two centuries of mortality decline in latin america: from hunger...

Alberto PalloniGuido PintoHiram Beltran-Sanchez

Two Centuries of MortalityDecline in Latin America: FromHunger to Longevity

— Monograph —

October 18, 2015

Chapter 2Estimation of Life Tables 1850-2010:Adjustments for Relative Completeness and AgeMisreporting

2.1 Introduction

This book is about mortality changes in the Latin American and Caribbean (LAC)region since, roughly, 1850, a period that, for most countries considered here, is inclose proximity to the end of the colonial stage, the aftermath of wars of indepen-dence from Spanish and Portuguese rule, and the establishment of nation states1.We cover approximately 160 years of history albeit not always with the same levelof detail nor in the same depth. There is abundant, but defective, information for theperiod following World War II, less so for the first half of the XX century, and rarelyfor the post-independence era (1830-1900). For the years before 1950 we estimatelevels and age patterns of mortality that reflect the experience of populations duringperiods never smaller than three years, in most cases of five to ten years and, whenother options are infeasible, over time intervals of 10 to 17 years. In contrast, for theperiod after 1950 we avail ourselves of yearly mortality data and it is possible, inprinciple at least, to estimate life tables reflecting mortality experiences over shortand contiguous periods of time.

The construction of a life table requires information on events (deaths by age) andexposure (population by age). When accurate vital statistics and population censusescounts (or even national sample surveys) are available, the requisite age specificmortality rates can be readily computed. These are then transformed into standardlife tables functions, such as conditional probabilities of dying, survival probabilitiesfrom birth to any age, and residual life expectancies. The best known among thesefunctions is the life expectancy at birth, the work-horse of all respectable mortalityanalyses.

The condition on which the computation of accurate life tables depends, accuratevital statistics and census counts, is satisfied only in a fairly recent period of human

1 Only two of the countries we study are in the Caribbean region, Cuba and Dominican Republic.The remaining members of our sample belong to Central-North America (Costa Rica, El Salvador,Guatemala, Honduras, Mexico, Nicaragua and Panama) or South America (Argentina, Bolivia,Brazil, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay and Venezuela).

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4 2 Adjustments for Relative Completeness and Age Misreporting

history. While it is true that past mortality trends in Western Europe and some partsof North America have been studied using genealogies and population reconstruc-tion from parish records, these are not sources readily available in LAC countries.As a consequence, attempts to reconstruct the past must begin sometime after theintroduction of population and vital records, a product of modern civilization, be-comes established as part of each nation-state’s bureaucratic agenda. Systematiccollection of relatively accurate national mortality statistics for national populationsdoes not begin anywhere before 1750 and, regrettably for us, it does not make itsappearance in the LAC region until well into the XX century, too late to offer us therequisite material to trace with some accuracy the history of mortality changes overa period longer than half a century. Thus, estimates of life tables for all countriesin LAC for an extensive stretch of time are obtained by deploying a broad array ofindirect techniques, applied to different periods, with country specific variants and,oftentimes, requiring separate treatment of different age groups. As a consequence,a crucial component of our analysis, one that we pursue obsessively throughout, isthe establishment of estimates that are consistent during the period examined andacross all countries that contribute to the mortality history of the region.

This chapter begins in Section 2.2 with a description of data sources. A dis-cussion of data flaws is in Section 2.3 whereas Section 2.4 describes methods fordetection and adjustment of errors and estimation of adult life tables. Section 2.5reviews methods employed to estimate adult life tables in the the period 1850-1950.Section 2.6 is about child mortality estimation, Section 2.7 is a step-by-step illustra-tion of the construction of final life tables, and Section 2.8 describes estimation ofmortality by causes of death.

2.2 Data sources: 1850-2010

The key ingredients for a life table are mortality rates, the ratios of deaths in one agegroup to the population exposed to the risk of death in that age group. An accuratedeath rate requires both accurate count of deaths (numerator) and of population(denominator) by age groups. Seldom are these found in the LAC region, even in themost recent period. In most cases we estimate life tables for the years following 1950from two external, independent sources, population censuses and vital statistics. Toestimate mortality at ages older than 5 during 1850-1950 we use a combinationof methods that rely on two censuses and vital statistics, or censuses and modelmortality patterns. Child mortality for the same period is estimated using a blend ofindependently obtained adjustment factors, vital statistics, and models of mortality.

2.2 Data sources: 1850-2010 5

2.2.1 Population censuses

Most LAC countries began collecting population censuses on a regular basis after1945-50 though a handful of them carried out national census as early as 1890,even if not part of a well-established, centralized, and systematic program of pop-ulation assessment. Census information is publicly available either directly fromcountry national statistics offices or in summary briefs and computer files from thePan American Health Organization, CELADE, OECD databases, the Population Di-vision of the United Nations, and the World Health Organization. Some of these datawere made public periodically through broadly accessible sources, including multi-ple editions of the UN yearbook as well as in publications from the Pan AmericanOrganization. The most readily accessible census population figures are in 5-yearage groups with separate information for the first two age groups, 0 and 1-4, and theopen age group defined at 85+, and are routinely tabulated by sex. For about half thecountries in our sample, data by single year of ages up until age 100 are accessiblethrough computerized data bases starting in 1950 or so (see Tables 2.1 and 2.2).

While all countries in LAC have undertaken national population censuses, thesehave not always been periodic, seldom attain full population coverage prior to 1950,and the information content (age reporting) is of unequal quality. Of particular im-portance to us is the completeness of census counts, that is, the fraction of the totalpopulation in each age group actually identified by the census. Except for recentperiods, post enumeration surveys to assess census counts, are unavailable. As aconsequence, assessment of the quality and degree of coverage of censuses in theregion can only be undertaken by identifying scattered ancillary studies, frequentlyof local rather than national scope, designed to evaluate census plans, field workprotocols, and ex-post judgments of the quality of the information collected. Whenavailable, we use these studies as secondary sources of quality control for our esti-mates. However, the cornerstone of the approach we use here consists of evaluatingthe relative quality of death rates rather than on making inferences about or adjustingseparately the coverage and precision of population and death counts.

Table 2.1 displays a complete list of censuses included in our data base. Evenif population censuses were of uniformly high quality, it is difficult to constructa continuous time series of mortality rates starting in 1850 since, with one or twoexceptions, intercensal periods are irregular and oftentimes stretch over lengthy timeintervals. Periodicity of census-based statistics is a recent trait, starting in most casesduring the 1950’s.

2.2.2 Vital statistics and death counts

As early as in the 1880’s LAC countries began establishing official, nationwide vi-tal statistics systems to collect and organize information on deaths, births, and mar-riages. As is the case of population counts, death counts are generally available byyear of death, grouped in 5 or 10-year intervals and gender, and include informa-


Table 2.1 Availability of population censuses 1850-2010.

