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.     

Probabilistic Projections of the Total Fertility Rate for All Countries

We describe a Bayesian projection model to produce country-specific projections of the total fertility rate (TFR) for all countries. The model decomposes the evolution of TFR into three phases: pre-transition high fertility, the fertility transition, and post-transition low fertility. The model for the fertility decline builds on the United Nations Population Division’s current deterministic projection methodology, which assumes that fertility will eventually fall below replacement level. It models the decline in TFR as the sum of two logistic functions that depend on the current TFR level, and a random term. A Bayesian hierarchical model is used to project future TFR based on both the country’s TFR history and the pattern of all countries. It is estimated from United Nations estimates of past TFR in all countries using a Markov chain Monte Carlo algorithm. The post-transition low fertility phase is modeled using an autoregressive model, in which long-term TFR projections converge toward and oscillate around replacement level. The method is evaluated using out-of-sample projections for the period since 1980 and the period since 1995, and is found to be well calibrated.     

New Cohort Fertility Forecasts for the Developed World: Rises,Falls, and Reversals

With period fertility having risen in many low‐fertility countries, an important emerging question is whether cohort fertility trends are also reversing. We produce new estimates of cohort fertility for 37 developed countries using a new, simple method that avoids the underestimation typical of previous approaches. Consistent with the idea that timing changes were largely responsible for the last decades' low period fertility, we find that family size has remained considerably higher than the period rates of 1.5 in many “low‐fertility” countries, averaging about 1.8 children. Our forecasts suggest that the long‐term decline in cohort fertility is flattening or reversing in many world regions previously characterized by low fertility. We document the marked increase of cohort fertility in the English‐speaking world and in Scandinavia; signs of an upward reversal in many low‐fertility countries, including Japan and Germany; and continued declines in countries such as Taiwan and Portugal. We include in our forecasts estimates of statistical uncertainty and the possible effects of the recent economic recession.     

[Estimating the error of random projection of discrete populations].]

The author estimates the error of stochastic projection of discrete populations by defining fertility and mortality as stochastic processes. Using Chebyshev's inequality and data for Huzhou City's population, the author estimates random fluctuations of fertility and mortality and develops an equation for estimating the relative error of projection of total population.     

A method for calculating rates of induced abortion

Accurate measurement of induced abortion is necessary for understanding the fertility dynamics of a population and for making projections about the future. Changes in abortion rates can amplify or dampen the impact of changes in contraception. This paper presents a methodology for calculating marital induced abortion rates from observed marital fertility and contraceptive prevalence and for modeling the impact of substituting contraception for abortion on future fertility. The methodology is validated against observed abortion complications in three populations, and the impact of substituting contraception for abortion on expected fertility is demonstrated.     

Annual fertility rates from census data on own children: Comparisons with vital statistics data for the United States

This paper begins by describing the procedure and data requirements for calculating annual fertility rates from census data on own children. Then, using data from the United States Censuses of 1960 and 1970, fully adjusted estimates are presented and compared with recorded vital statistics rates. Total fertility estimates derived from own children data for whites average less than two percent lower than the recorded rates- a difference that can be attributed partially to the fact that the estimates are adjusted for net census undercount but the recorded rates are not. Even without adjustments for mortality, children not living with their mothers, and net census undercount, the own children data estimates accurately replicate recorded trends (even though the levels are misspecified). The utility of own children data for the study of differential fertility is discussed.     

Census-derived estimates of fertility by duration since first marriage in the Republic of Korea

This paper estimates ever-married birth rates by age and duration since first marriage and ever-married total fertility rates for the Republic of Korea, derived by applying an extension of the own-children method of fertility estimation to the 1975 and 1980 censuses. Since each census provides annual estimates for the 15-year period previous to enumeration, there is a ten-year period of overlapping estimates that facilitates checks for consistency and accuracy. Comparisons are also made with estimates derived from the 1974 Korea National Fertility Survey, which was part of the World Fertility Survey. The method works well, except in its application to the 1975 Census where the evidence suggests considerable misreporting of age at first marriage because of the way the question was asked and coded. Results confirm that ever-married fertility fell substantially in Korea between 1961 and 1980, with a temporary resurgence in the late 1960s and early 1970s. Ever-married fertility rose at younger ages and shorter durations and fell at older ages and longer durations. Ever-married fertility differentials by urban-rural residence and by education were usually in the expected direction, with urban fertility generally lower than rural fertility and the fertility of those with more education usually lower than the fertility of those with less education. Differential ever-married fertility by urban-rural residence and education declined over the estimation period.     

