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.     

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.     

Analyzing fertility histories: Do restrictions bias results?

The straightforward tests we have conducted lead to two major conclusions. First, parameter estimates, such as the proportions that practice contraception or that breastfeed, can be biased in data restricted to the last closed and open interval. This is particularly true the further back in time one goes. However, the second conclusion is that these restrictions do not bias estimates of the structure of the relationships predicting fertility. This may seem surprising, and perhaps even magical. The reason is that multivariate life table techniques allow one to reach the same conclusion even if the proportions in various categories are altered by a criterion such as limiting the analysis to intervals begun by the last and next-to-last live births. Limiting the analysis in this way means that there are fewer short intervals and thus fewer cases of intervals with characteristics associated with short intervals (e.g., no contraceptive use, no breastfeeding, or infant mortality). As long as the model specified in the multivariate life table is an appropriate one, that is, it is not misspecified, and as long as the skew produced by the WFS restriction is not too extreme, then the multivariate life table procedures can produce unbiased estimates of the structure of the relationships predicting birth interval dynamics. Thus even though the WFS data are in fact inappropriate for some simple parameter estimation procedures, they appear to be adequate for the more complex multivariate procedures of the sort used here. Several caveats must be added to the foregoing results. First, we have performed this test in only one country, Korea; it is possible that the same results might not be obtained in other countries. We expect, however, that they would. Second, our procedure only looks at the first 40 months of experience in the birth interval. A procedure that incorporates the long tails of the birth interval distribution may obtain different results. In fact, we caution against analyzing the tail of the distributions using data from the normal WFS sample, since these would be most affected by the restriction to last closed and open intervals. Third, the extent to which these results are generalizable to other types of substantive problems is unknown at present. We suspect, however, that examining the determinants of lengths of breastfeeding will produce similar results. Finally, even with multivariate procedures, it would be highly misleading to impose the WFS restrictions and then examine trends in the length of birth intervals.(ABSTRACT TRUNCATED AT 400 WORDS)     

Starting age and subsequent birth intervals in cohabitational unions in current danish cohorts, 1975

This paper presents some main results of an investigation by life table methods of birth interval data in cohabitational unions (marriages as well as consensual unions) in current Danish cohorts. Our results confirm previous findings that an early age at the start of recorded exposure to childbearing risk is indicative of a rapid pace and high level of subsequent fertility. The analysis modifies previous results and adds several new details regarding cohort trends and the effect of parity at the start of reported cohabitation. For each parity within a period of cohabitation, fertility differentials by reported starting age seem to have diminished from our older cohorts (of age up to 49 years in 1975) to our younger ones (of age less than 30 years in 1975). There are indications of a dramatic change in childbearing behaviour following the arrival of novel attitudes to non-marital cohabitation and childbearing in Denmark about 1967.     

Population dynamics based on birth intervals and parity progression

Abstract The Chinese population policy of 'later-longer-fewer' suggests a formulation of population dynamics in terms of birth intervals and parity progression. This leads to population projections based on birth interval distributions and parity progression ratios and to an alternative to Lotka's stable population theory in which age is replaced by parity and interval since last birth. A numerical comparison with Lotka's model indicates similarities and differences between the two approaches: The formulation suggests an approach to the analysis of birth interval and parity progression statistics that effectively solves censoring and selection problems.     

Multidimensional Life Table Estimation of the Total Fertility Rate and Its Components

Using discrete-time survival models of parity progression and illustrative data from the Philippines, this article develops a multivariate multidimensional life table of nuptiality and fertility, the dimensions of which are age, parity, and duration in parity. The measures calculated from this life table include total fertility rate (TRF), total marital fertility rate (TMFR), parity progression ratios (PPR), age-specific fertility rates, mean and median ages at first marriage, mean and median closed birth intervals, and mean and median ages at childbearing by child’s birth order and for all birth orders combined. These measures are referred to collectively as “TFR and its components.” Because the multidimensional life table is multivariate, all measures derived from it are also multivariate in the sense that they can be tabulated by categories or selected values of one socioeconomic variable while controlling for other socioeconomic variables. The methodology is applied to birth history data, in the form of actual birth histories from a fertility survey or reconstructed birth histories derived from a census or household survey. The methodology yields period estimates as well as cohort estimates of the aforementioned measures.     