Country Before 1900 1900-1949 1950 and after

Argentina 1869, 1895 1914, 1947 1960, 1970, 1980, 1991, 2001, 2010Bolivia 1950, 1976, 1982, 2001, 2012Brazil 1872, 1890 1900, 1920, 1940 1950, 1960, 1970, 1980, 1991, 2000, 2010Chile 1854, 1865, 1875, 1895 1907, 1920, 1930, 1940 1952, 1960, 1970, 1982, 1992, 2002Colombia 1905, 1912, 1918, 1928, 1938 1951, 1964, 1973, 1985, 1993, 2005, 2012Costa Rica 1864, 1883, 1892 1927 1950, 1963, 1973, 1984, 2000, 2011Cuba 1841, 1861, 1877, 1887 1907, 1919, 1931, 1943 1953, 1970, 1981, 2002, 2012Ecuador 1950, 1962, 1974, 1982, 1990, 2001, 2010El Salvador 1930 1950, 1961, 1971, 1992, 2007Guatemala 1880, 1893 1921, 1940 1950, 1964, 1973, 1981, 1994, 2002, 2011Honduras 1930, 1935, 1940, 1945 1950, 1961, 1974, 1988, 2001Mexico 1895, 1900 1910, 1921, 1930, 1940 1950, 1960, 1970, 1980, 1990, 2000, 2010Nicaragua 1940 1950, 1963, 1971, 1995, 2005Panama 1911, 1920, 1930, 1940 1950, 1960, 1970, 1980, 1990, 2000, 2010Paraguay 1950, 1962, 1972, 1982, 1992, 2002, 2012Peru 1876 1940 1961, 1972, 1981, 1993, 2007Dominican Republic 1920, 1935 1950, 1960, 1970, 1981, 1993, 2002, 2010Uruguay 1900 1908 1963, 1975, 1985, 1996, 2004, 2011Venezuela 1926, 1936, 1941 1950, 1961, 1971, 1981, 1990, 2001, 2011

tion for infants (age 0), young children (aged between 1 and 4) and the populationolder than 70 or 75. In most cases, death counts by age and sex are available yearlyafter 1950 but more erratically during 1920-1950. When available, mortality statis-tics during this more distant period are highly scattered but, in some cases, they arenicely centered around census years.

Information about death counts is even more fragile than information about pop-ulation counts. Up until recently most countries of the region either did not haveestablished vital statistics systems in place or had one that recorded partially, imper-fectly, and selectively the occurrence of vital events. Although there are other defi-ciencies, such as inconsistency between timing of deaths and timing of registrationas well as age misreporting, the most important one is the lack of complete deathregistration. Before 1950 virtually no country in the region issued statistics that wereanywhere near complete. After 1950 there has been considerable improvement butthere are still laggards with deficient vital statistics systems in place (Bolivia), andmore than a handful of countries whose official statistics are irregularly producedand/or of questionable quality.

Table 2.2 summarizes the availability of death statistics in the LAC region. Thetable includes country-years for which the pertinent information on deaths is brokendown by gender and is available in age groups not coarser than ten year groups.

2.3 Data flaws

In what follows we describe two problems that complicate the creation of a con-tinuous, complete, and accurate set of mortality estimates. These affect depth andaccuracy of the information.

2.3 Data flaws 7

Table 2.2 Availability of vital statistics: 1880-2010.

Country Years

Argentina 1912-1915, 1944-1970, 1977-2010BoliviaBrazil 1974-2010Chile 1936-2010Colombia 1926-1928, 1936-1971, 1973-1975, 1982-2012Costa Rica 1940, 1950-2011Cuba 1927-1936, 1959-2010Ecuador 1957-2010El Salvador 1933-2009Guatemala 1933-2009Honduras 1933-1943, 1947-1971, 1973-1990Mexico 1930-2010Nicaragua 1933-1946, 1948-2010Panama 1941-1943, 1948-2010Paraguay 1936-1944, 1948, 1950-2010Peru 1939-2009Dominican Republic 1937-1976, 1979, 1982-2011Uruguay 1905-1907, 1909-1921, 1923, 1929-2010Venezuela 1933-1945, 1947-2010

2.3.1 Depth: time, content and resolution

The data we use contains three depth-related weaknesses. The first consists of short-comings affecting time depth or the density of estimates per year during a period oftime. Characterization of mortality levels and patterns before 1950 is mostly basedon scattered vital records and censuses that, with a few exceptions, take place atirregular and/or excessively long intervals of time. Because it is not always possi-ble to generate a set of continuous estimates, we document levels and patterns ofmortality with life tables centered in census years and/or covering an intercensalperiod not exceeding 15 years. As a result of this choice, the density of estimates forthe period before 1950 is on average between 0.38 and 0.89 life tables per countrydecade. Conditions improve substantially after 1950 when it becomes possible tocompute life tables on a yearly basis if one is willing to rely on interpolated figuresof population counts to inform intercensal periods. For the most part, the analysesin this book are based on life tables estimated for years centered on a populationcensus as well as on the middle of intercensal periods not exceeding 15 years. Onthe whole, the density of estimates for the most recent period increases to about 0.98life tables per country-year. Despite these limitations, we are able to generate yearlyestimates of mortality between 1960 and 2010 for the bulk of countries in our sam-ple. These estimates are used in Chapters 4 and 7. Table 2.3 identifies country-yearswith estimated life tables that are not based on interpolated population figures.

The second weakness is one of content depth: we do not aspire nor are we able tocharacterize mortality at levels of aggregation lower than the nation-state. The only


Table 2.3 Pivotal life tables centered on intercensal periods.

Country Before 1900 1900-1949 1950 & After

Argentina 1882 1904, 1914, 1930 1953, 1965, 1975, 1985, 1996, 2005Bolivia 1925 1963, 1984, 1996, 2006Brazil 1881, 1895 1910, 1930, 1945 1955, 1965, 1975, 1985, 1995, 2005Chile 1859, 1870, 1880, 1890 1901, 1913, 1925, 1935, 1946 1956, 1965, 1976, 1987, 1997, 2006Colombia 1908, 1915, 1923, 1938, 1944 1957, 1968, 1979, 1989, 1999, 2008Costa Rica 1873, 1887 1909, 1927, 1938 1956, 1968, 1978, 1992, 2005Cuba 1851, 1869, 1882, 1893 1903, 1907, 1913, 1925, 1937, 1948 1961, 1975, 1991, 2006Dominican Republic 1927, 1942 1955, 1965, 1975, 1987, 1997, 2006Ecuador 1956, 1968, 1978, 1986, 1995, 2005El Salvador 1940 1955, 1966, 1981, 1999, 2008Guatemala 1886 1907, 1930, 1945 1957, 1968, 1977, 1987, 1998, 2005Honduras 1932, 1937, 1942, 1947 1955, 1961, 1967, 1974, 1981, 1988, 1994, 2001Mexico 1897 1905, 1915, 1921, 1925, 1935, 1945 1955, 1965, 1975, 1985, 1995, 2005Nicaragua 1945 1956, 1967, 1983, 2000, 2007Panama 1915, 1925, 1935, 1945 1955, 1965, 1975, 1985, 1995, 2005Paraguay 1956, 1967, 1977, 1987, 1997, 2007Peru 1908 1950, 1966, 1976, 1987, 2000, 2008Uruguay 1904, 1908, 1935 1969, 1980, 1990, 2000, 2007Venezuela 1931, 1938, 1945 1955, 1966, 1976, 1985, 1995, 2006

exception to this is in the analysis of child mortality during the period 1950-2010(Chapter 6) where we compute estimates of mortality under age five for low levelsof aggregation, including individual households. Even in these cases, though, thegoal is not to characterize the mortality experience of an entire population but onlyits most youthful segment. Thus, our study is about national trends and is silent onregional or social class heterogeneity that may emerge with or be shaped by nationaltrends.

The third weakness is one of resolution depth: the estimates we compute areoftentimes based on information aggregated in five year age groups and containinga category for unknown age. The information on population and death counts isin five year age groups, starting at age 5 and ending in an open group at age 85.In the bulk of country-years of interest, the population younger than 5 is in twoage groups, 0 and 1-4. Although a descriptive study of mortality trends based onabbreviated life tables is not to be scorned at, some of our analyses and adjustmentprocedures demand more detail in the form of mortality rates by single years of age.To produce this information, and whenever the requisite data was not available, weproceed along two different routes. First, in the most recent period, after 1975 orso, we utilize data originally released (but not always published) in single years ofage. This data requires simple corrections to minimize the impact of age heaping.Second, when the original single-year of age data is unavailable we rely on theapplication of conventional cubic interpolation procedures (Sprague multipliers) tobreak five year age groups into their single year of age components. We do this onlyfor ages older than 5 and younger than 85. Our analyses do not require single year ofage rates below age 5 and figures for single years at ages older than 85 are estimatedusing specially tailored techniques described elsewhere in the book (Chapter 7).