The Pattern of Age-Specific fertility rates

A study of the pattern of age-specific fertility rates by five-year age-groups has revealed the possibility of using the Pearsonian type I curve as a graduating equation. Such distinctions have been examined for fifty countries having high, medium, and low fertility rates. Results have been found to be quite satisfactory, even when, for purposes of simplicity, the parameters have been restricted to depend on the first two moments, instead of the first four. The number of independent parameters has thus been reduced to only three, and method shave been suggested for their estimates, using ancillary information which is usually provided in the census reports. The findings seem to be particularly useful for countries lacking effective registration systems of vital events such as births related, as they are, to the age of the mother.In short, the determination of a series of age-specific fertility rates depends primarily on the modal fertile age (a1) and on one of the two exponents (m, or m2) of the type I distribution. It has been shown that the proportions of women married in the age groups 20-24 years and 25-29 years can be used for approximate evaluations of both of these parameters. While estimates of a, and m, are sufficient to generate a relative distribution of age-specific fertility rates, actual values can be obtained if, in addition, the annual number of births is either known or can be estimated from census age distributions of children.Modal distributions of relative values of age-specific fertility rates (such that the sum of these rates adds up to 100.0) have also been obtained for different combinations of al and m1, but are not shown here for lack of space. Once the estimates of al and m, have been made, the modal tables can always be referred to for the corresponding distributions of age-specific fertility rates.     

Estimation of vital rates by means of monte carlo simulation

Monte Carlo simulation has been used to estimate age-specific fertility and mortality rates for a small population,the French-derived isolate of Northside on St. Thomas, U. S. Virgin Islands. Estimates were based on data collected in a household census and genealogical survey and on birth, death, and marriage records for the years 1916to 1966. During this 50-year period (in which the population size increased from 202 to 657), the numbers of births and deaths were too, small to estimate age-specific rates directly, and in addition, death registration was incomplete. Mortality rates were estimated using a simulation program in which mortality was the only stochastic variable. A model mortality schedule was chosen which most accurately reproduced the growth pattern of the population over the 50-year period. To estimate fertility rates, a more complex simulation model was used in which fertility, nuptiality, and mortality were random variables with probability distributions. Preliminary estimates of fertility were made from the birth records and used as input to this simulation program. Birth probabilities were adjusted empirically from one set of simulation runs to the next, until population growth rates, as well as other demographic characteristics, were similar in the real and simulated populations. The birth rates which produced the best fit to the real population data were taken as the estimated age-specific fertility schedule. To reproduce the real population age structure more closely, secular changes in birth probabilities were applied.     

Measuring marital fertility control with CPA

Cohort Parity Analysis (CPA) is an indirect method for measuring the extent and timing of the adoption of fertility control within marriage. Cohort Parity Analysis uses 2 sets of parity distributions, 1 from the population under study (the "target" population) and 1 from a similar population in which fertility control is absent (the "model" population). Fertility control is defined as any set of behaviors in the target population that cause its parity progression rates to differ from those of a similar but noncontrolling population. The target population contains an unidentified mixture of controlling and noncontrolling couples for a given age range and marriage duration. The model population is similar in all respects except that it contains no controllers. Examples are taken from urban and rural Irish populations in the 1911 census data. Cohort parity analysis rests on 2 basic assumptions: 1) For every specified marriage duration and marriage age, there exists some parity above which controllers are never found; and 2) Parity progression rates of those in the mixed population who have never initiated fertility control are identical to those in the model population. Further, one must identify a cutoff parity beyond which controllers are presumed not to be found. In these urban Irish examples cutoff parity for each marriage age and duration is set at 20% of that of the noncontrolling rural Irish population. The upper and lower bounds on the percentage of controllers are derived by obtaining the lower and upper bounds on the size of the group at that parity who never controlled and subtracting these estimates from the observed percentage of the target population observed at that parity. Flow charts are used to present the methodology of cohort parity analysis in a simple format. the method is illustrated in the case of Ireland in 1911. Aggregate parity distributions for the county boroughs of Dublin, Belfast, Cork, LOndonderry, Limerick, and Waterford are taken as the target distribution; and parity distributions for those living outside the county boroughs are taken as the model distributions. The analysis shows that 28% of urban-dwelling Irish women who married before age 30 and were married between 4 and 29 years in 1911 used some effective means of fertility control. The parity distributions also show that women who married later controlled their fertility to a greater extent and that between 20.9 and 25.5% of women married 4 years in 1911 had already practiced fertility control by 1911, but only 15.5% of those married 15-19 years had used fertility control by 1911. Cohort Parity Analysis is thus a useful method of examining historical populations in the process of fertility transition.     

