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....The results are illustrated with U.S. age-specific mortality: (1) analyse age-specific mortality data from 1968-1985. In some cases the aggregate forecasts appear to be the more credible ones." This is a revised version of a paper originally presented at the 1990 Annual Meeting of the Population Association of America (see Population Index, Vol. 56, No. 3, Fall 1990, p. 407).     

Revised regional model life tables at very low levels of mortality

This paper presents and discusses new model life tables at very low mortality, which make use of age-specific death rates from the 1960s, 1970s, and 1980s. These life tables fit recorded death rates in very low mortality populations better than do the existing ones at expectations of life of 77.5 and 80 years. The old tables incorporate too-high mortality at the higher ages and in infancy and they incorporate regional differences that no longer exist. The new tables "close out" the mortality schedules above age 80 more realistically. The convergence of age patterns of mortality at very high life expectancies in populations that used to conform to different families is in itself of demographic interest. Some convergence may perhaps be expected. Sullivan (1973) found that, in Taiwan, the comparison of mortality at ages 1-5 to mortality at 5-35 in the late 1950s showed higher mortality at the younger ages relative to the ensuing 30-year age interval than was found in any of the models, including the South model, which has the highest relative mortality from ages 1-5 among the 4 regional patterns. Then, in the late 1960s, the relation of mortality at 1-5 to mortality at 5-35 in Taiwan fell to a position intermediate between the West and South tables. Sullivan found in data on mortality by cause of death a large reduction in mortality from diarrhea and enteritis, no doubt as a result of environmental sanitation. Mortality from these causes is concentrated among young children, and reduction in deaths from these causes would naturally diminish the excess mortality in this age interval. The East pattern, characterized by very high mortality in infancy (but not from 1-5), may be the result of the prevalence of early weaning or avoidance of breast feeding altogether in the populations characterized by this pattern. As health conditions have improved, evidenced by the overall design of mortality, these special factors are diminished or erased. Model life tables at these very low mortality levels have different uses from most applications of model life tables at higher mortality. The use of model tables to estimate accurate schedules of mortality when the basic data are incomplete or inaccurate is less relevant in this range of mortality levels.     

Comparison of information on death certificates and matching 1960 census records: age,marital status,race, nativity and country of origin

A sample of death certificates matched with 1960 Census records permitted comparison of response data for items asked on both records. Estimates of bias in death rates which are based on information from the two records are derived from the comparison data. Most of the comparisons yielded small discrepancies of inconsequential effect on the mortality rates. Some large inconsistencies, however, of potentially serious impact on the death rates were observed. The comparisons are examined and the implications of the results for the relevant mortality rates are discussed. In addition, age-specific death rates "corrected" for the disparities found in the age information on the two records are presented.     

Marriage selection and age patterns of mortality: a mathematical investigation

"In this paper, we use simulation models to demonstrate the complexity of the relationship between the marriage selection process and the resulting RMRs [relative mortality ratios]. In particular, we show that marriage selection alone can produce a relative mortality ratio which remains large and relatively constant at ages far beyond the marriage span....Our general objective...is to determine the range of age patterns of relative mortality which could, in theory, result from marriage selection on the basis of health characteristics. We also evaluate the effects of variations in the marriage selection mechanisms on the resulting mortality patterns....We develop and apply several simple mathematical models of the marriage selection process. In order to distinguish the potential consequences of marriage selection from marriage protection, we consider hypothetical populations in which causal effects are absent....We begin by considering an extremely simple marriage selection process and subsequently explore a more realistic selection model based on recent death and marriage rates for Japan."     

Differences in mortality by marital status in Finland from 1976 to 2000: analyses of changes in marital-status distributions, socio-demographic and household composition, and cause of death

Being currently not married is more common today than 25 years ago. Over this period relative differences in mortality by marital status have increased in several countries, mainly as a result of a sharp decline in mortality among the married. Using Finnish census data linked with death certificates, we show that these increases are not explained by the non-married population becoming more marginalized in socio-economic status or household composition. However, the increases in marital-status differences in mortality from accidental, violent, and alcohol-related causes of death in the 30-64 age group indicate that changes in the health-related behaviour of the non-married population may play a role. The public-health burden associated with not being married has also grown. At the end of the 1990s about 15 per cent of all deaths above the age of 30 would not have occurred if the non-married population had had the same age-specific mortality rates as the married population.     