A different dimension of the resolution depth problem is the magnitude of thefraction of population and death counts of age unknown, a quantity that varies by

2.3 Data flaws 9

country and becomes gradually smaller in the last two decades. Although we tested anumber of methods for redistributing unknown age counts, we finally settled for thesimplest one, namely a redistribution according to the observed age distribution ofcounts. Other methods, based on the use of stable populations and approximationsto population parameters yield results that are difficult to distinguish from those ofthe simple procedure but are considerably more costly and cumbersome to apply.

2.3.2 Accuracy: relative completeness and age reporting

The most important limitation of the mortality data assembled here is defective cov-erage and age misreporting. By and large, observed death counts are a fraction of the‘true’ number of deaths that take place at a particular time as they exclude eventsthat, for a number of reasons, are never recorded. Deficiencies are worst at veryyoung and old ages but also affect the population of working ages and differentiallyso by gender. It is only when national vital registration systems operate efficientlyand have a truly national reach, as it does so in the most recent period, that defi-ciencies in death counts are confined to issues of consistency between timing ofoccurrence and recording of events.

Since population censuses too are normally affected by coverage problems, mor-tality rates computed with the raw data may contain smaller net errors that would beexpected otherwise. In general, however, the observed mortality rates underestimatemortality levels, particularly at very young and old ages. Throughout, we will referto this as the relative completeness problem. We use the term relative completenessfactors when we speak of ratios of observed to true mortality rates. Table 2.4 of-fers a quick rendition of the nature of the problem: it displays estimates of relativecompleteness of adult (over 5 years of age), infant (age 0) and early child (ages 1-4)death registration for a sample of LAC countries over two different periods of time.The figures in this table suggest that the quality of the information is poorer at veryyoung ages and that, although there is a clear universal trend toward improvement,an important fraction of countries still show signs of deficient registration even dur-ing the most recent periods.

Defective relative completeness is not, however, the only or even the worst prob-lem we face. The accuracy of both census and death counts can be threatened by agemisreporting in either source. First, age heaping is a well-known problem of popula-tion counts though less so in death registration. It can be repaired, albeit imperfectly,using simple techniques for identifying preferred digits and then redistributing bothpopulation and deaths to follow a smoother trajectory. In most cases these simpleadjustments suffice to produce accurate summary measures and age patterns of mor-tality but not always with much precision at high levels of resolution.

Our general strategy for adjustments is to start out with population (death) countsin five year age groups beginning at age 5 and fit polynomials to obtain population(death) counts by single years of age between ages 5 and 84. We then calculate thequantities needed for adjustment in single years of age, obtain adjusted mortality


Table 2.4 Relative completeness of deaths registration in the LAC countries: 1920-2010.

Country Period 1900-1949 Period 1950 +Mid-Year Age 0 Age 1-4 Age 5+ Mid-Year Age 5+

Argentina 1914 0.968 0.865 0.939 1953 0.9742005 0.995

Brazil 1985 0.8852005 0.996

Chile 1925 0.867 0.829 0.852 1956 0.9611945 0.867 0.829 0.934 2006 0.980

Colombia 1944 0.821 0.815 0.749 1957 0.7902008 0.800

Costa Rica 1927 0.901 0.922 0.893 1956 0.9181938 0.901 0.922 0.893 2005 0.975

Cuba 1925 0.806 0.893 0.800 1961 0.8901948 0.806 0.893 0.870 2006 0.989

Dominican Republic 1942 0.476 0.451 0.487 1955 0.5002006 0.604

Ecuador 1956 0.7382005 0.805

El Salvador 1940 0.554 0.776 0.721 1955 0.7002008 0.714

Guatemala 1945 0.714 0.898 0.784 1957 0.8882005 0.940

Honduras 1942 0.542 0.551 0.495 1955 0.5181947 0.542 0.551 0.500 1989 0.750

Mexico 1925 0.843 0.822 0.752 1955 0.8601945 0.843 0.822 0.883 2005 0.959

Nicaragua 1945 0.526 0.545 0.498 1956 0.4562007 0.561

Panama 1945 0.837 0.757 0.829 1955 0.8392005 0.853

Paraguay 1956 0.6012006 0.681

Peru 1950 0.4902008 0.533

Uruguay 1908 0.844 0.822 0.879 1969 0.9602007 0.996

Venezuela 1938 0.833 0.857 0.846 1955 0.8661945 0.833 0.857 0.855 2006 0.895

rates, and estimate life tables in single years. Finally, we aggregate the adjusteddata into five year age groups and compute abbreviated life tables. In most casesthe resulting life tables do not contain the footprints of age heaping. When theydo, we proceed to smooth the mortality rates applying a local smoother with nomore than three contiguous age groups for support2. In cases when the data are

2 Because raw, five-year death and population counts are ultimately allocated into single years ofage, adjustment for heaping is redundant.

2.3 Data flaws 11

available already in single years of ages we apply smoothing techniques to removeage heaping and then proceed as described before.

A more insidious manifestation of errors of age declaration is systematic over(under) reporting. As we show later, vital and census statistics in LAC countries are,almost without exception, affected by age overstatement, particularly at ages over40 or 45. When the (true) age distribution of a population is roughly exponential innature —as it always is in stable and quasi stable populations—systematic age over-statement of populations induces downward biases in mortality rates at older ages.Unfortunately, these biases are not quite fully offset when there is an equal propen-sity to overstate ages at death. The reason these two type of errors do not canceleach other out is that while both adult mortality rates and adult population age dis-tributions are roughly exponential, one slopes upwards (mortality rates) whereas theother slopes downwards (population). Matters are made worse when, as is almost al-ways the case, the rate of decrease of population with age (natural rate of increase ina stable population) is several times lower than the rate of increase of adult mortalityrates (rate of senescence in Gompertz mortality regimes). The consequence is thatunless the propensity to overestimate ages at death is much higher than the propen-sity to overestimate ages of population, observed mortality rates will be downwardlybiased. If left uncorrected, the resulting life tables will offer a misleading portrayalof the curvature of mortality at older ages, suggesting the existence of slower ratesof senescence or heavy influence of selection due to changing frailty composition.As vital registration and census enumeration improves, the magnitude of these bi-ases tends to decrease and the entire history of observed life tables will erroneouslysuggest trends in old age patterns of mortality and even misleadingly generous rel-ative deceleration of the rates of mortality decline at older ages. Table 2.5 displaysestimated biases in mortality rates at ages over 45 in a sample of country-years usedin our analysis and the corresponding errors in life expectancy at age 60.

In Section 2.4 below we describe and evaluate a battery of procedures to com-pute adjusted life tables that minimize errors due to imperfect relative completenessand defective age reporting. The section focuses on developments and applicationsto compute adjustments for observed mortality during 1950-2010, the period withthe most complete information. It is then followed in Section 2.5 by a review andassessment of estimation methods employed to adjust adult mortality in 1850-1900.Section 2.6 describes the genesis of estimates of child mortality for the entire pe-riod under study. In all cases male and female populations as well as adult and childmortality are treated separately. Although we considered alternative definitions, itstrengthens the consistency of adjustments to define as “adult” the population aged5 and older and as “children” those younger than age 5. The weakness of this def-inition is that we lump together a youthful and a genuinely adult population (withpossibly sui generis mortality experiences) in one single category, the “adult popu-lation”. There is no escape from this awkward labelling. In fact, while it is possibleand desirable to estimate mortality separately for the population younger than 5,between 5 and 19, and older than 20, there are no specially tailored procedures toadjust mortality estimates for the population aged 5-19, even though it might be


Table 2.5 Biases due to age overstatement.

