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." The geographical focus is on the United States. (SUMMARY IN FRE)     

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.     

Heterogeneity,dependence among causes of death and Gompertz

Heterogeneity in a population with respect to mortality, or variation in “frailty”; among members of that population, which has been discussed extensively in the literature over the last decade and a half is essential to any realistic model of dependence among causes of death. The main problem then is the development of a mortality model incorporating heterogeneity and cause of death which is both realistic and of manageable proportions.

In a recent paper (J. H. Pollard, 1991), it has been shown that many life table results are remarkably insensitive to the strict shape of the mortality curve, at least for more developed populations, and that accurate approximations can in many cases be obtained knowing only the mortality rates at two representative ages (e.g. 50 and 70). These results and the Gompertz “law”; of mortality can be used to develop manageable approximate formulae for the expectation of life under heterogeneity and correlation among the causes of death. The formulae are confirmed by simulation.

Numerical results indicate, somewhat surprisingly, that the effects of correlation among causes of death, even at quite high levels, on expectation of life and changes on expectation of life when particular causes of death are reduced or eliminated are relatively minor.     

Empirical Prediction Intervals for County Population Forecasts

Population forecasts entail a significant amount of uncertainty, especially for long-range horizons and for places with small or rapidly changing populations. This uncertainty can be dealt with by presenting a range of projections or by developing statistical prediction intervals. The latter can be based on models that incorporate the stochastic nature of the forecasting process, on empirical analyses of past forecast errors, or on a combination of the two. In this article, we develop and test prediction intervals based on empirical analyses of past forecast errors for counties in the United States. Using decennial census data from 1900 to 2000, we apply trend extrapolation techniques to develop a set of county population forecasts; calculate forecast errors by comparing forecasts to subsequent census counts; and use the distribution of errors to construct empirical prediction intervals. We find that empirically-based prediction intervals provide reasonably accurate predictions of the precision of population forecasts, but provide little guidance regarding their tendency to be too high or too low. We believe the construction of empirically-based prediction intervals will help users of small-area population forecasts measure and evaluate the uncertainty inherent in population forecasts and plan more effectively for the future.     

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.     

Coherent Mortality Forecasting: The Product-Ratio Method With Functional Time Series Models

When independence is assumed, forecasts of mortality for subpopulations are almost always divergent in the long term. We propose a method for coherent forecasting of mortality rates for two or more subpopulations, based on functional principal components models of simple and interpretable functions of rates. The product-ratio functional forecasting method models and forecasts the geometric mean of subpopulation rates and the ratio of subpopulation rates to product rates. Coherence is imposed by constraining the forecast ratio function through stationary time series models. The method is applied to sex-specific data for Sweden and state-specific data for Australia. Based on out-of-sample forecasts, the coherent forecasts are at least as accurate in overall terms as comparable independent forecasts, and forecast accuracy is homogenized across subpopulations.     

Multistate life‐tables and regression models

A survey is given of the use of modern statistical techniques in event history analysis, and in particular in the study of multi‐state life‐tables in demography. Emphasis is placed on the interplay between partial likelihood and nonparametric maximum likelihood based methods, a) when analysing semi‐Markov models or models with repeated spells, and b) in frailty models for inobservable heterogeneity.     

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.     

European Demographic Forecasts Have Not Become More Accurate Over the Past 25 Years

Nowadays, demographers, population statisticians, and population forecasters have richer data, more refined theories of demographic behavior, and more sophisticated methods of analysis than they had two or three decades ago. This scientific progress should have made it easier to predict demographic behavior. But analyses of the errors in older forecasts show that demographic forecasts published by statistical agencies in 14 European countries have not become more accurate over the past 25 years. The findings demonstrate that scientific progress in population studies during the previous two to three decades has not kept up with the trend toward less predictable demographic behavior of populations in European countries. There is no reason to be more optimistic about US Census Bureau forecasts. Population forecasts are intrinsically uncertain, hence should be couched in probabilistic terms.     

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.     

Changes in mortality and life expectancy: Some methodological issues

Measuring and explaining the effects of mortality changes on life expectancy has been discussed for the past three decades. Different approaches have been proposed using discrete or continuous methods. Two basic ideas underlie these approaches. The first compares two different mortality schedules and quantifies the contribution of each age group to the increase in life expectancy. The second analyzes how the progress in the mortality schedule translates into progress in life expectancy. This paper discusses and compares the approaches proposed by the United Nations (1982), Arriaga (1984), Pollard (1982, 1988), and Vaupel (1986), identifying their problems, advantages, and the types of situations where each one can best be applied.     

Point and interval forecasts of age-specific fertility rates: a comparison of functional principal component methods

Accurate forecasts of age-specific fertility rates are critical for government policy, planning and decision making. With the availability of the Human Fertility Database (2011), the paper compares the empirical accuracy of the point and interval forecasts, obtained by the approach of Hyndman and Ullah (Comput Stat Data Anal 51(10), 4942?C4956, 2007) and its variants for forecasting age-specific fertility rates. The analyses are carried out using the age-specific fertility data of 15 mostly developed countries. Based on the one-step-ahead to 20-step-ahead forecast error measures, the weighted Hyndman-Ullah method provides the most accurate point and interval forecasts for forecasting age-specific fertility rates, among all the methods we investigated.     

