A simplified robust estimate of the birth rate

It is shown that other estimates of the birthrate can be derived from Coale's robust birthrate estimate. Coale's estimate is nearly equal to the birthrate obtainable from reverse survival or reverse projection of the proportion of a population under age 15 (both sexes), or C(15), using a life table corresponding to l5. As a sequel to this, a birth rate estimate was obtained that does not require reference to stable population models and results in computational economy and ease. Taking advantage of the strong linear relation between l5 and 15L0, a simple robust estimate was derived of the birthrate that does not depend upon model stable populations or model life tables. After presenting these methods, their use is illustrated with data from several Asian and African countries. Coale (1981) suggested using the observed C(15) for both sexes and l5 to locate an appropriate stable population from a family of stable models to represent the observed population and to use its birthrate as an estimate of the population under study. The estimate of l5 can be obtained by any of the indirect methods like the Brass method. Coale observed that such methods yield birthrates that are not much affected even when the populations are not stable. He also suggested an adjustment for the stable birthrate for nonstability. To obtain the birthrate, one needs the denominator, namely, the number of persons that lived. This is obtained by using the rate of increase, r, which differs for a stable and a nonstable or observed population. Various methods can be used to obtain the time reference of the mortality estimate, l5, by providing years prior to the survey or census to which the l5 estimate is applicable.     

Examination of the generalized age distribution

The formula for the age distribution and other relationships that follow from it for any (non-stable) population presented by Preston and Coale are significant contributions to demography. The formulas summarize the relationships among various demographic measures precisely, and are formally analogous to the relationships that hold for stable populations. The significance of these formulas cannot be overstated; they allow us to understand clearly the relationships among demographic measures in any arbitrary population. However, when it comes to using them for estimating demographic measures when census data are defective, the method of estimation is still affected by defective data. The reason is that the series of age-specific growth rates reflects the observed census age distributions exactly so that any defects in the census data are summarized in the growth rates. This paper begins with the formulation of the discrete version of the "new synthesis" developed by Preston and Coale. With the discrete formulation, the three kinds of errors introduced when the continuous time formulas are applied to real data can be avoided. Then it is pointed out that when two accurate census data are available, the Preston-Coale procedure of "estimating" the age distribution at the second census is equivalent to checking the identity of the age distribution formula. Also "estimating" mortality by the procedure of Preston-Coale is shown to be equivalent to obtaining mortality directly from intercensal survival rates. That the procedure which involves the age-specific growth rates is equivalent to those that involve the intercensal survival rates may have escaped notice because there are no a priori constraints for patterns of age-specific growth rates to follow. The irregularity in growth rates due to defective data are not distinguishable from true irregularity that exists in the population, contrary to the well-known regularity in the pattern of survival rates in human populations.     

Simplified birth rate estimates under nonstable conditions

Coale's robust birth rate estimate, obtained by adjusting the birth rate of a stable population selected by matching the observed population of both sexes under the age of 15, C(15), and the probability of survival of births to age 5, l5, is shown to be equal to the birth rate obtainable by reverse surviving the proportion under the age of 15. Variations of matching the criterion of Coale's method to the rate of increase and C(15) or l5 are shown to lead to variants of Coale's birth rate estimate that are also nearly equal to the reverse survival birth rate based on C(15). A simplified birth rate estimate that does not require reference to models of life tables or stable populations is suggested and some of its applications are illustrated.     

Mortality estimation from registered deaths in less developed countries

Age-specific population growth rates were introduced to demographic analysis in earlier work by Bennett and Horiuchi (1981) and Preston and Coale (1982). In this paper, we derive a method which uses these growth rates to transform what may be a set of incompletely recorded deaths by age into a life table that accurately reflects the true mortality experience of the population under study. The method does not rely on the assumption of stability and, for example, in contrast to intercensal cohort survival techniques, is simple to implement when presented with nontraditional intercensal interval lengths. Thus we can obtain mortality estimates for less developed countries with defective data, despite departures from stability. Further, we assess the sensitivity of the method to violations in various assumptions underlying the procedure: error in estimated growth rates, existence of non-zero net intercensal migration, age dependence in the completeness of death registration, and misreporting of age at death and age in the population. We demonstrate the use of the method in an application to data referring to Argentine females during the period 1960 to 1970.     