Country Mid-Year Unadjusted Adjusted*E(45) E(60) E(45) E(60)

Argentina 1953 25.96 15.39 25.29 14.552005 30.02 17.96 29.33 17.15

Brazil 1985 28.55 17.61 27.62 16.512005 31.27 19.77 30.23 18.58

Chile 1956 24.44 14.57 23.72 13.642006 33.20 20.45 32.16 19.33

Colombia 1957 27.34 16.68 26.46 15.672008 35.09 22.29 33.86 20.96

Costa Rica 1956 29.08 17.55 28.10 16.462005 34.96 22.40 33.78 21.13

Cuba 1961 30.13 18.15 29.18 17.082006 33.46 20.94 32.56 19.95

Dominican Republic 1955 33.62 22.44 31.91 20.522006 38.35 25.76 36.41 23.68

Ecuador 1956 28.75 17.98 27.77 16.832005 37.42 25.23 35.94 23.62

El Salvador 1955 27.64 17.54 26.69 16.422008 32.79 21.74 31.85 20.62

Guatemala 1957 24.44 15.06 23.68 14.072005 31.39 20.22 30.42 19.10

Honduras 1955 30.55 20.37 29.14 18.641989 37.33 25.06 35.61 23.17

Mexico 1955 26.57 16.69 25.80 15.712005 33.04 21.13 31.97 19.95

Nicaragua 1956 32.09 21.05 30.61 19.372007 36.23 24.05 34.71 22.41

Panama 1955 28.93 17.67 27.87 16.452005 35.92 23.18 34.65 21.81

Paraguay 1956 32.97 20.81 31.73 19.442006 34.84 22.17 33.60 20.84

Peru 1950 30.61 20.64 29.47 19.252008 39.37 26.32 37.66 24.52

Uruguay 1969 26.72 15.47 26.11 14.692007 30.35 18.17 29.85 17.57

Venezuela 1955 27.49 16.81 26.47 15.642006 32.75 20.94 31.53 19.59

* For age misreporting

preferable to do so. Finally, the older adult population, those older than 60 years, isstudied separately in Chapter 7.

2.4 Adjustments of adult mortality for the period 1950-2010 13

2.4 Adjustments of adult mortality for the period 1950-2010

As should be clear from the above description, the nature of problems faced is highlyheterogeneous: they vary by country, time period, age groups and, lastly, by gender.This state of affairs is complicated even more by the fact that there are multipleprocedures, each relying on specialized assumptions, to adjust for the errors thatexist in the data. The strategy we follow to make these deficiencies tractable is intwo stages. In the first stage we develop an evaluation study designed to identifythe optimal adjustment procedure for relative completeness and age misreporting.In the second stage we apply those procedures to all country years, doing as little vi-olence as possible to each country (or period) peculiarities. These two strategies aredescribed below. Section 2.4.1 contains the details of the evaluation study, section2.4.2 identifies adjustment techniques, and section 2.4.3 is a summary of results.

2.4.1 Evaluation study

This section3 describes an evaluation study designed to assess the performance andestablish a ranking of alternative methods to correct for errors due to under (over)counting of population and deaths by age groups and age misreporting after 1950.This is a period during which all LAC countries (except Bolivia) release periodicvital statistics data and official population counts. As documented in Table 2.3 wecompute approximately 100 pivotal life tables in years comprised between 1950 and2010. A pivotal life table is defined as one centered within an intercensal period. Alllife tables were constructed adjusting separately mortality rates between ages 5 and85, on one hand, and mortality rates between ages 0 and 5, on the other, and thenjoining the two sets of estimates to generate a complete life table (see Section 2.7).Once pivotal life tables are established (and if the information on death counts isavailable by year), we proceed to compute adjusted year-by-year life tables that relyon interpolation of population census figures.

2.4.1.1 Evaluation study: a summary of goals

Over the last two to three decades, but mostly in the late seventies and eighties, de-mographers developed a large number of techniques to adjust faulty data from cen-suses, vital statistics and population surveys to estimate both fertility and mortality.There are nearly 15 different, albeit not completely independent methods, to esti-mate adjusted adult mortality and associated life tables, each with advantages andshortcomings and each depending on sets of non-identical but overlapping assump-

3 The investigations that follow were first documented elsewhere [Palloni and Pinto (2004)]. Un-less otherwise specified, all techniques to detect or adjust for errors are applied separately bygender.


tions. The work of Hill and colleagues established some solid findings about theperformance of a subset of these methods [Hill et al. (2009), Hill and Choi (2004),Hill et al. (2005)]. We use this work and build on it to elaborate a more generalassessment of a set of 13 methods.

Less investigated is a second problem facing the estimation of age patterns ofmortality, namely, age misreporting4. It is well-known that census and death countsby age are influenced by digit preference (‘heaping’) and biases due to propensity toincrease (decrease) the true age. Although problematic in its own right, age heapingcan be repaired because in most cases it is possible to restore the original age distri-bution in the neighborhood of preferred digits using computations that rely on safeassumptions. Systematic age misstatement is altogether different since it is harder todiagnose and its treatment requires additional knowledge of at least two functions:(a) the conditional (on age and gender) propensity of individuals to exaggerate (de-crease) the true age and (b) the conditional (on age and gender) distribution of thedifference between the correct and declared age. To solve this problem we proposegeneralizations of an existing procedure to identify the presence of age misstate-ment, formulate a new method to estimate functions (a) and (b) from observables,and define an algorithm that adjusts observed adult mortality rates for both faultycoverage and systematic age misreporting.

Neither adjustment for faulty coverage nor detection and correction of biases dueto age misreporting are feasible in the absence of well-established criteria to decidewhich of the many (for coverage) or fewer (for age misreporting) candidate meth-ods performs optimally. To fill this gap we carry out a systematic evaluation studyof the performance of extant methods using a range of conditions similar to thoseexperienced in countries under study. Our objective is to assess the sensitivity ofalternative techniques to violations of assumptions on which they are based, partic-ularly those that are most likely to misrepresent historical conditions. We first sim-ulate populations representing different demographic profiles (stable, quasi-stableand non-stable) driven by combinations of (a) constant fertility and mortality, (b)constant fertility and declining mortality, and (c) declining fertility and decliningmortality. We then combine these profiles with different patterns of distortions dueto faulty coverage of population and death counts and age misreporting. A batteryof 12 techniques is deployed and in each case we compute multiple measures of per-formance comparing the true parameter(s) with those retrieved by each technique.We rank the performance of techniques for each combination of conditions violatingassumptions on which the techniques rely. Finally, we score techniques according totheir sensitivity to violation of combinations of assumptions. The optimal techniqueis then paired with a new procedure to adjust for age misreporting and, jointly, theyare used in an algorithm to make final adjustments to observed adult mortality rates.A crucial issue discussed at length below is the order in which these techniques, onefor adjustment of coverage and one for age misreporting, must be deployed and thejustification for that order.

4 The simulations by Hill and colleagues did include some forms of age misreporting. We augmentand generalize this aspect to capture patterns of age misreporting that are more typical of LACcountries.


Sections 2.4.1.2 through 2.4.1.5 describe simulated population regimes,patternsof coverage errors and of age misreporting.

2.4.1.2 Simulated populations: four classes of demographic profiles

The first step of the evaluation study is to simulate a large number of populationsspanning a broad range of fertility and mortality regimes that come close to repro-ducing age-specific counts of deaths and populations that would have been observedover an interval of about 100 years in the absence of errors in the data. We start outwith a stable age distribution in single years of age, e.g., Pxt0 ,x = 0, ...100, to repre-sent an average population in 1900 and then project it forward for 100 years usingschedules of probabilities of surviving, e.g. (Sx = 0,100), and of fertility rates, e.g.,(Fx = 15,50)5. We chose four different trajectories of mortality and fertility roughlyreproducing four classes of demographic regimes experienced by Argentina, CostaRica, Guatemala and Mexico respectively [Palloni (1990)]. All trajectories are de-fined by choosing values of life expectancy at birth (E0), and Gross ReproductionRate (GRR) thus defining the rate of natural increase (r) for every decade between1900 and 2000. With the exception of the first trajectory (corresponding to the ex-periences of Argentina and Uruguay), we assume an initial stable populations withr and E0 equal to those observed in the first population census before 1940. In thecase of the Argentina/Uruguay profile we use the observed average age distributionin the population censuses within the period 1850-1910. We assume linear intra-decade changes in the two key population parameters, r and E0 and, additionally,that each type of demographic transition profile preserves the age patterns of mor-tality and fertility. We chose the West model in the Coale-Demeny family of lifetables and an age pattern of fertility identical to the one used in the computationsof the Coale-Demeny stable population models [Coale et al. (1983)]. Informationon the four classes of demographic transitions used here are in Appendix 16. Fol-lowing routine population projection calculations we produce 505 populations andassociated distributions of births and deaths by single calendar year and single yearsof age. The simulated populations represent a very broad set of experiences, fromthose preserving population stability up until 1950 or thereabouts, to those shiftingto quasi-stability from 1930 up to 1980, to those with little or no stability at all fromthe start7.