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)     

Survival of related individuals: an extension of some fundamental results of heterogeneity analysis

"Many ideas in the analysis of heterogeneous mortality are based on the relationship between individual and observed hazard rates. This connection is established with the help of conditional averaging procedure: The observed risk of death at age x is calculated among those who survive this age. The analogy of this result for bivariate survival model with correlated individual hazards is derived. In the case of correlated frailty model the parametric specification of the mean, variance and correlation coefficient of the bivariate frailty distribution among survivors is obtained. The relationship between local association measure and the characteristics of the bivariate frailty distribution among survivors is established." (SUMMARY IN FRE)     

US mortality in an international context: age variations

Compared to other developed countries, the United States ranks poorly in terms of life expectancy at age 50. We seek to shed light on the US's low life expectancy ranking by comparing the age-specific death rates of 18 developed countries at older ages. A striking pattern emerges: between ages 40 and 75, US all-cause mortality rates are among the poorest in the set of comparison countries. The US position improves dramatically after age 75 for both males and females. We consider four possible explanations of the age patterns revealed by this analysis: (1) access to health insurance; (2) international differences in patterns of smoking; (3) age patterns of health care system performance; and (4) selection processes. We find that health insurance and smoking are not plausible sources of this age pattern. While we cannot rule out selection, we present suggestive evidence that an unusually vigorous deployment of life-saving technologies by the US health care system at very old ages is contributing to the age-pattern of US mortality rankings. Differences in obesity distributions are likely to be making a moderate contribution to the pattern but uncertainty about the risks associated with obesity prevents a precise assessment.     

On the Far Eastern pattern of mortality

Since the early 1980s, it has been accepted widely that there is a Far Eastern pattern of mortality, a pattern characterized by excessively high death rates among older men relative to death rates among younger men and among women. It has been regarded as a unique regional mortality pattern, applying primarily to Far Eastern populations. A re-examination of the mortality data of some Far Eastern populations reveals that changes in both age patterns of and sex differentials in mortality have been widely observed. Further, mortality patterns similar to the so-called Far Eastern mortality model have been found in many other populations.     

On the Far Eastern pattern of mortality

Since the early 1980s, it has been accepted widely that there is a Far Eastern pattern of mortality, a pattern characterized by excessively high death rates among older men relative to death rates among younger men and among women. It has been regarded as a unique regional mortality pattern, applying primarily to Far Eastern populations. A re-examination of the mortality data of some Far Eastern populations reveals that changes in both age patterns of and sex differentials in mortality have been widely observed. Further, mortality patterns similar to the so-called Far Eastern mortality model have been found in many other populations.     

Methods for combining ancillary data in stochastic compartment models of cancer mortality: generalization of heterogeneity models

"We present a mortality model where nationally representative survey data on risk factor distributions are combined with data on cohort mortality rates to increase information, i.e., a fixed marginal risk factor distribution is combined with a cohort model representing unobserved individual risk heterogeneity. The model is applied to lung cancer mortality in nine U.S. white male cohorts aged 30 to 70 in 1950 and followed 38 years. Estimates of the cohort specific proportions of smokers were made from the National Health Interview Survey. Comparisons are made for models with different patterns of changes with age of individual heterogeneity." (SUMMARY IN FRE)     

African-american mortality at older ages: Results of a matching study

In this paper we investigate the quality of age reporting on death certificates of elderly African-Americans. We link a sample of death certificates of persons age 65+ in 1985 to records for the same individuals in U.S. censuses of 1900, 1910, and 1920 and to records of the Social Security Administration. The ages at death reported on death certificates are too young on average. Errors are greater for women than for men. Despite systematic underreporting of age at death, too many deaths are registered at ages 95+. This excess reflects an age distribution of deaths that declines steeply with age, so that the base for upward transfers into an age category is much larger than the base for transfers downward and out. When corrected ages at death are used to estimate age-specific death rates, African-American mortality rates increase substantially above age 85 and the racial “crossover” in mortality disappears. Uncertainty about white rates at ages 95+, however, prevents a decisive racial comparison at the very oldest ages.     

Population dynamics in Germany: the role of immigration and population momentum

The effects of changes in rates of mortality, fertility, and migration depend not only on the age-specific patterns and levels of these rates, but on the age structure of the population. In order to remove the influences of the age structure and concentrate on the effects of the demographic rates themselves, a common practice is to analyze the influences of the rates for a standard age structure. This paper analyzes current and future population changes in Germany, using a stationary population equivalent model (SPE) that shows long-term effects of current fertility, mortality, and international migration patterns. Results indicate that the German population will eventually decline because of below replacement fertility, if net immigration does not counteract this decrease. This means, for instance, that the long-term stationary population levels for Germany will decrease by approximately 6.5 million during a decade in which current fertility, mortality, and international migration levels prevail. The paper also reports how various other assumptions for mortality, fertility, and international migration affect the SPE model for Germany.     

