12Geographic Differences in Life Expectancy at Age 50 in the United States Compared with Other High-Income Countries

INTRODUCTION

Just as mortality differs across countries, it also differs geographically within countries. In the United States, for example, the range of life expectancy at birth (e₀) for the years 1999-2001 extended from 72.3 for the District of Columbia (lowest) and 73.6 for Mississippi (second lowest) to 79.0 for Minnesota (second highest) and 79.7 for Hawaii (highest).¹ Life expectancy at age 50 (e₅₀) for the same years reflected a similar hierarchy: from 28.0 for both the District of Columbia and Mississippi to 31.4 for Minnesota and 32.4 for Hawaii. These ranges are smaller than those found across a broad group of high- and middle-income countries in 2000 (see Table 12-1). They are, however, larger once we exclude countries of Eastern Europe and the former Soviet Union from the comparison set.

The geographic variation of life expectancy at age 50 in the United States is illustrated here in Figure 12-1, which shows results separated by sex (men and women) and by administrative unit (states and counties). The broad pattern of geographic variation is similar across the four panels of Figure 12-1: relatively low values of e₅₀ in the District of Columbia and across a large area of the Southeast, extending northward into Appalachia and to a lesser extent into parts of the Great Lakes region; and relatively high values of e₅₀ across the far north central region of the country, extending into the mountain states as well.

Despite an increasing trend in life expectancy during the latter half of the 20th century at all ages and for all states (plus the District of Columbia), the rankings of the various states or regions in this geographic hierarchy have changed rather little over this time period (National Center for Health Statistics, 1975, 1998). Moreover, in a recent investigation at the county level, Ezzati and colleagues uncovered an even greater range of disparities in life expectancy at birth in the United States, of around 13 years for women and 18 years for men in 1999 (Ezzati et al., 2008). The authors point out that, whereas geographic variability diminished during the 1960s and 1970s, the distribution of e₀ by county in the United States started to diverge from the early 1980s onward. They demonstrated that this divergence—which was more pronounced for women than for men—was due to disparate trends affecting the more and the less advantaged areas of the country, as the former experienced a continuous rise in longevity while the latter experienced stagnation and, in the most extreme cases, a partial reversal of gains achieved in previous decades.

The divergence of the geographic distribution of mean longevity in the United States during the last two decades of the 20th century coincided with a rapid fall in the country's position in international rankings with respect to various measures of mortality or longevity. The deterioration of the U.S. position is well documented with regard to infant mortality (for a recent discussion on this topic, see in particular MacDorman and Mathews, 2008) but appears to be less well known regarding mortality at older ages. In 1980, among the full set of comparison countries used here,² values of life expectancy at age 50 extended from 24.2 for Hungary and the Czech Republic to 29.6 for Iceland, and the United States ranked 10th out of 33 with an e₅₀ of 28.0 (Human Mortality Database, see http://www.mortality.org [accessed November 13, 2009]).³ By 2006, the level of e₅₀ for the United States had risen to 31.3, a gain of 3.3 years. Over the same period, however, other countries experienced an even faster pace of improvement. Japan, with an e₅₀ of 34.4 in 2006, had moved into the top position by gaining 5.5 years since 1980. As a result of its relatively poor performance during these years, the position of the United States fell to 20th among the 34 comparison countries with data available in 2006. In fact, only Taiwan, Denmark, and the 12 countries of Eastern Europe and the former Soviet Union fared worse than the United States at that time.

Figure 12-2 illustrates the change in international ranking for e₅₀ among a more limited collection of comparison countries (excluding countries of Eastern Europe and the former Soviet Union, where mortality trends have been consistently less favorable than in the United States since 1980). The figure shows that, whereas until 1994 the United States was positioned among the upper 50 percent of the countries (not weighted by population size) with a rank that fluctuated between 9th and 12th, it lost position rapidly thereafter, falling to 13th in 1996, 14th in 1997, 18th in 1999, and 20th in 2005 and 2006 (just above Denmark in the list of 21 countries with data available for the most recent years).⁴ Although the difference in e₅₀ between the United States and the highest-ranking country was just 1.6 years in 1980, it grew to 2.2 years in 1995, 3.0 years in 2000, and 3.1 years in 2006.

