Estimating the goodman,keyfitz, pullum kinship equations: An alternative procedure∗

As is often the case in demography, Goodman, Keyfitz and Pullum (1974) developed their theory of the interrelationships of fertility, mortality and kinship numbers by means of continuous mathematics [integrals], but resorted to ad hoc finite approximations for calculating results in concrete empirical cases. Their reason: ‘Ordinarily, we cannot evaluate the l(x) and m(x) functions for arbitrary values of x, since the data are usually collected for five‐year age intervals’ [p. 24]. Recent developments in computer software now provide an alternative, two‐step procedure that avoids extensive programming of finite approximation algorithms: 1) using a popular scientific curve‐fitting package, functions are found to represent particular sets of discrete data on fertility and mortality, 2) the resulting functions and parameter estimates are then inserted directly into the kinship equations, and the integrals evaluated numerically using readily available mathematics software. This procedure has potentially wide application in other areas of population mathematics where theory is given by integrals and other continuous expressions, but data are for discrete age groups.     

A nine‐parameter version of the Heligman‐Pollard formula

In this paper we outline and evaluate a nine‐parameter version of the Heligman‐Pollard formula. In our applications, using mortality data for five European countries we found that this version provides closer fits to empirical mortality data than the classical eight‐parameter formula, thus eliminating a source of systematic error in this latter formula.     

A relational technique for estimating the age‐specific mortality pattern from grouped data

This paper presents a new technique of expanding an abridged life table. This technique is a non‐parametric one, which relates the probabilities of dying of the abridged table to those of a standard complete table. In order to evaluate the accuracy of the new technique we use it, as well as two other techniques, for expanding empirical abridged data sets. According to the results of our calculations the new method proves very efficient in producing complete life tables from grouped data, in many cases producing more accurate results than the other two methods in spite of its simplicity.     

A semi‐Markovian approach to using event history data in multiregional demography

As a first step towards infusing event‐history analysis into multiregional demography, this paper introduces a semi‐Markovian framework and outlines its salient features as differentiated from a pure Markovian framework. Specifically, what differentiates the former from the latter is an explicit consideration of duration‐dependence in migrating from one region to another. This duration‐dependence is one of the complexities involved in using event‐history data in multiregional demography. The use of event‐history data lends itself easily to defining basic probabilities involved in a semi‐Markovian framework directly on sample paths of individuals. The underlying concepts of a semi‐Markov process in the special case of time‐homogeneity or age‐independence of transition probabilities are given in a coherent and concise form. Illustrations of empirical applications to the event‐history data on migration as provided by the Korean National Migration Survey conducted in 1983, and of distinct features of the semi‐Markovian analysis through a parametrization of the basic probabilities are also given in this paper.     

Statistical modeling of selected aspects of the childbearing process with application to world fertility survey countries∗

The childbearing process should be monitored in developing countries experiencing high population growth rates and high levels of maternal and infant mortality. A mathematical model for estimation of certain aspects of the childbearing process, which requires only data on age‐specific fertility rates, is developed. Synthetic maternal childbearing indices, namely, mean ages at first and last birth, length of reproductive life span, inter‐birth spacing, and proportion of childless women, in addition to the well‐known mean age at childbearing, for the WFS countries are obtained using the proposed model. The indices are free from age truncation effects, and, under certain assumptions, provide information about a cohort's completed fertility before the women stop reproducing. The effects of women's residence and education on fertility are also examined.     

Mixed estimation of old‐age mortality

The estimation of the mortality of the “oldest old”; is subject to considerable random error, but important prior information exists that can be used to make the estimates more robust. Mixed estimation is a method of incorporating auxiliary information into the statistical estimation of linear models. We extend the method to cover general maximum likelihood estimation, and show that the mixed estimator can be represented approximately as a weighted average of the purely data based estimator and the auxiliary estimator. The methods can be applied to the analysis of the old‐age mortality via logistic and Poisson regression. A major advantage of the mixed estimator is the simplicity with which it can incorporate partial prior information. Moreover, no special software is needed in the fitting. We show how the targeting methods of Coale and Kisker can be represented as mixed estimation in a natural way that is more flexible than the original proposal. We also derive empirical estimates of the target information based on pooled data from several countries with high quality data. We consider the mortality of Finland at ages 80 +, study the reliability of the evidence of mortality crossover, and derive estimates of life expectancy at age 100.     

Comparison of Four Methods for Estimating Complete Life Tables from Abridged Life Tables Using Mortality Data Supplied to EUROCARE-3

ABSTRACT

To estimate mortality due to cancer, it is necessary to have mortality data by year of age in the population of cancer patients. When such data are not available, estimating one-year (complete) life tables from five-year (abridged) life tables is necessary. Four such methods—Elandt–Johnson, Kostaki, Brass logit, and Akima spline methods—are compared with respect to 782 empirical complete life tables pertaining to 19 European regions or countries, from 1954 to 2000. Abridged life tables are first derived from the empirical ones, then used to produce one-year-life tables by each of the four methods. These reconstituted complete life tables are then compared with the empirical complete life tables. Among the four methods, the Elandt–Johnson demographic method produces the best reconstitutions at adult ages, specifically those ages at which observed cancer survival needs to be corrected.     