Estimating marital fertility rates by educational attainment using a survey of new mothers

Marital fertility rates by educational attainment of mother are estimated for the United States for 1963. These calculations are based upon information collected in a probability sample survey of women having births in 1963 and are prepared by relating birth estimates for educational attainment groups to estimates of married women in corresponding groups.The rates do not display a negative association between educational attainment and the annual level of fertility, thus differing from the pattern observed in other measures of period fertility. Women who completed some high school but did not graduate and women with one or more years of college had higher annual fertility rates than women in other attainment classes.Alternative estimation procedures are discussed which illustrate difficulties in obtaining satisfactory correspondence between two independent surveys which are used to obtain the rate calculation components.     

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.     

On the derivation of a two-sex stable population model

It is well known that the intrinsic rates of growth derived separately for the males and females in a population, when one assumes the continuation of their respective mortality and fertility experiences, usually turn out to be different. Noting that the phenomenon of human reproduction is a product of the cooperation between the two sexes, we have attempted in this paper to define the sex-age-specific fertility rates as a function not only of age but also of time, where the latter is implicitly introduced in the model through the sex composition of the reproductive population. It has been shown that a stable model can then be defined based on such changing sex-age-specific fertility rates and given sets of unchanging mortality rates. The fertility rates stabilize with time, and the common intrinsic rates of growth for the two sexes are found to lie in the interval generated by the corresponding rates of the two one-sex models. Several other interesting relationships among the parameters of this model have been presented in the paper. Among other alternatives, a least square solution has been presented for the values of sex-age-specific fertility rates that are minimally discrepant with the observed rates but are consistent in terms of the parametric estimates they generate. It is interesting to note that a relatively modest adjustment in the sex-age-specific fertility rates is all that it takes to eliminate the inconsistencies generated by the separate one-sex models.     

Constructing fertility tables for Soviet populations

Because the 1970 Soviet Union census does not provide information on the age structure of men and women separately by sex and according to their ethnic affiliation, the 1959 USSR census data serve as the basis to infer knowledge about ethnic fertility. The model takes into account (1) the total number of births in 1960, estimated from the child-woman ratio in 1959, (2) the age structure of women in 1959, and (3) the assumed pattern of age-specific birth rates structured in terms of the modal age at childbearing and the length of the fertility age span. The results show that Ukrainians among the Slav populations ranked as the lowest with 2.07 children born per woman. Their total fertility contrasts with that of Kazakhs native to Central Asia, who reportedly according to Soviet sources had 7.46 children per woman in 1958-1959, and whose estimated rate is around 8.59 children. Extreme variations appear in the estimates of fertility among nationalities of the Caucasus region, Volga Basin, and to a lesser degree in Siberia. Official Soviet calculations of crude birth rates and age-specific rates for 15 Union Republics in 1967-1968 are transcribed and compared with the estimates for nationalities in 1959-1960. The same theoretical model used to generate the Soviet rates may be adapted under different assumptions to non-Soviet populations in other situations where the data are scanty or incomplete.     

Developing strategies for deriving small population fertility rates

Many agencies require population estimates and projections by ethnic group. These projections need ethnic-specific, age-specific fertility rates (ASFRs) but their inclusion is challenging since ethnicity is not recorded at birth registration. In this paper maternity data are used in a case study of electoral wards in Bradford, West Yorkshire, to develop fertility rates for small populations for a 1991 based projection. The challenge is to capture local variations in fertility by ethnic group when data are sparse. Small areas were grouped together using cluster analysis to define combinations with similar sociodemographic and fertility experiences so that sparse data could be aggregated to estimate reliable ethnic-specific fertility rates. For comparison, the data were aggregated into the 1991 Office for National Statistics area type classification. Fertility rates by single year of age for all area types were smoothed using the Hadwiger function. For the White ethnic group there were sufficient births to create ethnic-specific, ward-level ASFRs. For other ethnicities grouping of areas was necessary. The accuracy of the ASFRs in predicting births was assessed using mean absolute percentage error. Results show that for some minority groups district-level ethnic-specific fertility rates produced the most accurate birth estimates even though they were based on a larger area. This implies that rates created may be informative about the local area for White ethnic type but not in the same way for smaller ethnic groups. In terms of grouping strategies we recommend that existing classifications are assessed to determine how well variations in rates are stratified before embarking on a custom scheme. Where population sub-groups are small in some areas, it may be more reliable to use rates derived for larger areas and apply these to local populations. Inevitably, the rates used in a projection are a compromise but hopefully will still capture important dimensions of population change.     