Like the geographic divergence in the United States, the loss of position by the country in these rankings has been much more severe for women than for men. From 1980 to 2006, the ranking of U.S. women in terms of e₅₀ fell from 11th to 20th (out of 21 countries) and for U.S. men from 10th to 15th.⁵ Among all 21 countries on Figure 12-2, only Danish women had shorter lives, on average, after age 50 than U.S. women in 2006. Furthermore, the gap that separates the United States from other high-income countries is growing: whereas in 1980 women in the United States lived an average of just 1.1 years less on average than women in Iceland (who had the highest value of e₅₀ at that time), in 2006 they lived 4.1 years less than women in Japan. Men in the United States are doing better in international rankings and have also been more successful than women at progressively narrowing the gap that separates them from the top-ranking countries (Iceland in 1980, Australia in 2006) in terms of e₅₀, reducing this difference from 2.5 years in 1980 to 1.3 years in 2006.

Given the coincidence of timing (from the early 1980s until recently) and the shared characteristic of a greater impact on women, it is natural to inquire whether the increasing geographic disparity observed by Ezzati and colleagues is related in some causal fashion to the reduced pace of increase in values of life expectancy for the United States and thus to the country's loss of position in international rankings for this key indicator of population health. In the simple model of change proposed here (see the section on Methods), narrowing the gap between the most and the least advantaged areas of a country tends to accelerate a rise in longevity, whereas widening this gap tends to slow down and may even halt or reverse an increasing trend.

In this chapter we compare levels and trends in the variability of life expectancy at age 50 in the United States and four other countries (Canada, France, Germany, and Japan) and across an aggregate of countries or subnational areas of Western Europe. Our main purpose is to determine whether the increasing disparity in values of life expectancy in the United States may have contributed in a mechanical or otherwise causal fashion to the country's deteriorating position in international comparisons.

THEORETICAL FRAMEWORK

Although social and economic inequality is often cited as an explanation for the poor ranking of the United States in international comparisons of mortality or longevity, the exact nature of the connection is by no means obvious. In some studies, mortality or longevity is viewed as a response variable that can be expressed as a function of a stimulus variable, such as income.⁶ In this framework, an important question is whether variability in income (or some other stimulus) is negatively associated with levels of average longevity: in other words, can one attribute a lower level of life expectancy for some population to its higher level of income inequality? The correct answer is not necessarily “yes.” In fact, if the functional relationship between stimulus and response is linear, a symmetrical increase of variability in the stimulus induces no change in the response, as gains in longevity for those at the top of the income distribution are balanced exactly by losses for those at the bottom (see Duleep, 1995; Rodgers, 1979).

In the specific case of income, however, the functional relationship with longevity is distinctly nonlinear: as demonstrated using both aggregate-and individual-level data, gains in longevity decelerate sharply as income rises (Preston, 2007; Preston and Taubman, 1994; Antonovsky, 1967). In general, if a positive relationship between stimulus and response becomes weaker at higher levels of the stimulus, a symmetrical increase of inequality in the stimulus leads necessarily to a decrease of the mean response: in our example, gains in longevity by those at the top of the distribution have less impact on the mean longevity of the population than losses by those at the bottom.⁷

The problem posed here, however, is somewhat different, as we are studying the relationship between trends in the mean and the variance of a single variable, with no model of stimulus and response. In this situation, quantifying the contribution of changes in variance to changes in mean requires choosing a reference group in the population, which could be the highest-ranking half, third, fifth, etc., in terms of the variable of interest (here, life expectancy at age 50). By thus identifying a “leading group” in the population, we develop in the next section a simple means of quantifying the contribution of changes in the geographic distribution of longevity in a population to changes in mean longevity.

In this way we are able to obtain some key insights about the role of changing geographic disparities in e₅₀ to trends in e₅₀ itself for the United States and other high-income countries. Using this framework, convergence of subnational levels of e₅₀ helps to accelerate the national trend, as the less advantaged locations catch up to the leaders;⁸ conversely, divergence slows down the national increase, as the laggards fall farther and farther behind the more advantaged locations.