5 Throughout we use conventional notation and when referring to discrete functions we employsubscripts, e.g. Px, whereas for continuous functions we use parentheses enclosing the function’sargument, e.g. P(x).6 We also construct a fifth population profile corresponding to a stable population with naturalrate of increase and fertility pattern equivalent to the average of LAC populations in the interval1950-60, e.g. not yet heavily perturbed by large scale net migration as in Argentina, Brazil, Cuba,and Uruguay, or fertility changes, as in Argentina and Uruguay.7 To compute single years of age stable populations we first generate single years of age life tablesby strictly adhering to the separator factors adopted by Coale and Demeny and then use standardstable population expressions. The precise routine followed is in a STATA do file available onrequest from authors.


2.4.1.3 Simulated distortions I: faulty coverage and completeness factors

Distortions due to population or death coverage can be implemented in a straightfor-ward matter. We define observed population (or death) counts by age as a fractionof the simulated (true) quantities:

Poxt1 = C1Ps

xt1

Poxt2 = C2Ps

xt2 ; t2 < t1Do

xt = C3Dsxt ; t = t1, t1 +1, . . .≤ t2

for x ≥ 5, where Poxt1 is the observed (distorted) population at age (x,x+ 1] at

time t1, Poxt2 is the observed (distorted) population at age (x,x+ 1] at time t2, and

Doxt is the observed (distorted) number of deaths in year t; Ps

xt1 ,Psxt2 and Ds

xt are thesimulated (true) quantities and C1,C2 and C3 are the fractions of total events actuallyobserved (completeness factors). The completeness factors for censuses were setat values in the range 0.80-1.0 in intervals of 0.5 whereas the death completenessfactors varied between 0.70 and 1.0 in intervals of 0.5. Altogether we produce atotal of 875 (175*5) patterns of including distorted and true demographic profiles.These definitions are sufficient to evaluate adjustment methods that require only onecensus and one to three years of deaths centered on the census or, alternatively, thosethat demand as inputs two population censuses and an array of intercensal deaths.

This set up contains a massive assumption, namely, that completeness of bothpopulation and death counts is age invariant. At least within the age range in whichthe techniques are deployed (5-85), the assumption is unlikely to be met, particu-larly for population counts. To complete the set of reasonable distortions we addtwo different patterns of age varying completeness generating a total of 2,625 sim-ulated populations. We show later, however, that as long as the difference betweenmaximum and minimum completeness stays below 10% of the mean value of com-pleteness, variability of completeness by age does not have a strong impact on ourpreferred strategy (Section 2.4.3).

2.4.1.4 Simulated distortions II: systematic age misreporting

To describe the model of age misreporting we begin with a few basic definitions.Let θ o

x be the average conditional probability that individuals aged x overstatetheir age in a census and θ u

x the conditional probability of understating their age.Then (1−θ o

x −θ ux ) is the probability of an accurate age statement. Individuals who

over(under) state their age do so by choosing, not always randomly, the age de-clared and observed in the census. This age could be n > 0 years removed fromthe true age. As we show below, it suffices to let n range between 1 and 10+ sincethe frequencies for values of 10 years and above are exceedingly small, e.g. indi-viduals rarely over(understate) their age by more that ten digits. Let ρo

x (n) be theaverage conditional probability that individuals aged x who overstate ages will doso by n years with an analogous definition for the probabilities for age understate-


ment, ρux (n) and with ∑n ρu

x (n) = ∑n ρox (n) = 1. To compute the observed number

at age y, Poy we consider the true number at that age PT

y , and apply the conditionalprobabilities defined above:

Poy = PT

y (1−θox −θ

ux )+

n=10

∑n=1

PTy− jρ

oy− j( j)θ o

y− j +n=10

∑n=1

PTy+ jρ

uy+ j( j)θ u

y+ j (2.1)

This expression can be generalized for all ages between 0 and 100 in compactmatrix notation:

Πo =ΘΠ

T (2.2)

where Π o is the (101x1) observed population vector, Π T is the (101x1) truepopulation vector and Θ is a 101x101 square matrix of “transition” probabilities,e.g. the probabilities of migration into or out of single year age-groups. In particular,the diagonal of Θ contains the probabilities of correctly declaring ages, (1− θ o

x −θ u

x ), and entries in the off-diagonal row k for columns k−1,k−2, ...,k−10 are thevalues ρo

y−1( j)θ oy−1, ...,ρ

oy−10( j)θ o

y−10 whereas those in columns k+1,k+2, ...k+10are the values ρu

y+1( j)θ uy+1, ...,ρ

uy+10( j)θ u

y+10. One can retrieve the matrix with thetrue age distribution of the population after pre-multiplying the previous expressionby the inverse of Θ−1, that is

Θ−1

Πo = Π

T (2.3)

an operation that requires full knowledge of the matrix Θ . As we show below,demographers have only superficial information about the nature of this matrix inLAC countries or anywhere else for that matter (but see [Bhat (1990)]). In the ab-sence of precise knowledge of the probabilities contained in the matrix one couldadopt shortcuts, simplifications that circumvent knowledge gaps but that, as shownbelow, lead to identification problems, most of which translate into inability to spec-ify an invertible matrix of transition probabilities.

What do we know about age misreporting in population and death counts inLAC and in other countries? There is an extensive literature on general errors inage reporting [Ewbank (1981), Chidambaram and Sathar (1984), Kamps E. (1976),Nunez (1984)] as well as on systematic age misstatement, mostly adult age over-statement, in population counts. And while a fair number of these studies uncoverevidence of overstatement in low income countries [Mazess and Forman (1979),Grushka (1996), Bhat (1987), Bhat (1990), Del Popolo (2000), Dechter and Preston (1991)]or in US migrant (Hispanic or Hispanic origins) groups [Rosenwaike and Preston (1984),Spencer (1984)], there is a body of literature that identifies patterns of age over-statemet in high income countries as well [Horiuchi and Coale (1985), Coale and Kisker (1986),Condran et al. (1991), Preston et al. (2003), Elo and Preston (1994)]. In the US, forexample, age overstatement is one of the factors that could explain the so called


Black-White mortality crossover, whereby African American mortality rates dipbelow those of their White counterparts at very old ages (over 70). And whilethe conjecture of selection due to frailty has not been completely discarded, themost recent investigations suggest that overstatement of ages in the population (andalso deaths) among African American more so than among Whites accounts fora substantial part of the mortality crossover [Elo and Preston (1994)]. The Black-White mortality crossover is just an extreme example of the damage that age mis-reporting can inflict on estimates of adult mortality. As others before us have done[Dechter and Preston (1991), Grushka (1996), Bhat (1987), Bhat (1990)], we willshow that age overstatement also an important source of error in LAC countries.