Optimization models for degrouping population data

In certain countries population data are available in grouped form only, usually as quinquennial age groups plus a large open-ended range for the elderly. However, official statistics call for data by individual age since many statistical operations, such as the calculation of demographic indicators, require the use of ungrouped population data. In this paper a number of mathematical models are proposed which, starting from population data given in age groups, enable these ranges to be degrouped into age-specific population values without leaving a fractional part. Unlike other existing procedures for disaggregating demographic data, ours makes it possible to process several years' data simultaneously in a coherent way, and provides accurate results longitudinally as well as transversally. This procedure is also shown to be helpful in dealing with degrouped population data affected by noise, such as those affected by the age-heaping phenomenon.     

Age patterns of mortality for older women: An analysis using the age‐specific rate of mortality change with age

The age‐specific rate of mortality change with age, defined by k(x) = d Inμ(x)/dx, where μ(x) is the age‐specific death rate at exact age x, is estimated for middle and old ages in ten selected populations that are considered to have relatively accurate age data. For females in each of the study populations, k(x) follows a bell‐shaped curve that usually peaks around age 75. In some of the populations, the age pattern of k(x) for males is confounded with substantial cohort variations, which seem to reflect long‐term impacts of their World War I experiences.

Among the mathematical models proposed by Gompertz, Makeham, Perks and Beard, only the Perks model is consistent with the bell‐shaped pattern of k(x). It is shown that, if the risk of death for every individual follows the Makeham equation and if the individual frailty is gamma‐distributed, then the age‐specific death rate follows the Perks equation.     

Effect of aggregation on the estimation of trend in mortality

There are three approaches to analyzing and forecasting age‐specific mortality: (1) analyze age‐specific data directly, (2) analyze each cause‐specific mortality series separately and add the results, (3) analyze cause‐specific mortality series jointly and add the results. We show that if linear models are used for cause‐specific mortality, then the three approaches often give close results even when cause‐specific series are correlated. This result holds for cross‐correlations arising from random misclassification of deaths by cause, and also for certain patterns of systematic misclassification. It need not hold, if one or more causes serve as “leading indicators”; for the remaining causes, or if outside information is incorporated into forecasting either through expert judgment or formal statistical modeling. Under highly nonlinear models or in the presence of modeling error the result may also fail. The results are illustrated with U.S. age‐specific mortality data from 1968–1985. In some cases the aggregate forecasts appear to be the more credible ones.     

Migration bias in indirect estimates of regional childhood mortality levels

Demographers often use Brass‐style indirect methods to obtain childhood mortality estimates for regions within developing countries. Regional populations are not closed to migration, however, and mortality reports of women resident in a certain region on the survey date may contain information on events and exposure that occurred elsewhere as the mother migrated. Including this “imported”; mortality information may cause significant bias in regional estimates. In this paper the authors: (1) investigate the possible magnitude of migration bias using a multiregional simulation model, (2) propose a modification to standard methods which should reduce bias in many circumstances, and (3) apply the modified technique to data from Brazil's 1980 Census. We find that migration bias can indeed be significant, and that in the specific case of São Paulo state, imported mortality information may result in overestimates of local mortality levels of 10–15% when using Brass‐style methods.     

Mortality modeling: A review

Looking at survival in terms of biological indicators of aging has given rise to various models of mortality, some of which we review here. The most notable models are that of Strehler and Mildvan, which relates the force of mortality to the ability of organisms to compensate for stress, and that of Sacher and Trucco, which describes the role played by homeostatic forces in shaping the age‐specific pattern of mortality. The analysis of longitudinal data in aging studies now incorporates the notions of heterogeneity and frailty, as well as that of changes in the “repair capacity”; of organisms. Furthermore, attention is now being paid to evolutionary theory and to models of senescence. These models and directions for further research are discussed.     

Populations with constant immigration

We consider a Leslie‐type model of a one‐sex (female) population of natives with constant immigration. The fertility and mortality schedule of the natives may be below or above replacement level. Immigrants retain their fertility and mortality, their children adopt the fertility and mortality of the natives. It is shown how this model may be written in a homogeneous form (without additive term) with a Leslie‐type matrix. Reproductive values of individuals in each age group are discussed in terms of a left eigenvector of this matrix. The homogeneous form of our projection model permits the transformation into a Markov chain with transient and recurrent states. The Markov chain is the basis for the definition of genealogies, which incorporate immigration. It is shown that genealogies describe the life histories of individuals in a population with immigration. We calculate absorption times of the Markov chain and relate them to genealogies. This extends the theory originally designed for closed populations to populations with immigratioa     

Effects of targets and aggregation on the propagation of error in mortality forecasts

Official forecasts of mortality depend on assumptions about target values for the future rates of decline in mortality rates. Smooth functions connect the jump‐off (base‐year) mortality to the level implied by the targets. Three alternative sets of targets are assumed, leading to high, middle, and low forecasts. We show that this process can be closely modeled using simple linear statistical models. These explicit models allow us to analyze the error structure of the forecasts. We show that the current assumption of perfect correlation between errors in different ages, at different forecast years, and for different causes of death, is erroneous. An alternative correlation structure is suggested, and we show how its parameters can be estimated from the past data.