Although we are focusing here on the variation of mean longevity across geographically delimited population groups, it is also possible to analyze levels and trends of inequality among individual members of a population. Studies of the “compression of mortality” or “rectangularization of the survival curve” address the issue of internal variability for a given population as described by the distribution of deaths in a life table (Wilmoth and Horiuchi, 1999; Edwards and Tuljapurkar, 2005): we may call this intrapopulation variability. In contrast, the approach we are following here consists of studying inequality across countries or their geographic subunits: interpopulation variability. Both of these notions of variability or inequality of longevity are valid, and a more comprehensive analysis of the effects of changes in inequality on changes in mean longevity would take both perspectives into account. In this chapter, however, we focus on aggregate geographic differences as a means of gaining some preliminary insights into this matter.

DATA

Mortality indicators for selected years from 1950 to 2006 were collected for the United States, Canada, Japan, and 19 national or large subnational areas of Western Europe, namely Austria, Belgium, Denmark, England and Wales, Finland, France, West Germany, Iceland, Republic of Ireland, Northern Ireland, Italy, Luxembourg, the Netherlands, Norway, Portugal, Scotland, Spain, Sweden, and Switzerland. These estimates were obtained from the Human Mortality Database (see http://www.mortality.org [accessed July 26, 2009]) and are based on information from vital registration, censuses, and when available, population registers. These data series begin in 1950 or earlier for all except two countries, West Germany (1956) and Luxembourg (1960).

We gathered regional data on mortality and longevity for five countries for which such information was readily available to us; in addition to the United States, this group includes Canada, France, Germany, and Japan. Whenever possible, we collected full life tables from the available published sources. However, in some cases we collected only values for the expectation of life, as this is the main indicator used for this analysis. Annex A contains a detailed accounting of the data sources used.

The size of basic geographic units varies enormously by country, as does presumably their heterogeneity, reflecting national traditions with regard to administrative divisions and political functions. Thus, we obtained data for states and counties of the United States, for prefectures of Japan, for departments of France, for provinces of Canada, and for federal states of Germany. The underlying idea was that counties of the United States could be compared with relatively smaller administrative units in other countries, whereas states of the United States could be compared with larger administrative units within countries and with countries of Western Europe.

At one level of aggregation, the United States is composed of 50 states plus the District of Columbia, with an average population (in 2000) of 5.5 million persons and an average surface area of 189 square kilometers. At another level the country can be divided into 3,141 counties; however, since many of these counties are too small for reliable mortality estimation, we have adopted the practice of Ezzati and colleagues by analyzing data for 2,068 individual counties or merged county aggregates, with an average population (in 2000) of 136,000 people and an average surface area of around 5 square kilometers.

For two of the comparison countries, the internal geographic divisions used here are relatively detailed and thus similar in some respects to U.S. counties. The 47 prefectures of Japan and the 96 departments of France are roughly similar in physical size although much more populous on average than U.S. counties (see Table 12-2). For the other two comparison countries, available data refer to much larger geographic subunits. In terms of average population, the 10 provinces of Canada and the 15 federal states of Germany resemble states of the United States (see Table 12-2).

In many cases the estimates of life expectancy used here refer to multiyear time periods rather than a single calendar year. To simplify the exposition, we often refer to multiyear estimates in terms of the middle year. For states of the United States, data refer to 3-year time periods around census years: 1939-1941, 1949-1951, …, 1999-2001. Note that for 1939-1941 and 1949-1951, the life table values are available only for whites and for nonwhites separately and for men and women separately as well. The life tables for 1959-1961 include estimates for the total population with sexes combined, but not separately by sex. Using various assumptions (see Annex A), we have approximated some missing pieces of information for purposes of this analysis. For U.S. counties, data refer to 5-year intervals around single calendar years from 1961 to 2003. Thus, it should be understood that when we cite estimates of life expectancy for states or counties in, say, 1990, the data refer to 1989-1991 for states and 1988-1992 for counties. Some of the data for French departments also refer to multiyear time periods.