The geometric mean of the age-specific death rates as a summary index of mortality

Even though a single summary index of mortality can never replace the set of age-specific death rates, it has been found to be extremely useful for a wide variety of purposes. Such indexes are generally one of two types: aggregative indexes, such as directly standardized rates which reflect absolute differences between corresponding age-specific mortality rates; and average of relatives indexes which reflect proportional differences between those rates. The choice of index depends upon the purposes for which it is to be used, and is important as different indexes can produce very different results. While directly standardized rates are widely used, they depend upon the selection of an appropriate standard population and give disproportionately heavy weight to the high ages. Average of relatives indexes give equal weight to all ages, but are infrequently used as no index of that type has gained wide acceptability. This paper recommends the use of the geometric mean of the age-specific mortality rates as such an index, and shows that this index is readily calculable, unbiased, needs no standard population, is directly comparable to all other indexes so calculated, and accurately reflects exponential mortality patterns.     

Development,inequality, health care,and mortality at the older ages: a cross-national analysis

We examine mortality at ages 50 and above in female populations of 38 countries and control for variation in quality of the mortality data. We find that economic development, economic distributional inequality, and basic primary health care have independent cross-national effects on cause of death structures and that these effects are not uniform across the age intervals of interest. As improvements occur in level of living and heath care, age-specific death rates decline except at the oldest ages, at which point they may increase. Our results are interpreted in terms of their relevance for mortality research, theory, and policy.     

Separating the Signal From the Noise: Evidence for Deceleration in Old-Age Death Rates

Widespread population aging has made it critical to understand death rates at old ages. However, studying mortality at old ages is challenging because the data are sparse: numbers of survivors and deaths get smaller and smaller with age. I show how to address this challenge by using principled model selection techniques to empirically evaluate theoretical mortality models. I test nine models of old-age death rates by fitting them to 360 high-quality data sets on cohort mortality after age 80. Models that allow for the possibility of decelerating death rates tend to fit better than models that assume exponentially increasing death rates. No single model is capable of universally explaining observed old-age mortality patterns, but the log-quadratic model most consistently predicts well. Patterns of model fit differ by country and sex. I discuss possible mechanisms, including sample size, period effects, and regional or cultural factors that may be important keys to understanding patterns of old-age mortality. I introduce mortfit, a freely available R package that enables researchers to extend the analysis to other models, age ranges, and data sources.     

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.     

Effect of mortality change on stable population parameters

The stable population model is used to establish formulas expressing the effects of mortality change on population growth rates, birth rates, and age composition. The change in the intrinsic growth rate is shown to be quite accurately approximated by the average decline in age-specific death rates between age zero and the mean age at childbearing in the stable population. This change is essentially independent of the initial level of fertility in the population. Changes in birth rates and age composition are shown to be simple functions of the age pattern of cumulative changes in mortality rates relative to an appropriately defined “neutral” standard.     

A method for deriving mortality estimates from incomplete vital Statistics

Summary Although they are available in many developing countries vital registration records are very little used for mortality estimation which is still mainly based on census returns. However, defective death records may yield accurate estimations of mortality. This procedure requires few data only; a sex-age distribution of the population (preferably at the middle of a period) and a sexage distribution of deaths, either derived from vital records or from census returns to questions relating to deaths during the preceding twelve months. This method is based on the observation that for a fixed age structure of the population, there is a one-one relation between the age structure of deaths (measured by the proportion of deaths at older ages) and the level of mortality (measured by the death rate above a certain minimum age). It is assumed that at ages above this minimum the rate of underregistration of deaths does not vary significantly with age. Therefore, the age distribution of registered deaths makes it possible to estimate the true proportion of deaths at older ages. This in its turn will permit the estimation of the true level of mortality, because of the relation which exists between age structure of deaths and level of mortality. The true level is then compared with the observed, to estimate the rate of underregistration, and observed age-specific death rates can be adjusted in the light of this knowledge.     

On the theory of stable populations: A new and elementary proof of the theorems under weaker assumptions

Expositions and elementary proofs are given for the basic theorems of stable population theory: That a population subjected to vital rates (not necessarily constant over time) satisfying certain postulates will eventually "forget" its original age distribution and take on one (not necessarily constant over time) which depends only on its history of agespecific vital rates, a process called "weak ergodicity." That consequently the subsequent birth, death, and growth rates (none of these necessarily constant over time) depend only on the history of age-specific vital rates and not on the original age distribution. And, in particular, that these results apply to the special case, herein called "classic" stable population theory, in which the age-specific vital rates are constant over time, and in which after the "forgetting" takes place the subsequent age distribution and birth, death, and growth rates all become constant. This formulation of the theory differs from previous ones in two respects: First, the postulates required are weaker, and hence the theorems more general, than previously; in particular, this formulation permits the highest age of childbearing to change from cohort to cohort, which is important for populations practicing contraception. Second, none of the advanced mathematics used in previous formulations is needed; only the manipulation of sums and inequalities from high school algebra and the concept of "limit" from freshman calculus are required.