The effect of the level of aggregation on the accuracy of mortality forecasts is considered. It is not clear whether or not age‐ and cause‐specific analyses have been more accurate in the past than analyses based on age‐specific mortality alone would have been. The major contribution of forecasting mortality by cause appears to have been in allowing for easier incorporation of expert opinion rather than in making the. data analysis more accurate or the statistical models less biased.     

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.     

Turbulent dynamics in a Xviith century population

A reconstruction of the population of the Pays de Caux (1589–1700) yields the time series of a fertility behavior indicator, the overall Coale index If. In spite of the noisy appearance of its evolution, the trajectory of If looks ordered, as if it were confined alternatively to two given zones, looping in each of them for a while, then suddenly jumping from the low one to the higher one, or slowly whirling down from the high to the low one.

An attempt is made to explain this general temporal structure by using a simulation model based on the autoregulation model (the so‐called European Marriage Pattern), putting into play a choice of the spouse function, a fertility function, modalities of marriage and remarriage, under the environmental forcing of the reconstructed mortality conditions.

The correspondence between reconstruction and simulation turns out to be quite good, not only for the population size or the Coale index, but also for the marriage series, quite independently of the reconstructioa

A second simulation with simulated mortality conditions shows a bifurcation point: as the mean frequency of crisis increases, the state of the system leaves the lower level and concentrates more and more in the higher level.

Thus, not only does the autoregulator model appear validated by empirical data, but its bi‐modal structure is revealed, depicting the dynamic response of a traditional community both to the environment and to the endogeneous demographic process.     

Modeling mortality at old age with time-varying parameters

Several models of old age mortality with time-varying parameters are expressed in a single formula. In these models, the existence of an age threshold above which mortality increases over time and below which mortality decreases over time is problematic. The conditions of appearance of this threshold are expressed and shown on logistic and exponential models with empirical data. The conditions of appearance of the threshold reflect actual situations in developed countries. Richards’ curve avoids the appearance of the threshold with empirical data.     

Continued Reductions in Mortality at Advanced Ages

Mortality data for 30 mostly developed countries available in the Kannisto–Thatcher Database on Old‐Age Mortality (KTDB) are drawn on to assess the pace of decline in death rates at ages 80 years and above. As of 2004 this database recorded 37 million persons at these ages, including 130,000 centenarians (more than double the number in 1990). For men, the probability of surviving from age 80 to age 90 has risen from 12 percent in 1950 to 26 percent in 2002; for women, the increase has been from 16 percent to 38 percent. In the lowest‐mortality country, Japan, life expectancy at age 80 in 2006 is estimated to be 6.5 years for men and 11.3 years for women. For selected countries, average annual percent declines in age‐specific death rates over the preceding ten years are calculated for single‐year age groups 80 to 99 and the years 1970 to 2004. The results are presented in Lexis maps showing the patterns of change in old‐age mortality by cohort and period, and separately for men and women. The trends are not favorable in all countries: for example, old‐age mortality in the United States has stagnated since 1980. But countries with exceptionally low mortality, like Japan and France, do not show a deceleration in death rate declines. It is argued that life expectancy at advanced ages may continue to increase at the same pace as in the past.     

Inequality in Life Spans and a New Perspective on Mortality Convergence Across Industrialized Countries

The second half of the twentieth century witnessed substantial convergence in life expectancy around the world. We examine differences in the age pattern of mortality in industrialized countries over time to show that inequality in adult life spans, which we measure with the standard deviation of life table ages at death above age 10 years, S₁₀, is increasingly responsible for the remaining divergence in mortality. We report striking differences in level and trend of S₁₀ across industrialized countries since 1960, which cannot be explained by aggregate socioeconomic inequality or differential external‐cause mortality. Rather, S₁₀ reflects both within‐ and between‐group inequalities in life spans and conveys new information about their combined magnitudes and trends. These findings suggest that the challenge for health policies in this century is to reduce inequality, not just lengthen life.     

On the use of the truncated Gompertz distribution and other models to represent the parity progression functions of high fertility populations

The Gompertz distribution, developed from the mortality “law”; long used by actuaries and demographers promises to be a useful distribution for many other demographic purposes as well. The continuous distribution can also be adapted to represent discrete data commonly encountered in demographic work, and maximum likelihood estimates of the two parameters are easily calculated using formulae developed in this paper, whether those data be continuous or discrete, truncated below or provided with observations in a final open‐ended interval.

The distribution is unimodel. The use of the truncated form of the distribution, however, allows the researcher to fit it to a wider range of observed distributions, including many for which the density function is monotonic decreasing.

Empirical studies using parity progression data of two high fertility populations indicate that the truncated Gompertz distribution in its discrete form provides a good overall picture of the parity distribution. Interestingly, the simple method of partial sums, commonly employed to fit the Gompertz function, appears to provide parameter estimates which are close to those estimated by maximum likelihood.