METHODS

The analysis of geographic variability presented here is based entirely on period values of life expectancy at age 50, e₅₀, measured at both national and subnational levels. Life expectancy at age 50 was chosen as the main indicator of mortality at older ages to comport with the other studies in this volume. Some of our methods of presenting and manipulating this measure of mean longevity are standard and require no explanation. For example, Figure 12-3 presents the level of female versus male e₅₀ by state in 1950 and 2000 in the form of a simple scatter plot. Other methods are somewhat less traditional and require additional documentation.

The ellipses of Figure 12-4 were derived by the method of principal components. As explained in more detail in Annex B, the axes of each ellipse are aligned with the first and second principal components of the bivariate distribution of male and female e₅₀ for a given population and time period. The size of the ellipse is the minimum required in order to include at least 90 percent of the data points. The method is similar though not identical to that used by Coale and colleagues in their historical analysis of the decline of fertility in Europe (Coale and Treadway, 1986).

As a global measure of the geographic variability of life expectancy in a population, we computed the standard deviation across N population subunits, taking into account their relative sizes. For each population and time period, the weighted standard deviation of e₅₀ was computed as follows:

where x₁, x₂, …, x_N represent the values of e₅₀ across N subunits, and w₁,w₂, …, w_N are weights proportional to population size (scaled so that $\text{∑}_{i=1}^N{w}_i=1$, and $\overline{x}=\text{∑}_{i=1}^N{w}_i{x}_i$ is a weighted mean.⁹ Trends in this measure of variability are presented in Figure 12-5.

Both the quintile trends of Figures 12-6 and 12-7 and the analysis of convergence effects in Table 12-3 and 12-4 require the computation of percentiles for empirical distributions of e₅₀ across geographic space. The input for such calculations includes not only the value of e₅₀ but also the associated population size for all geographic subunits. Using the same notation as above but specifying that the values of e₅₀ for population subunits (x₁, x₂, …, x_N) are in increasing order, the value of the 100p-th percentile of e₅₀ equals x_k, where k is the smallest integer (between 1 and N) such that

In standard usage, the term “quintile” may refer either to the value of cut points located at the 20th, 40th, 60th, and 80th percentiles or to each of five equal-sized groups of ordered observations (where some observations are split in appropriate proportions across adjacent groups). For this analysis, a quintile has the latter meaning. A key set of results (see Figures 12-6 and 12-7) consists of trends in the average value of e₅₀ within the five quintiles of a given population.¹⁰

The results presented in Tables 12-3 and 12-4 involve dividing the various populations into two equal-sized groups of ordered observations and, as before, computing the average value of e₅₀ for each half in (or around) the years 1980 and 2000.¹¹ The focus of this analysis is the mean change in values of e₅₀ between these 2 calendar years (in years per annum) for the population as a whole and for the two halves, as described in columns (a), (b), and (c) of Table 12-3. The mean change for the entire population is the average of mean changes for the two halves.¹²

We define a convergence effect to be the difference between the mean changes for the total population and for the upper half of the geographic distribution, as shown in column (d) of Table 12-3; this effect also equals one-half the difference between the mean changes for the lower and upper halves of the distribution. So defined, the convergence effect represents the increased rate of change for the total population that is attributable to faster change in the lower half of the geographic distribution compared with the upper half. If change is faster in the upper half, the value is negative and thus represents a divergence effect. Finally, column (e) of Table 12-3 gives the magnitude of the convergence effect as a fraction of the total change.

The values of Table 12-4, which are derived directly from those of Table 12-3, indicate how the differential pace of increase in e₅₀ from 1980 to 2000 (for the United States compared with the other populations in the study) can be apportioned to each of three components. A slower pace of improvement for the United States can result from (1) a difference in the trends of e₅₀ for the upper 50 percent of the geographic distribution in the United States versus the upper 50 percent of the geographic distribution for the comparison population, (2) a divergence between the lower and the upper 50 percent of the geographic distribution for the United States, and (3) a convergence between the lower and the upper 50 percent of the geographic distribution in the comparison area. The first component can be interpreted as the portion of the differential increase that is attributable to factors affecting all states (or counties) of the United States in a similar fashion, whereas the second and third components measure the portions attributable to increasing geographic variability in the United States or declining variability in the comparison population. The sum of the second and third components represents the portion of the differential increase due to different trends in geographic variability.