Birth spacing and fertility limitation: a behavioral analysis of a nineteenth century frontier population

Our analysis of changing birth interval distributions over the course of a fertility transition from natural to controlled fertility has examined three closely related propositions. First, within both natural fertility populations (identified at the aggregate level) and cohorts following the onset of fertility limitation, we hypothesized that substantial groups of women with long birth intervals across the individually specified childbearing careers could be identified. That is, even during periods when fertility behavior at the aggregate level is consistent with a natural fertility regime, birth intervals at all parities are inversely related to completed family size. Our tabular analysis enables us to conclude that birth spacing patterns are parity dependent; there is stability in CEB-parity specific mean and birth interval variance over the entire transition. Our evidence does not suggest that the early group of women limiting and spacing births was marked by infecundity. Secondly, the transition appears to be associated with an increasingly larger proportion of women shifting to the same spacing schedules associated with smaller families in earlier cohorts. Thirdly, variations in birth spacing by age of marriage indicate that changes in birth intervals over time are at least indirectly associated with age of marriage, indicating an additional compositional effect. The evidence we have presented on spacing behavior does not negate the argument that parity-dependent stopping behavior was a powerful factor in the fertility transition. Our data also provide evidence of attempts to truncate childbearing. Specifically, the smaller the completed family size, the longer the ultimate birth interval; and ultimate birth intervals increase across cohorts controlling CEB and parity. But spacing appears to represent an additional strategy of fertility limitation. Thus, it may be necessary to distinguish spacing and stopping behavior if one wishes to clarify behavioral patterns within a population (Edlefsen, 1981; Friedlander et al., 1980; Rodriguez and Hobcraft, 1980). Because fertility transition theories imply increased attempts to limit family sizes, it is important to examine differential behavior within subgroups achieving different family sizes. It is this level of analysis which we have attempted to achieve in utilizing parity-specific birth intervals controlled by children ever born.(ABSTRACT TRUNCATED AT 400 WORDS)     

Infant mortality,birth order,and sibship size: The role of heterogeneous risk and the previous‐death effect

This paper derives an analytic model to study biases in infant mortality estimates by birth order and sibship size, which occur when the death of an infant tends to shorten the next birth interval and mortality risk varies among families. We find that order‐specific and sibship‐size‐specific estimates are biased by a selection for high‐risk women across birth orders, since women with higher risk will tend to have shorter intervals, and more births, within a given period of time. Sibship‐size‐specific estimates are, in addition, biased by a selection of women who have experienced deaths, even if there is no heterogeneity in risk. Numerical examples based on data from Matlab, Bangladesh, are used to illustrate the possible magnitude of these biases. The results resemble patterns of infant mortality by birth order and sibship size which are often observed empirically.     

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.     

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.     

The effect of child mortality experience on subsequent fertility: in Pakistan and Bangladesh

Summary This paper presents an empirical analysis of the effects, behavioural and biological, of child mortality experience on subsequent fertility in two South Asian Islamic nations. Data for the investigation came from retrospective pregnancy histories of 2,910 currently married women interviewed in the Pakistan National Impact Survey (1968-69) and from longitudinal vital registration data (1966-2070) of 5,236 women residing in a rural area of Bangladesh collected by the Cholera Research Laboratory. The aim of this study was to assess the importance of the child-replacement motivational response to child death experience after biological effects have been controlled adequately. A common approach employed previously has been to examine cumulative fertility according to child death experience. In Pakistan and Bangladesh, a consistently positive relationship was demonstrated between the number of children ever born and the number of child deaths. This method, however, did not exclude the inverse relationship, the influence of fertility on mortality, nor did it dissect out behavioural from biological effects. Utilizing a measure of subsequent fertility, live-birth-to-live-birth intervals, the study further illustrated another common pitfall. Since the risk of infant death, which leads to shorter birth intervals, is associated with the mother's reproductive history, women with child mortality experience are more likely to experience shorter intervals because of the biological effect of subsequent infant death. Behavioural influences may, therefore, be observed by considering only those birth intervals in which the first-born child survives to the end of the interval. With these limitations controlled, very few, if any, behavioural influences were noted in the Pakistan and Bangladesh data. Median birth intervals in Pakistan varied between 35-43 and 41-42 months, increasing with parity. Within each parity group, no consistent difference was observed between women with and without previous child loss. In Bangladesh, the median birth interval for all women with a surviving infant was 37-2 months. This was shortened to 24-31 months by an infant death. When intervals with infant deaths were excluded, little or no behavioural influence was detected among women of the same parity, but with varying levels of previous child loss. Even without behavioural effects, elimination of infant mortality in Bangladesh would reduce fertility by prolonging the average period of post-partum sterility. In the Bangladesh setting, however, the size of the effect was only about four per cent. This modest effect, more-over, was counterbalanced by an overall increase of net reproduction by seven per cent due to better survivorship of infants.     