Empirical bayes estimation of demographic schedules for small areas

In this article, we analyze empirical Bayes (EB) methods for estimating small-area rate schedules. We develop EB methods that treat schedules as vectors and use adaptive neighborhoods to keep estimates appropriately local. This method estimates demographic rates for local subpopulations by borrowing strength not only from similar individuals elsewhere but also from other groups in the same area and from regularities in schedules across locations. EB is substantially better than standard methods when rates have strong spatial and age patterns. We illustrate this method with estimates of age-specific fertility schedules for over 3,800 Brazilian municipalities.     

Drawing statistical inferences from historical census data, 1850–1950

Virtually all quantitative microdata used by social scientists derive from samples that incorporate clustering, strati cation, and weighting adjustments (Kish 1965, 1992). Such data can yield standard error estimates that differ dramatically from those derived from a simple random sample of the same size. Researchers using historical U.S. census microdata, however, usually apply methods designed for simple random samples. The resulting p values and confidence intervals could be inaccurate and could lead to erroneous research conclusions. Because U.S. census microdata samples are among the most widely used sources for social science and policy research, the need for reliable standard error estimation is critical. We evaluate the historical microdata samples of the Integrated Public Use Microdata Series (IPUMS) project from 1850 to 1950 in order to determine (1) the impact of sample design on standard error estimates, and (2) how to apply modern standard error estimation software to historical census samples. We exploit a unique new data source from the 1880 census to validate our methods for standard error estimation, and then we apply this approach to the 1850–1870 and 1900–1950 decennial censuses. We conclude that Taylor series estimation can be used effectively with the historical decennial census microdata samples and should be applied in research analyses that have the potential for substantial clustering effects.     

Age,biological factors,and socioeconomic determinants of fertility: A new measure of cumulative fertility for use in the empirical analysis of family size

To influence the number of children ever born to a woman, socioeconomic variables must operate through behavioral and biological mechanisms such as the age at marriage, the level of fertility in the absence of deliberate fertility control, and the level of control exerted to reduce fertility within marriage. In this paper, we propose a new measure of cumulative fertility which is standardized for the age-fecundity relationship and for exposure to the risk of conception associated with duration of marriage. A simple model of fertility behavior which incorporates some of the mechanisms through which socioeconomic factors may affect fertility is developed and applied to data from the United States to demonstrate the properties of alternative measures of family size. The results indicate that use of the new measure allows more precise estimates of socioeconomic fertility relationships than would be obtained with children ever born or by sample stratification.     

Malthusian Dynamics in a Diverging Europe: Northern Italy, 1650–1881

Recent empirical research questions the validity of using Malthusian theory in preindustrial England. Using real wage and vital rate data for the years 1650–1881, I provide empirical estimates for a different region: Northern Italy. The empirical methodology is theoretically underpinned by a simple Malthusian model, in which population, real wages, and vital rates are determined endogenously. My findings strongly support the existence of a Malthusian economy wherein population growth decreased living standards, which in turn influenced vital rates. However, these results also demonstrate how the system is best characterized as one of weak homeostasis. Furthermore, there is no evidence of Boserupian effects given that increases in population failed to spur any sustained technological progress.     

应用队列累计生育率分析我国生育水平变动趋势——兼与郭志刚教授讨论

本文对反映生育水平的两个基本指标——总和生育率和队列累计生育率进行分析,肯定多年来多个调查所得到的队列累计生育率的数据质量。尽管队列累计生育率反映的是"过去"而不是"当前"的实际生育水平,但历次调查所反映出的1990年代以来生育水平变化趋势是持续下降,并推断近几年的总和生育率已经下降到1.6以下。