A reply to Kim's comment

This reply criticizes Kim's note as incorrectly characterizing the essential feature of the method proposed for life table construction. The method suggested for estimating N(a), the number attaining age 2 during the intercensal period, is to make a separate estimate of the contribution to N (a) made by each single-year cohort that attains 'a' during the period between censuses. Each cohort estimate is constructed by interpolation, utilizing as data the recorded number in the relevant single-year cohorts in the 2 censuses. 2 methods of interpolation were proposed. 1 is an iterative procedure that constructs a preliminary life table by linear interpolation for each cohort and then derives more refined interpolation factors from this preliminary life table. The other procedure derives interpolation factors on the assumption that the proportionate distribution of deaths by age as each cohort moves from the earlier to the later census date is the same as the proportionate distribution of deaths by age over the same age range in a model life table. The advantage of the proposed procedure is that it supplies better estimates of N (a) than do alternative methods. The author concedes that a life table calculated from accurately recorded deaths and an accurately enumerated population would ordinarily be superior. However, he also notes that in the absence of registered deaths data, there is no precise enough conventional method to yield accurate values of average intercensal single-year age-specific mortality rates from nothing more than 2 accurate censuses 11 years apart. A common procedure for calculating life expectation at a very advanced age is to calculate the reciprocal of the death rate among persons over the age in question.     

A simple model for linking life tables by survival mortality ratios

Patterns of variation in mortality can be studied by measuring changes in selected life table functions. A model is proposed in which the rate of change over time in the life table survivorship probability at any age has been assumed as proportional to the product of its own value and its complementary probability or the probability of dying by that age, where the proportion is the same for all ages and depends only on the time duration between successive life tables. The end result is that the logit functions of the survivorship probabilities at two points in time are linearly related with a slope of one. The projecting power of the model has been tested by using U.S. life tables for the years 1950 and 1970 as well as Coale and Demeny's regional model life tables. In the latter case, the model produced surprisingly close matches even when the expectations of life differed by as much as 20 years.     

Life table construction on the basis of two enumerations of a closed population

The author demonstrates that an accurate detailed life table that represents average mortality experience between two censuses can be constructed if the censuses provide accurate records of the single-year age distribution of a closed population. This life table can begin at age zero if accurate data on the annual number of births during the inter-censal period are available; otherwise the first age in the life table must equal the duration of time between the censuses. "The estimation technique involves the calculation of the number of persons attaining each age during the period between the censuses and the determination of the average rate of increase in the number at each individual age. The success of the technique comes from the use of interpolation to calculate how many in each cohort attain each exact age the cohort passes through between the censuses." The estimation technique is tested using two alternative methods of interpolation. Some illustrations based on data for Sweden and China are included.     

The index of overall headship: A simple measure of household complexity standardized for age and sex

This paper develops and tests an age-sex standardized measure of household complexity, defined broadly as the tendency of adults (other than spouses) to head their own households or to share households. The aim is a measure of household complexity which can be computed with a minimum of demographic data, namely, data on number of households and on the population by age and sex. The procedure is similar to that of Coale for fertility measurement (Coale, 1969); it is a form of indirect standardization in which the actual number of households is related to the number that would exist if maximum age-sex-specific household headship rates were to apply. Various forms of this indirectly standardized measure show a correlation of better than 0.9 with directly standardized measures for a sample of 33 nations for which requisite data are available. The new measure promises to extend considerably the geographical and temporal range of comparable empirical measures of household complexity.     

A model of the age patterns of births by parity in natural fertility populations

This research develops a convolution model to express the age patterns of fertility at each birth order in natural fertility populations in terms of six parameters, directly representing the proximate determinants of fertility, and a series of parity level indicators. The parity level indicators at each birth order are simply the proportions of women in a cohort who will eventually have births at each birth order it the age‐related fecundity decline is controlled. The Coale‐McNeil nuptiality model is adopted to represent the age pattern of first marriage rates and the natural fertility schedule employed in the Coale‐Trussell fertility model is incorporated to adjust age effects. The fast Fourier transform is used in solving the model numerically. It proves that the model is able to provide excellent fits to fertility for rural Chinese women in the 1950s.     

New estimates of fertility and child mortality in Africa, south of the Sahara

Summary Earlier work by Page and Coale has estimated demographic indices of fertility and mortality for parts of Africa using the Sullivan modification of Brass's technique. The present paper presents modified and more accurate estimates of fertility and child mortality, not only for the sub-national units covered by Page and Coale but also for areas not covered by them. The present analysis which employs Trussell's refinement of Brass and Sullivan's techniques also includes improvements overlooked in earlier estimates. The salient finding that emerges is that while the Brass mortality technique is very powerful, his equally ingenious fertility technique is very weak and should not be relied on for estimating fertility parameters.     

On the measurement of naturalization

This paper proposes a new way of measuring naturalization, which takes into account both emigration and death. I argue that the new method corrects for underestimation and thus provides a more accurate measure of the concept. Using data from six groups of the 1973 immigrant cohort and multiple-decrement life table techniques, I estimated and compared naturalization measures derived from new and old methods. The results show that failure to control for emigration has a significant effect on the measurement of naturalization, particularly if an immigrant group has relatively high rate of emigration. Some further substantive implications of this new method are also explored.     