The TFR and birth control]

Total fertility rate if (TFR) is a simple an straight forward measure of women's fertility. However, it is difficult to use the TFR as a target measure in FP programs. If TFR level is set as a target for a particular year, how can women's fertility be regulated to achieve this target? The following analysis suggests a simple model to control the proportion of birth parity. First, the TFR is decomposed into a parity- specific TFR. The parity-specific TFR can be worked out using coefficients of the regression models calculated from data of previous fertility surveys. Once the TFR is given, the parity-specific total fertility can be calculated using a model with coefficients from empirical data. Then the number of births of each parity may be calculated from the parity-specific TFR using the female age structure in a particular year, the survival probability, and the standard fertility model for each parity. When the number of surviving children of each women at child-bearing age is known, the desired proportion of births of each parity can be calculated using the standard birth probability during a years. From these models, it would be possible to calculate how many women can have their first child/year, and how many can have the second. Thus, family planning organizations would be in a position to formulate a birth quota on the basis of the above information.     

Regression-like analysis of birth interval sequences

Employing a family of generalized linear models for the gamma distribution, the structure of birth interval data derived from three historical populations is examined. The relative importance of such factors as current age, parity of the mother, and duration of marriage in determining the length of succeeding intervals is investigated. The results suggest that the three data sets have a common, rather simple structure and various applications of such empirical models are considered.     

High foetal mortality and birth intervals

Abstract Studies of birth intervals have generally separated the process into: (1) the period of infertility following a pregnancy termination; (2) the length ofthe interval from the end of the infertile period to the next conception (which reflects the probability of conception), and (3) the average duration of pregnancy for both live births and foetal deaths.(1) Empirically derived intervals between successive live births average from 19 to 30 lunar months(2) (Table 1), depending upon the population studied and the birth order.(3) Estimates of conception rates using birth intervals data require some estimate of the extent to which foetal wastage increases the birth interval.     

A Study Of Differential Fertility In Bombay

This study is based on special tabulations of the "order of pregnancy," as reported on the birth certificate, according to mother's age, locality, and religion and father's state of birth and duration of stay in Bombay, India. The sample consists of 50 percent of the births registered in 1960 in Bombay (where birth registration has a fairly high degree of completeness).Despite the limitations of statistical information on the complex of factors influencing fertility, it wasfound that three indicators derived from the 1961 census could numerically account for 51 percent of the total variance of the parity averages in the sections of Bombay. Overcrowding (which is an indicator of the level of living) had the strongest influence and accounted alone for 44 percent of the variance. Knowledge of thereligious patternraised thepercentage to49, whileinclusionof theilliteracy measure added only another two to this percentage.An assessment of fertility differences by religious groups, as well as by place of origin of the father, is then carried out. The data show significant differences among religious groups, with Moslems and Buddhists having the highest parity averages, followed by Hindus and Jains. A considerably lower level is shown by Christians, but the lowest parity level is that of the small Parsi community. Mothers whose husbands were born in the south and east were found to show the lowest parity level, while those who have north-born husbands showed the highest level.Finally, the data show consistent increase in age-standardized parity averages with increase of the duration of stay in Bombay. This pattern is observed for each of the three main geographic regions of origin. While this increase is difficult to explain on the basis of the available data, the fact that the pattern of parity differences among regions persists from one duration group to the next lends support to the conclusion that little assimilation seems to have been taking place.     

Measuring change and continuity in parity distributions

Procedures are developed to allocate the change in mean fertility to the change in specific parities or groups of parities. One procedure uses the proportion at each parity and another uses parity progression ratios. Both are based on the delta method for approximating change in a function of several variables. Drawing on an analogy to survival in a life table, the relational logit model is applied to parity progression. This method allows several parity distributions to be synthesized and to have differences summarized with two parameters. The three procedures are applied to successive cohorts of white U.S. women who completed their childbearing between 1920 and 1980.     

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.     

On the human survivorship function and life table construction

The construction of life tables is often marred by one or more of the following restrictions: (1) assumptions that are either unjustifiable or of questionable generality; (2) rough approximations; (3) exacting data requirements. This paper recommends instead a simple method which regards the force of mortality as constant within each age interval. The reasoning is readily comprehensible and all life table functions are easily calculated from the age-specific death rates without any need for further assumptions, approximations, or data. Furthermore, this method produces numerical results that are close to those obtained by other methods.     

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.     

Infant death clustering in families: magnitude, causes, and the influence of better health services, Bangladesh 1982-2005

This analysis of infant mortality in Bangladesh focuses on explaining death clustering within families, using prospective data from a rural region in Bangladesh, split into areas with and without extensive health services (the area covered by the International Centre for Diarrhoeal Disease Research and the comparison area, respectively). The modelling framework distinguishes between two explanations of death clustering: (observed and unobserved) heterogeneity across families and a causal 'scarring' effect of the death of one infant on the survival chances of the next to be born. Keeping observed and unobserved characteristics constant, we find scarring in the comparison area only. There the likelihood of infant death is about 29 per cent greater if the previous sibling died in infancy than otherwise. This effect mainly works through birth intervals: infant deaths are followed by shorter birth intervals, which increases the risk of infant death for the next child.