RESULTS

In this section we first describe trends in U.S. life expectancy at age 50 by sex and race as well as changes in the degree of geographic variation, at both state and county levels. We then describe changes in regional disparities among the other high-income countries in the study before presenting the results of our analysis relating changes in regional variability within countries to changes in variability between countries.

DISCUSSION

To summarize our main results, every population in our study experienced gains in e₅₀ from 1980 to 2000 at a pace of at least half a year per decade. In general, improvements in longevity during this period benefited men more than women, so that the gender gap has been progressively closing. The United States has made smaller progress than all the other populations with, for women, a gain of 1.1 years between 1980 and 2000 compared with 2.1 years in Canada, 3.2 in Western Europe, 3.4 in France, 4.3 in Germany, and 4.9 in Japan, and for men, a gain of 3.0 years compared with 3.5 years in Canada, 3.6 in Western Europe, 3.7 in France, 4.8 in Germany, and 3.1 in Japan. A substantial drop in the U.S. position in international rankings of e₅₀ reflects this relatively slow improvement.

Our analysis has demonstrated that the slower progress achieved by the United States is partially due to its increasing regional variability compared with other high-income countries. Indeed, whereas internal disparities in the United States, whether measured at the state or at the county level, tended to decline up to the early 1980s, they have increased since then, in contrast to most other populations in the study (with the notable exception of women in Western Europe taken as a whole), which have experienced stability or an ongoing decline of geographic variability. For men, the difference of trends in regional disparities explains up to 50 percent of the relatively slower pace of increase in e₅₀ for the United States compared with three of the four countries examined here (as noted earlier, this comparison is not meaningful in the case of Japan, since the pace of change in male e₅₀ was nearly the same as in the United States over this time period). For women, however, rather little (under 10 percent in the most relevant cases) of the slow progress recorded by the United States in e₅₀ compared with other countries can be attributed to differential trends in regional disparities. Indeed, the difference between the United States and the other countries in the number of years of life gained after age 50 over the last 20 years of the 20th century was not much different when comparing only the better-off 50 percent of each population than when comparing the worse-off 50 percent.

Thus, although the relatively less favorable trend in life expectancy at age 50 for the United States was due in part to increasing geographic disparities in the country during 1980-2000, combined with a general reduction of such disparities in the other countries examined, most of the slower pace of improvement must be attributed to policies, practices, and behaviors that are characteristic of the nation as a whole. This conclusion is consistent with the findings by Banks and colleagues (Banks et al., 2006), who showed that, even within similar income strata, the English are in much better health than their U.S. counterparts with regard to seven key health indicators (diabetes, hypertension, heart disease, myocardial infarctions, strokes, diseases of the lung, and cancer). These researchers also found that the gradient of mortality differentials by socioeconomic status (measured by years of education and household income) is substantial in both countries but steeper in the United States. They noted that neither individual behaviors, such as smoking, alcohol consumption or diet, nor access to medical care, measured by whether respondents had health insurance, explained much of the difference between the two countries.

In conclusion, we think that this analysis helps to rule out an increase in geographic disparities as a dominant explanation for the deteriorating position of the United States in international rankings of life expectancy, especially for women. Any proposed explanation of the divergence in levels and trends of life expectancy observed among high-income countries in recent decades needs to acknowledge that even the most advantaged areas of the United States (at the state or county level) have been falling behind in international comparisons.

ACKNOWLEDGMENTS

This project received financial support from the National Institute on Aging, grants no. R01-AG011552 and 2P30AG012839, and from the Institut National d'Études Démographiques, INED research project no. P05-3-7.