The Epidemiologic Transition Revisited: Compositional Models for Causes of Death by Age and Sex

For decades, researchers have noted systematic shifts in cause‐of‐death patterns as mortality levels change. The notion of the “epidemiologic transition” has influenced thinking about the evolution of health in different societies and the response of the health system to these changes. This article re‐examines the epidemiologic transition in terms of empirical regularities in the cause composition of mortality by age and sex since 1950, and considers whether the theory of epidemiologic transition presents a durable framework for understanding more recent patterns. Age‐sex‐specific mortality rates from three broad cause groups are analyzed: Group 1 (communicable diseases, maternal and perinatal causes, and nutritional deficiencies); Group 2 (noncommunicable diseases); and Group 3 (injuries), using the most extensive international database on mortality by cause, including 1,576 country‐years of observation, and new statistical models for compositional data. The analyses relate changes in cause‐of‐death patterns to changing levels of all‐cause mortality and income per capita. The results confirm that declines in overall mortality are accompanied by systematic changes in the composition of causes in many age groups. These changes are most pronounced among children, for whom Group 1 causes decline as overall mortality falls, and in younger adults, where strikingly different patterns are found for men (shift from Group 3 to Group 2) compared to women (shift toward Group 2 then Group 3). The underlying patterns that emerge from this analysis offer insights into the epidemiologic transition from high‐mortality to low‐mortality settings.     

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.     

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.     

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.     

The Predictive Validity of Subjective Mortality Expectations: Evidence From the Health and Retirement Study

Several recent studies suggest that individual subjective survival forecasts are powerful predictors of both mortality and behavior. Using 15 years of longitudinal data from the Health and Retirement Study, I present an alternative view. Across a wide range of ages, predictions of in-sample mortality rates based on subjective forecasts are substantially less accurate than predictions based on population life tables. Subjective forecasts also fail to capture fundamental properties of senescence, including increases in yearly mortality rates with age. To shed light on the mechanisms underlying these biases, I develop and estimate a latent-factor model of how individuals form subjective forecasts. The estimates of this model’s parameters imply that these forecasts incorporate several important sources of measurement error that arguably swamp the useful information they convey.     

A stochastic version of the malthusian trap model: Consequences for the empirical relationship between economic growth and population growth in LDC's

A stochastic version of the Malthusian trap model relating the growth rate of income per capita to the population growth rate of a given country is described. This model is applied to the a priori evaluation of the cross‐sectional correlation between these two growth rates under two additional assumptions: i) the relations in the model at national levels include country‐specific and time‐invariant random components, and ii) these growth rates are measured with a certain degree of temporal aggregation. It is shown that these two assumptions can explain near‐zero correlations between the two growth rates even if there exist a strongly negative effect of population growth on economic growth. However it is not clear whether these assumptions fully explain such insignificant correlations. Indeed, the implementation of the model is complicated by the structural shifts which are likely to occur in the equations over the course of the demographic transition.     

Mortality in England and Wales from 1848 to 1947

This is a survey of the changing causes of death in England and Wales during the past 100 years. Based on the published mortality statistics of the General Register Office the framework of the survey is a series of specially prepared tables of death rates by sex, age and cause of death for the periods 1848–72, 1901–10, 1921, 1931, 1939 and 1947. Adjustments were made wherever necessary to compensate for changes in medical nomenclature and in the statistical classification of disease.

After allowance has been made for the changing age structure of the population, the male death rate at all ages in 1947 was 42% of the rate in 1846–50, and the female rate 35 %. Maximum improvement was among girls aged 5–9 years, whose death rate in 1947 was 9% of the rate 100 years before.

In 1848–72 the group to which were allocated the largest proportion of the deaths at all ages were the infectious diseases with one-third of the total; and these were followed by the respiratory, nervous and digestive diseases. In 1947, on the other hand, diseases of the circulatory system came first with rather more than one-third of the total at all ages, and these were followed by cancer.

Changes in proportionate mortality rates from various causes have been examined at successive ages from infancy to old age. There was a decline in proportionate mortality from the infectious diseases other than tuberculosis, but increased mortality from tuberculosis in the younger age groups and from violence, circulatory diseases and cancer.

The trends of absolute mortality from the various causes were also studied. The reduction in total mortality was such that whereas there were half a million deaths of civilians registered in England and Wales in 1947, the total would have been over a million had the death rates of 1848–72 still prevailed.

The article concludes with a brief review of the factors responsible for the changes that have taken place.     

A dynamic multistate model of robustness and frailty

Dynamic multistate models can show realistic population dynamics over time, model complex cycles, and encompass the life history of a cohort. This paper uses a recently developed approach to obtain the analytic solution of a time‐dependent multidimensional differential equation. The illustrative robust/frail model presented shows that the assumption of fixed individual frailty can be abandoned in a two living state model that allows transitions between health statuses and nonproportional hazards of death.