Reconstructing the size of the African American population by age and sex, 1930–1990

We estimate the size of the African American population in five-year age groups at census dates from 1930 to 1990 using a three-part strategy. For cohorts born after 1935, we follow the U.S. Census Bureau in using classical demographic analysis. To estimate the size of cohorts born before 1895, we use extinct-generation estimates. For remaining cohorts, we implement an age/period/cohort model of census counts. All approaches are applied to a data set in which the age distribution of deaths has been corrected for age misreporting. Results provide strong confirmation of the basic validity of Census Bureau estimates of census undercounts for African Americans while extending estimates to new cohorts and periods. Our estimates are less consistent with an historical series prepared by Coale and Rives (1973).     

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.     

Estimation of demographic measures for India, 1881-1961, based on census age distributions

Abstract India is one of the very few developing countries which have a relatively long history of population censuses. The first census was taken in 1872, the second in 1881 and since then there has been a census every ten years, the latest in 1971. Yet the registration of births and deaths in India, even at the present time, is too inadequate to be of much help in estimating fertility and mortality conditions in the country. From time to time Indian census actuaries have indirectly constructed life tables by comparing one census age distribution with the preceding one. Official life tables are available for all the decades from 1872-1881 to 1951-1961, except for 1911-1921 and 1931-1941. Kingsley Davis(1) filled in the gap by constructing life tables for the latter two decades. He also estimated the birth and death rates ofIndia for the decades from 1881-1891 to 1931-1941. Estimates of these rates for the following two decades, 1941-1951 and 1951-1961, were made by Indian census actuaries. The birth rates of Davis and the Indian actuaries were obtained basically by the reverse survival method from the age distribution and the computed life table of the population. Coale and Hoover(2), however, estimated the birth and death rates and the life table of the Indian population in 1951 by applying stable population theory. The most recent estimates of the birth rate and death rate for 1963-1964 are based on the results of the National Sample Survey. All these estimates are presented in summary form in Table 1.     

Estimation of interprovincial migration for Canada from place of birth by residence data, 1951–1961

In Canada, unlike many other countries, birth-residence data by age and sex are available in each of the decennial censuses from 1931 to 1961 which permit the estimation of intercensal net migration for the provinces and regions. After a brief discussion of the basic measures of migration from birth-residence data the paper focusses on the problems and procedures in estimating interprovincial net migration, 1951–1961 for Canada using “the place of birth survival ratio method, ” and it evaluates the estimates thus obtained. The evaluation of the estimates, taking into consideration the inherent limitations of the method and its merits compared with period migration estimates by the census survival ratio method and life table survival ratio method, suggests that the net migration estimates for the Canadian born by the place of birth survival ratio method are probably more reliable than those by the other two methods. One striking finding was that the net migration curves by age obtained from the census survival ratio and place of birth survival ratio estimates were smoother than the curve obtained with the use of the more accurate life table survival ratios. Furthermore, whatever the relative accuracy of net migration may be, the birth-residence approach is capable of furnishing more details about the net migration of the native born than by the standard survival-ratio methods. For the population under age 10 intercensal estimates were directly derived from the place of birth and residence distributions by age.     

An assessment of China’s fertility level using the variable-<Emphasis Type="Italic">r</Emphasis> method

The fertility level in China is a matter of uncertainty and controversy. This paper applies Preston and Coale's (1982) variable-r method to assess the fertility level in China. By using data from China's 1990 and 2000 censuses as well as annual population change surveys, the variable-r method confirms that Chinese fertility has reached a level well below replacement.     

The use of model life tables to estimate mortality for the United States in the late nineteenth century

This paper seeks to extend our knowledge about mortality in the late nineteenth century United States by using census mortality data for older children and teenagers to fit model tables. The same method can also be used with partially underregistered death data. The most commonly used model tables, the Coale and Demeny West Model, apparently do not adequately depict the changing shape of mortality over the period 1850--1910. An alternative model life table system is presented, based on the Brass two parameter logit system and available reliable life tables from the period 1850--1910. The two parameter system must be reduced to a one parameter system by means of estimated relationships between the parameters so that the fitting procedure can be used. The resulting model system is, however, heavily dependent on the experience of northern, industrial states, especially Massachusetts.     

A Mathematical Model Relating Cohort And Period Mortality

This paper presents a mathematical model for changing mortality in functional form. This model may be used to obtain cohort forces of mortality and cohort survivorship functions from a period force of mortality and a period life table under conditions of gradually changing mortality if an estimate of the amount of change in mortality is available. An example is given to show how the cohort functions are derived from the period functions.     

Practical aspects on the estimation of the parameters in Coale’s model for marital fertility

This paper demonstrates the estimation of the parameters in the Coale model for marital fertility by the maximum likelihood method, under the assumption of a simple Poisson process model. The necessary calculations are easily performed in the statistical computer program package GLIM, and the necessary commands are noted. Without access to GLIM, or any equivalent, it is still possible to fit the model approximately by the use of any weighted linear regression program. In both cases, goodness of fit tests are available.     

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.