REFERENCES

1. Antonovsky A. Social class, life expectancy, and overall mortality. The Milbank Memorial Fund Quarterly. 1967;45(2 pt. 1):31–73. [PubMed: 6034566]
2. Banks J, Marmot M, Oldfield Z, Smith JP. Disease and disadvantage in the United States and in England. Journal of the American Medical Association. 2006;295(17):2037–2045. [PubMed: 16670412]
3. Canadian Human Mortality Database. Introduction. Department of Demography, Université de Montréal (Canada); [March 2009]. Available http://www​.demo.umontreal.ca/chmd/
4. Coale AJ, Treadway R. A summary of the changing distribution of overall fertility, marital fertility, and the proportion married in the provinces of Europe. In: Coale AJ, Watkins SC, editors. The Decline of Fertility in Europe. Princeton, NJ: Princeton University Press; 1986. pp. 31–181.
5. Daguet F. Espérance de vie à certains âges par département et région. Données de démographie régionale 1954-1999. Insee Résultats Société 2006 2005 décembre;(n^o 49)
6. Duleep HO. Mortality and income inequality among economically developed countries. Social Security Bulletin. 1995;58(2):34–50. [PubMed: 7482130]
7. Edwards RD, Tuljapurkar S. Inequality in life spans and a new perspective on mortality convergence across industrialized countries. Population and Development Review. 2005;31(4):645–674.
8. Ezzati M, Friedman AB, Kulkarni SC, Murray CJL. The reversal of fortunes: Trends in county mortality and cross-county mortality disparities in the United States. PloS Medicine. 2008;5(4):0557–0568. [PMC free article: PMC2323303] [PubMed: 18433290]
9. Federal Statistical Office, Germany. Death Counts by Age and Year of Birth by Federal States (Table S09). various years.
10. Federal Statistical Office, Germany. Population as of 31st December by Age and Year of Year of Birth (Table B15). various years.
11. Hetzel AM. History and Organization of the Vital Statistics System. Hyattsville, MD: National Center for Health Statistics; 1997.
12. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Human Mortality Database, Overview. [June 2010]. Available: http://www​.mortality.org/ http://www​.humanmortality.de/
13. MacDorman MF, Mathews MS. Recent Trends in Infant Mortality in the United States. NCHS Data Brief. 2008;(9) [PubMed: 19389323]
14. Ministry of Health, Labour and Welfare (Japan). Prefectural Life Tables. various years.
15. National Center for Health Statistics. State Life Tables: 1959-1961. Washington, DC: U.S. Department of Health, Education, and Welfare, Public Health Service; 1966. [June 2010]. PHS Publication HRA 1252. Available: http://www​.cdc.gov/nchs​/data/life-tables/life59_2_1-26.pdf or http://www​.cdc.gov/nchs​/data/lifetables/life59_2_27-51.pdf.
16. National Center for Health Statistics. State Life Tables: 1969-1971. Rockville, MD: U.S. Department of Health, Education, and Welfare, Public Health Service; 1975. [June 2010]. DHEW Publication HRA 75-1151-1. Available: http://www​.cdc.gov/nchs​/data/life-tables/life69_2_12-6.pdf http://www​.cdc.gov/nchs​/data/lifetables/life69_2_27-51.pdf.
17. National Center for Health Statistics. U.S. Decennial Life Tables for 1979-1981. Hyattsville, MD: U.S. Department of Health, Education, and Welfare, Public Health Service; 1986. [June 2010]. DHHS Publication PHS 86-1151-1-1. Available: http://www​.cdc.gov/nchs​/products/life_tables.htm.
18. National Center for Health Statistics. U.S. Decennial Life Tables for 1989-1991. Vol. II, State Life Tables 1-51. Hyattsville, MD: U.S. Department of Health, Education, and Welfare, Public Health Service; 1998. DHHS Publication PHS-98-1151-1.
19. National Office of Vital Statistics. State and Regional Life Tables: 1939-41. Life Tables for the White Population of the United States, Each State, and Certain Groups of States, by Sex. Washington, DC: U.S. Federal Security Agency, Public Health Service; 1948. [June 2010]. Available: http://www​.cdc.gov/nchs​/data/lifetables/life39-41.pdf.
20. National Office of Vital Statistics. State and Regional Life Tables: 1949-1951. Life Tables for the White Population of Each State, and for the Nonwhite Population of 16 Southern States and the District of Columbia, Vital Statistics-Special Reports. Washington, DC: U.S. Department of Health, Education, and Welfare, Public Health Service; 1956. [June 2010]. Supplement. Available: http://www​.cdc.gov/nchs​/data/lifetables/life49-51_41supp.pdf.
21. Preston SH. The changing relation between mortality and level of economic development. International Journal of Epidemiology. 2007;36(3):484–490. [PubMed: 17550952]
22. Preston SH, Taubman P. Committee on Population. Division of Behavioral and Social Sciences and Education. Socioeconomic differences in adult mortality and health status. In: National Research CouncilMartin LG, Preston SH, editors. Demography of Aging. Washington, DC: National Academy Press; 1994. pp. 279–318.
23. Rodgers GB. Income and inequality as determinants of mortality: An international cross-section analysis. Population Studies. 1979;33(2):343–351. [PubMed: 12055146]
24. Smith JP. Healthy bodies and thick wallets: the dual relation between health and economic status. Journal of Economic Perspectives. 1999;13(2):144–166. [PMC free article: PMC3697076] [PubMed: 15179962]
25. Statistics Bureau (Japan), Ministry of Internal Affairs and Communications. Population Census of Japan. various years.
26. Wilmoth JR, Horiuchi S. Rectangularization revisited: Variability of age at death within human populations. Demography. 1999;36(4):475–495. [PubMed: 10604076]