Partial information on the matrix Θ has been obtained mostly from studies in-volving record linkages [Elo and Preston (1994), Preston et al. (1996), Rosenwaike and Preston (1984),Rosenwaike (1987)], post enumeration surveys [Ortega and Garcia (1985)] and com-parisons of two independently gathered data sources that should produce the sameoutcomes [Bhat (1990)]. In all these studies, however, the information is either ag-gregated in five-year age groups or applies to populations with levels of educationthat are much higher than those in LAC countries. Lack of age detail is problematicsince computation of conditional probabilities in coarse age groups rests on approx-imations that, if violated, are generally harmful to the accuracy of estimates. Usinga transition matrix appropriate for a population with higher or lower levels of edu-cation or literacy than the target one may lead to distortions since age misstatementis strongly associated with levels of education.

To circumvent these problems we take advantage of a 2002 evaluation studylaunched by the Central American Center for Population at the University of CostaRica. The program was designed to assess the quality of information of death regis-tration and the accuracy of the 2000 census counts8. One of the components of thisstudy was a linkage of an age stratified sample of 9,113 individual census recordswith the national voter registers, a database that contains age information from birthcertificates. A total of 7,426 records were matched corresponding to 81.5% of theoriginal sample and 86.6 % of the non foreign born part of the sample. The finaldata set contains individuals classified by gender, education and other traits, and by‘true’ and declared age. To estimate the entries of matrix Θ we proceed in two steps:

i Estimation of probabilities of age over and understatement, θ ox (V ) and θ u

x (V )where V is a vector of individual characteristics, including age: We first estimatea logistic model for a binary variable set to 1 when there is over (under) statementand zero otherwise. Initially the model specifies a vector of covariates includingage, age squared, urban/rural residence, gender, and education. The sample in-cludes individuals aged 50 and over since at younger ages there are only tracesof systematic age misstatement (mostly in the form of heaping). Because genderand age are the only covariates that can be used at a national level, we simplifythe model to include only these two traits as predictors. Finally, after verifyingthat the effects of age squared and gender were statistically insignificant, the final

8 We are grateful to Drs. Gilbert Brenes and Luis Rosero-Bixby from the Central American Popu-lation Center at the University of Costa Rica for having provided tabulations we use in this study.


model conditions only on ‘true’ age of individuals. Table 2.6 displays estimatedparameters for over and under stating ages using the weighted sample.

ii Estimation of conditional probabilities of over(under) stating ages by 1 < n≤ 10years, ρo

x ( j) and ρux ( j): We estimate a multinomial model with 9 categories that

includes gender and (true) continuous age as independent variable. The resultingestimates reveal that the effects of gender are always statistically insignificant,that those of age show no clear pattern and, in addition, that their magnitudeis quite small in 6 out of 8 cases for overstatement models and in 5 out of 8contrasts for age understatement. To simplify we estimate a null model predictingthe average conditional probabilities of exaggerating (or diminishing) by n yearsapplicable to all ages older than 50 and both genders. The values of the predictedprobabilities of over and understating the true age are in Table 2.6B.

Although it is now possible to compute an estimator of the target mobility ma-trix, Θ , there remains a knotty problem of identification that cannot be resolvedwithout additional simplifications. Suppose, for example, we seek to estimate mor-tality trends in a country with much lower levels of education than in Costa Rica.Replacing Θ for the true matrix in 2.2, we will obtain a true distribution of agesbut only under the very strong assumption that age misstatement is identical acrosscountries. This contradicts accumulated knowledge showing that the severity of agemisstatement increases as levels of education drop. A less constraining assumptionis to argue that while the age pattern of age misstatement is identical across coun-tries, the levels may differ. To express this one could think of multiplying the condi-tional probabilities of over and under stating ages (or a monotonic transform of it) bysome constant, say φ o and φ u for over and understatement respectively. While thisis a reasonable strategy it generates an additional problem, namely, that a unique so-lution for equation 2.2 may no longer be possible since different combinations of φ o

and φ u embedded in the transition matrix could plausibly yield identical results. Tocircumvent this new difficulty we propose a standard pattern of probabilities of netage overstatement as ϕS

x = θ ox −θ u

x and then apply to it the conditional probabilitiesof overstating one’s age by n years (the ρo

x ( j) values defined before). Under theseconditions the off-diagonal cells of the matrix defined by ϕS

x , Θ S, simplify as allentries involving age understatement become zeros. This makes identification morelikely and the search for a unique solution of φ no, a parameter measuring the mag-nitude of the net overstatement (no) relative to the standard pattern, a more feasibleenterprise.

There are two conditions required for this standard pattern to play a meaningfuland helpful role. The first is that the probabilities of age overstatement always belarger than the probabilities of age understatement. The second is that the condi-tional distribution of n, the integer number of years by which individuals exaggerate(diminish) their true age, be approximately the same among those who over and un-derstate ages. Figure 2.1A displays predicted probabilities of over and understatingages by age, θ o

x ,θux , Figure 2.1B displays the differences ϕS

x = θ ox −θ u

x , and Figure2.1C shows predicted conditional probabilities of over stating ages by n years with0 < n ≤ 10 or ρo

x ( j). These figures show that the first condition is always satisfiedwhereas the second is only approximately met in these data. However, differences


are minor and are found mostly at higher values of n, where the probabilities ofover(under) stating are small. We define these two items, the pair of age-specific dif-ferences between predicted probabilities of over and under statement (Table 2.6A)and the associated conditional probabilities of overstating by n years (Table 2.6B),to be the standard pattern of age net overstatement9.

Table 2.6 A. Estimated parameters of best logistic models for age misreporting.

Variable Overreporting Coeff(se) Underreporting Coeff(se)

True age1 0.014(.0036) 0.002(.0040)Constant -2.127(.271) -1.846(.297)N 6290 6290

1 Regressions estimated using sampling weights. Sample includes population with true age 60 andolder and excludes ambiguous cases and foreign citizens.

Table 2.6 B. Average (conditional) probabilities of overreporting ages.

Probability1

n Overstating Understating

1 0.621 0.5102 0.191 0.1283 0.079 0.0914 0.040 0.0525 0.023 0.0416 0.015 0.0357 0.009 0.0288 0.007 0.0269 0.005 0.01310+ 0.009 0.060

1Predicted values computed from a null multinomial logistic model with 10 categories, n=1786(males and females). Estimation using sampling weights. Figures may not add up to 1 due torounding errors.

The developments above only refer to age misreporting in population counts.However, it is known that mortality rates are also influenced by age misreporting ofages at death [Rosenwaike (1987)]. The nature of the problem in this case is some-what different since it is not the decedent that declares the age at death but a kinor someone else unrelated to the decedent. A handful of studies based on record

9 The representation we use throughout suggests that patterns of age misreporting in any countryare a multiple of the standard pattern. Although this helps the algebra and derivation of proofs, wecheat in our computations and follow a roundabout algorithm. In fact, we generate new patterns ofvalues from the standard by defining the function logit(ϕ i

x) = α +βlogit(ϕSx ), set the value

of β equal to 1, and then identify the level of age overstatement in a population i by fixing α sothat ϕ i

x ∼ φ oϕSx , where φ o is the desired level of age over reporting.


Fig. 2.1 A. Predicted probabilities of over(under) stating ages.

Age

Pro

babi

lity

0.15

0.20

0.25

0.30

45 50 55 60 65 70 75 80 85 90 95 100

0.15

0.20

0.25

0.30

● ● ●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

Probability of overstatingProbability of understating

●

●

Source: Costa Rica Special study of 2000 population census.

linkages show that there is age misreporting of ages at death as well, albeit of lowermagnitude than that found in population counts, and that it also tends to be in thedirection of overstatement [Rosenwaike and Preston (1984)]. This is confirmed bythe application of indirect techniques designed to detect age at death overstatementin a number of low and high income countries (see below). It follows that expres-sions analogous to 2.1 and 2.2 must be applicable for death counts as well. To makethe problem tractable one needs an empirical approximation to a matrix analogousto Θ but now specialized to ages at death. To our knowledge no such matrix hasever been estimated in LAC or anywhere else and we are unaware of any extantnational data that could be used for such purpose. In what follows we assume thatthe standard age pattern of age misstatement of death counts is identical to thatof age misstatement of population counts, although its level may be different. Thisassumption enables us to define the final model of age misreporting as a set of twoequations with two unknown parameters:

Πo = φ

noΘ

SΠ

T (2.4)∆

o = λno

ΘS∆

T (2.5)

where ∆ T and ∆ O are the true and observed distributions of death counts and λ no

is the magnitude of net overstatement of ages at death relative to the standard pattern.


