvre au Perche, 1665-1765) demonstrate that the model is, indeed, consistent with observed reproductive behaviour in actual populations.     

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.     

Modeling Seasonality in Fecundability,Conceptions, and Births

This paper develops a model of seasonal fluctuations in fecundability, conceptions, and births. We begin with a model of individual fecundability that combines behavioral and biological components, with particular attention to the roles of coital frequency, sperm concentration, fetal loss, and contraception. The individual-level model is then expanded into a model of seasonal fluctuations in births at the population level, which accounts explicitly for seasonal fluctuations in the size of the susceptible population. We illustrate the use of the model by analyzing proposed explanations of birth seasonality that rely on extreme summer heat.     

An analytic simulation model of human reproduction with demographic and biological components

Abstract This Monte Carlo model for simulating the reproductive history of a cohort of women is described in detail. The model provides for patterns of survival, sterility, formation and dissolution of sexual unions, fecundability, lactation, foetal wastage, family planning practices etc. Natality indices specific for marital status, for duration of marriage and for age, as well as analyses of birth spacing patterns are among the results that may be obtained. In the model, the experimental unit is an individual woman. The complete life history of a woman is generated and recorded before the history of the next woman is generated. The data for the whole cohort are analyzed at the end of the programme. The model includes two kinds of states into which a woman may pass, namely: (1) permanent changes of status such as death, sterility, or becoming a family planner, and (2) temporary states, each with a probability distribution of length of stay. The probabilities of the various events or changes of state may vary from age, parity, and other features of a woman's status or history. Natural fecundability at any age may also vary from woman to woman. In this programme natality patterns and specific indices such as age-specific fertility rates are produced, in a quasi-realistic fashion, by the interplay of the demographic and biological parameters postulated for any cohort. Consequently, the effect of changes in anyone factor can be studied, as well as the interaction resulting from changes in several factors. The purposes and potentials of the model are both substantive and methodological. As an illustration, a series of computer runs attempting to simulate the reproductive patterns of Indian women is presented. These results, as well as some additional ones, indicate some effects of changes in marital patterns, levels of fecundability, duration of post-partum non-susceptibility, age incidence of sterility and foetal wastage. In the final section of the paper, the advantages and possible applications of the model are discussed together with the limitations encountered to date in the efforts to apply the model.     

A probability model applicable to the study of inter-live birth intervals and random segments of the same

Abstract This paper deals with an analytical study of two types of birth intervals, viz. 'closed intervals' and 'open intervals' through the application of simple probability theory. The 'closed interval' stands for the time interval between two successive live births of a woman, and the 'open interval' denotes the interval between the date of last live birth and the date of survey for a married woman in the reproductive age group surveyed at a point of time. The study considers the 'closed interval' as the sum of independent random variables, each representing a particular component like post-partum amenorrhea, waiting time in the susceptible state, etc. Approximations to the patterns of distributions of these component random variables are made from the available data collected in fertility surveys at Gandhigram. The 'open interval' for any parity is studied separately for two different (mutually exclusive) categories of women, viz. those who have at least one more live birth at some time or other during their reproductive period and those who cease childbearing. In the first case the 'open interval' is considered as a random segment or partition of the corresponding 'closed interval'; in the second as a random segment of the interval between the date of birth of the last child and date at which the woman attains 45 years of age. The mean and variance of the 'open interval' is obtained separately in each case, and the moments of the 'open interval' distribution for women chosen at random from the population are obtained as an appropriate mixture of the two types.