ANNEX 12A. DOCUMENTATION OF DATA SOURCES

All mortality data at the national level were obtained from the Human Mortality Database (see http://www.mortality.org [accessed July 26, 2009]). Regional data come from a variety of sources and present different issues depending on the country involved, as explained below.

ANNEX 12B. USE OF PRINCIPAL COMPONENTS FOR CREATING GRAPHICAL ELLIPSES

As illustrated here in Figures 12-3 and 12-4, male and female values of life expectancy at age 50 for a given time period have a strong positive correlation across states and counties of the United States An efficient means of characterizing the two-dimensional distribution of male-female values is to draw ellipses that contain most or all of the data points. A simple method for creating such ellipses in a different application was described by Coale and Treadway (1986). Here, we employ an alternative approach based on principal components analysis (PCA).

In words, we begin by centering the data points around their mean values, identifying their two principal axes and projecting the points onto the new basis (i.e., computing coordinates of the data points in relation to their principal axes), and then rescaling each point using standard deviations of projected abscissa and ordinate values. This series of calculations turns the original ellipsoidal scatterplot into a circular collection of points centered on the origin. To reduce the influence of outliers, we approximate the circular distribution while ignoring the outer 10 percent of data points; that is, we find a minimum radius r such that a circle with this radius (centered on the origin) contains 90 percent of the observed points. This centering, projecting, and scaling process is then reversed, so that the points on the circle with radius r are remapped (i.e., scaled, projected, and centered) so that they are comparable to the original values of male and female life expectancy, forming an ellipsoid that contains 90 percent of the data points.

In formulas, let x₁ = (x₁₁,x₂₁, …, x_n1)^T and x₂ = (x₁₂,x₂₂, …, x_n2)^T be column vectors containing male and female values of life expectancy at age 50 by state or country for a given year, and let x = (x₁, x₂) be an n by 2 matrix. Suppose that μ₁ and μ₂ are the mean values of x₁ and x₂, and let Y = (x₁ − μ₁, (μ₁, x₂), x₂ − μ₂) be an n by 2 matrix whose columns contain the recentered values of male and female life expectancy. The sample covariance matrix, ${\mathbf{\text{S}}}^2=\frac{1}{n}{\mathbf{\text{Y}}}^T\mathbf{\text{Y}}$, can be decomposed using PCA by invoking the spectral decomposition theorem (Mardia, Kent, and Bibby, 1979, pp. 213ff, 469): ${\mathbf{\text{S}}}^2=\frac{1}{n}{\mathbf{\text{Y}}}^T\mathbf{\text{Y}}=\mathbf{\text{U}}\mathrm{\Lambda }{\mathbf{\text{U}}}^T$, where the columns of U = [u₁, u₂] comprise an orthonormal basis, and Λ = diag(λ₁, λ₂) is a diagonal matrix with positive elements λ₁ > λ₂ > 0. By computing $\mathbf{\text{Z}}=\mathbf{\text{YU}}{\mathrm{\Lambda }}^{-\frac{1}{2}}$, we project the original data points onto the span of u₁ and u₂ and simultaneously rescale them so that the variation along each axis is now unity: $\frac{1}{n}{\mathbf{\text{Z}}}^T\mathbf{\text{Z}}=\mathbf{\text{I}}$. Note that the matrix, Z, like X and Y previously, contains two column vectors, z ₁ = (z₁₁,z ₂₁, …, z_n1)^T and z₂ = (z₁₂,z₂₂, …, z_n2)^T, both of length n.