Fig. 2.1 B. Predicted probabilities of net overstating ages.

Age

Net

pro

babi

lity

of o

vers

tatin

g

0.05

0.10

0.15

45 50 55 60 65 70 75 80 85 90 95 100

0.05

0.10

0.15

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In closed populations equations 2.4 and 2.5 are naturally related (see below) and itis unlikely that there is always a unique solutions for φ no and λ no unless we eitherfix the value of one of them or, alternatively, retrieve solely their ratio. A brief proofof lack of identification is in Appendix 2 and solutions for empirical estimation arein section 2.4.2.3.

2.4.1.5 Simulated distortions III: combining age misreporting and faultycoverage

We now have all the ingredients to generate distorted populations using as bench-marks the four demographic profiles described above. The defective populationswere defined considering each demographic profile separately, letting C1 and C2take on values between 0.80 and 1.0 in intervals of 0.05 whereas C3 takes on valuesbetween 0.75 and 1.0 in intervals of 0.05 and, finally, assigning values to φ no andλ no ranging from 0 to 2.5 in intervals of 0.50. We use all possible combinationsof these parameters and generate a total of 6300 populations per demographic pro-file for a population space containing total of 31500 observations or populations insingle years of age traced for a total of 100 years. This population space, completewith embedded coverage errors and age misreporting, is the cornerstone of our eval-uation study of indirect techniques for completeness and age misreporting and the


Fig. 2.1 C. Conditional probabilities of overstating age by n years.

Number of years over(under)statement

Pro

babi

lity

of o

vers

tatin

g n

year

s

0.00

0.20

0.40

0.60

1 2 3 4 5 6 7 8 9 10

0.00

0.20

0.40

0.60●

●

●

●●

● ● ● ● ●


foundation of adjustment procedures to reconstruct the history of secular mortal-ity in LAC countries10. Section 2.4.2 briefly describes these techniques and section2.4.3 summarize results of the evaluation study.

2.4.2 Estimation techniques

2.4.2.1 Techniques to identify and adjust for defective relative completeness ofdeath registration

The most important techniques to detect and adjust for faulty completeness eval-uated in this study are summarized in Table 2.711. The table identifies techniquesusing the names of researcher(s) who proposed them or modified an original ver-sion. The table highlights (a) key assumptions on which the techniques rely, and (b)information required to implement each of them. These methods share important

10 Later in this chapter we test for sensitivity to violations of the assumption of age invariantrelative completeness, add two patterns of deviations and generate a space of 94,500 populations.11 We reviewed a longer list of techniques and, with two exceptions, chose to test only those thatdid not rely on the assumption of stability or quasi-stability.


commonalities and all but two (Brass No 1 and Preston-Hill No 1) abstain from in-voking the assumption of stability. Yet they differ in at least one feature that, undersuitable empirical conditions, grants them an advantage over competing methods.

Table 2.7 Methods to adjust for completeness of death registration: assumptions and requireddata.1

Method Assumptions Required Data

Brass (B) 1-2-3-4-5 BBrass-Hill (BHill2) 2-3-4 ABrass-Martin (BMartin3) 1-2-3-4-6 BBennet-Horiuchi No 1 (BH 1) 1-2-3-4 ABennet-Horiuchi No 2 (BH 2) 1-2-3-4 ABennet-Horicuhi No 3 (BH 3) 1-2-3-4 ABennet-Horiuchi No 4 (BH 4) 1-2-3-4 ABennet-Horiuchi No 5 (2SBH 4) 1-2-3-4 APreston-Hill No 1 (PH 1) 1-2-3-4-5 BPreston-Hill No 2 (PH 2) 1-2-3-4 APreston-Bennet (PB) 1-2-3-4 APreston-Lahiri No 1 (PL 1) 1-2-3-4 APreston-Lahiri No 2 (PL 2) 1-2-3-4 A

1See appendix 5 for definitions of the four variants of Bennet-Horiuchi method and the twovariants of Preston-Lahiri method.2BHill is a method we use to retrieve estimates of the ratio of completeness of the first relative tothe second census.3BMartin is a variant of Brass classic method that relaxes the assumption of stability and assumesinstead past mortality decline.

KEYS FOR ASSUMPTIONS1. Identical completeness of census counts in both census2. Closed to migration3. No age misreporting4. Invariant completeness by age5. Stability6. Quasi stability

KEYS FOR REQUIRED DATAA. Two censuses and intercensal deathsB. One census and one to three years of deaths by age

• Computation of rates of growth: with two exceptions (Preston-Hill No1 and Brass)all methods require computation of age specific rates of growth in an inter-censal period. Because observed rates may be perturbed by differential cen-sus completeness, the estimates of the main parameter (relative completenessof death registration) could be biased if the method is sensitive to differen-tial census completeness. A way around this is to first adjust for relative com-pleteness of census registration and then apply any of the techniques usingadjusted age specific rates of growth. This idea was first put forward by Hill


[Hill and Choi (2004), Hill et al. (2009)] who suggests that one of the methodslisted in the table (Brass-Hill) be used to retrieve a robust estimate of the ratio ofcompleteness of both censuses.

• Population closed to migration: none of the methods in Table 2.7 works well inthe presence of significant intercensal migration. If information on net migrationis available, it must be used to adjust the observed rates of intercensal growth12

• Absence of age misreporting: all methods assume either no age misreporting or,alternatively, age misreporting that perturbs only trivially the figures of cumula-tive population above adult ages. This poses a conundrum: if, as asserted before,LAC population and mortality counts are heavily affected by age overstatement,how can one expect to obtain precise estimates of relative completeness usingtechniques that are vulnerable when there is age misreporting? There are twoconditions that provide a escape from this trap. The first is that the type of agemisreporting that predominates in LAC is net age overstatement. When usingcumulative populations over some age x the damage done to the target quantityby age misreporting only depends on population flows across age x originatingat younger ages. It is insensitive to transfers of population above age x. Further-more, the relative volume of flows, e.g. the relative error of the target quantity, isgenerally low for late adulthood and early old ages (less that 65 or 70) though itbegins to mount after age 75 or so. Since in all cases computations only requireto employ observations up to ages 70 or 75, the impact of age overstatement willbe minor13.The second favorable condition that circumvents the problem is thatthe optimal method (Bennett-Horiuchi No 4) is also the least sensitive to agemisreporting of the type encountered in LAC (see below).

• Age invariant relative completeness of death registration: all techniques rely onthe assumption that the relative completeness of death registration is age invari-ant. However, as we show later, when there are mild violations of the assumptionthe optimal method we choose (Bennett-Horiuchi IV) performs best.

• Estimation of life expectancy at older ages: all methods adopt ad hoc proceduresto handle the open age group. These procedures rely on exogenous computationsof parameters relating the quantity of interest, life expectancy at age 75 or 70and selected observed quantities in the data at hand. The relations are estimatedusing model life tables, stable population expressions, numerical approximationsor a combinations of all these. In the applications implemented here we followthe methods suggested by the authors in each case. Thus, some of the variabil-ity in performance that we uncover, albeit a small part, is due to heterogeneousstrategies to handle the open age group.

12 Hill and colleagues also investigated the effects of intercensal migration [Hill et al. (2009)]. Inthe simulations performed here we do not include consideration of migration but its effects arepartially but bluntly captured via differential censuses completeness.13 This is because even with heavy age overstatement the population at any particular age y < x,where x is below 65 or so, is a small fraction of the population above age x. These ratios increaseas x increases due to exponential decrease of population at older ages.