Let $C_r=\left\{\left(z_{i1},z_{i2}\right):z_{i1}^2+z_{i2}^2=r^2\text{for all}i\right\}$ be a set of points that lie on a circle of radius r centered on the origin, and let Z = [z₁,z₂] be a matrix containing these points (the number of points is arbitrary and can be adjusted upward or downward to obtain any desired level of precision for drawing the circle or corresponding ellipse). We find the minimum radius r such that 90 percent of the transformed data points lie inside the circle.

Computing $\mathbf{\text{Y}}=\mathbf{\text{Z}}{(\mathbf{\text{U}}{\mathrm{\Lambda }}^{\frac{1}{2}})}^T=\mathbf{\text{Z}}{\mathrm{\Lambda }}^{\frac{1}{2}}{\mathbf{\text{U}}}^T$ and X = (y₁ + μ₁, y₂ + μ₂), where y₁ and y₂ are the columns of Y, each point in the circle is mapped back onto the original basis. The points corresponding to rows of X form an ellipse that encloses 90 percent of the original data points.

Footnotes

1

Estimates of life expectancy in 1999-2001 for states of the United States were computed by the authors using vital registration and census data (see Annex A regarding data sources).

2

The set includes Western Europe (see the notes to Table 12-1) and other high-income countries (Australia, Canada, Japan, New Zealand, and Taiwan), plus certain countries of Eastern Europe and the former Soviet Union (Belarus, Bulgaria, the Czech Republic, Estonia, East Germany, Hungary, Latvia, Lithuania, Poland, Russia, Slovakia, and Ukraine).

3

The country with the highest life expectancy is ranked first.

4

See the notes to Table 12-1 for a list of the countries included in the comparison.

5

If we include Taiwan and countries of Eastern Europe and the former Soviet Union in this comparison, the ranking of U.S. women fell from 11th (out of 33) to 22nd (out of 34) and for U.S. men from 10th to 15th.

6

It is well known that the direction of this causal relationship is more complex than depicted here (e.g., Smith, 1999). Nevertheless, we limit our discussion to this simplistic example in order to focus attention on other topics.

7

Substituting either “negative” for “positive” or “stronger” for “weaker” reverses the conclusion. Changing both at the same time leaves the conclusion unchanged.

8

Although in theory the deterioration of a country's international position could equally be achieved by a convergence resulting from the more advantaged locations regressing to the level of the less advantaged ones, this situation is not observed in the data presented here.

9

The denominator of the formula for the weighted standard deviation ensures an unbiased estimate under standard statistical assumptions. Note that if [inline] for all observations, this formula reduces to the usual one with N – 1 in the denominator.

10

Similar results could be obtained using tertiles, quartiles, deciles, etc. After some experimentation, we concluded that quintiles offer an adequate level of detail without making the graph so cluttered that it becomes difficult to read.

11

Note that a given subpopulation may be included in different halves of the geographic distribution in 1980 and 2000.

12

One complication encountered here arises from the fact that e₅₀ for the total population does not equal the weighted average of e₅₀ for geographic subunits. Since it is typically quite small, we ignore this difference in practice and express our results in terms of the weighted average of e₅₀ for the population subunits.