2.4.2.2 Techniques to identify age misreporting

A key component of our analysis is the detection and identification of patterns of agemisstatement in the population and death counts. As shown in a previous section,the distortions associated with age misreporting in population and death counts ismore complex than those involving only faulty completeness. Detection of the prob-lem is difficult since its manifestations are quite subtle and, in the absence of overtand striking phenomena such as the US Black-White cross over, is likely to remainconcealed and undetected. There are two well-tested methods to identify the exis-tence of age over(under) statement in either population or death counts. The firstmethod requires an external data source with correct dates of birth or ages in a pop-ulation at a particular time that can be compared to age-specific census counts atapproximately the same time. An example of this is the utilization of Medicare datain the US, a source of information that, as a rule, contains both population exposedand mortality data. Because Medicare data are linked to Social Security records andthese are known to register age with high precision, mortality rates computed fromMedicare data are a gold standard against which conventional mortality rates couldbe contrasted and their quality evaluated [Elo and Preston (1994)]. If one ignoresthe existence of a population not covered by Medicare records, it is also feasible tolink individual census records to Medicare records and investigate more preciselythe nature of patterns of age misreporting in census counts. If, in addition, Medicarerecords are linked to the US National Death Index (NDI) it is then possible to re-peat the same operations and assess the quality of reporting of age at deaths. In allcases one must assume that the coverage of population in both sources is completeor, if incomplete, identical14. Record linkage from multiple sources such as thoseillustrated above has rarely been used as it is expensive and involves resolution ofcomplicated confidentiality issues.

A second method is less data demanding, considerably less expensive, and sim-ple to apply but can only reveal the existence of age misreporting in one of the twosources and provides few clues about its nature. The procedure was proposed byPreston and colleagues [Rosenwaike and Preston (1984), Elo and Preston (1994),Bhat (1990), Grushka (1996)] and has been applied in countries of North Amer-ica, Western Europe and in Latin America [Condran et al. (1991), Grushka (1996),Dechter and Preston (1991), Palloni and Pinto (2004), Del Popolo (2000)]. In a nut-shell the method consists of comparing cumulative population counts in a census inyear t1 to the expected cumulative population counts in a second population censusin year t2. The computation of expected quantities requires both an initial censusopening the intercensal interval, a second census counts at time t2 closing the inter-censal interval, and age specific deaths counts in the intercensal period spanning aninterval of k = (t2− t1+1) years. The ratio of observed to expected population is anindicator of age misstatement:

14 The assumption is more restrictive that we made it sound: if population coverage is not completein either source, then the subpopulations missed in each census must be random relative to theirtrue and reported age.


cmRox,[t1,t2]

=cmPo

x+k,t2/cmPo

x,t1

1− (cmDox,[t1,t2]

/cmPox,t1)

(2.6)

where cmPox,t1 and cmPo

x,t2 are cumulative populations over ages x and x+ k inthe first and second census respectively and cmDo

x,[t1,t2]is the cumulative deaths after

age x during the intercensal period. This expression is a simple contrast between twodifferent estimates of the same underlying quantity (population parameter), namely,the cumulative survival ratio: the denominator uses the complement of the observedratio of (cumulative) intercensal deaths to (cumulative) population in the first censuswhereas the numerator expresses it as the survival ratio computed from the cumu-lative counts in two successive population censuses. It is useful to express 2.6 in alogarithmic form, namely,

ln(cmRx,[t1,t2]) = ln(SNox,x+k)− ln(SDo

x,x+k) (2.7)

where SNox,x+k is the ‘survival ratio’ computed from two censuses and SDo

x,x+k

is the survival ratio computed from intercensal deaths15. In the absence of migra-tion, age misstatement and imperfect completeness of census and death counts, bothestimators should yield the same number, the ratio in 2.6 should be 1, and the logexpression in 2.7 should be 0 for all adult ages.

To shed light on the meaning of expressions 2.6 or 2.7 and to simplify notationand terminology we will speak of net age misreporting to refer to the net resultof both age over and under statement. Furthermore, because we, as well as pastresearch, uncover systematic net age overstatement of adult ages in LAC countries,we will speak of ‘age overstatement’ or ‘age overreporting’ even though we refer tothe net result of age under and over reporting. In Appendix 3 we show that when theassumption of absence of age misreporting is violated, we can approximate 2.7 as

ln(cmRx,[t1,t2])∼ ln(

h(x+ k)h(x)

)−(

g(x)h(x)−1)(

1+ ITx,x+k

)(2.8)

where ITx,x+k is a true integrated hazard analogue between ages x and x+ k (and

hence strictly positive), h(x) is an increasing function of age that depends on ageoverstatement of populations and g(x) is an increasing function of age that dependsonly on overstatement of ages at death. Both h(x) and g(x) are functions of thepropensity to overstate and the underlying population and deaths age distribution.Assume now that the propensity to overstate ages (of populations or deaths) is ageinvariant or increases with age and that the following three conditions hold: (a) the(true) age distribution slopes sharply downward, (b) the age distribution of deathsincreases with age, and (c) the rate of decrease of population with age is smallerthat the rate of increase of deaths with age. Under these three conditions, almostuniversally verified in all human populations, the ratio h(x+ k)/h(x) will always

15 In Appendix 3 we provide terminology and a full justification for the use of this index.


be larger than 1 and will increase with age, g(x) will always be larger than 1 andincrease with age, and the rate of increase in g(x) will exceed the rate of increasein h(x) so that g(x) > h(x) almost everywhere in the age span. The following arepossible scenarios16:

1. When there is systematic age overstatement of population counts ONLY, h(x)> 1and g(x) = 1, then expression 10 reduces to

ln(cmRx,[t1,t2]) = ln(

h(x+ k)h(x)

)+(h−1(x)−1)(1+ IT

x,x+k)< 0

The inequality results because the positive term in the expression, that is, thedistortion of the survival ratio based on population counts, will be smaller thanthe negative term influenced by the distortion in the second estimator based onintercensal death rates.

2. When there is systematic age overstatement of death counts ONLY, h(x) = 1 andg(x)> 1, the expression becomes


h(x+ k)h(x)

)+(g(x)−1)(1+ IT

x,x+k)> 0

and the positive sign results from fact that all terms in the expression are positive.3. When there is systematic overstatement of BOTH population and death counts,

g(x)> h(x)> 1, then


h(x+ k)h(x)

)+

(g(x)h(x)−1)(1+ IT

x,x+k)> 0

because, by assumption, all terms are positive.

Before we can use the above to diagnose conditions in an empirical case, two is-sues must be resolved. First, it is possible that there are empirical patterns of ageoverstatement of deaths and populations that offset each other and produce ratiosclose to 1 even though the underlying data are incorrect. That is, scenario (3) issuch that the log of the ratio is 0 at all ages even when there is net age overstate-ment. Because of this possibility, a diagnostic of observed conditions based on theindex (or the log of the index) can only detect consistency (including error consis-tency) of age declaration in population and death counts, rather than suggest accu-racy [Dechter and Preston (1991)]. Second, throughout we assumed that both cen-sus and death counts had perfect coverage. When one allows for defective censuscoverage, an identification problem is created since now we will have

ln(cmRx,[t1,t2])∼ ln(

C2

C1

)+ ln

(f (x+ k)

f (x)

)−(

C3g(x)C1h(x)

−1)(1+ IT

x,x+k) (2.9)

16 The impact of age misreporting predicted analytically in these scenarios has been confirmed bysimulations studies [Condran et al. (1991), Palloni and Pinto (2004), Grushka (1996)]. In section2.4.3 we show that our simulations also accord with analytic predictions.


and it is clear that we can no longer separate the role of age overstatement andcompleteness. In particular, even if there is no age misreporting, expression 2.9 canyield non-zero values and mimic increasing or decreasing patterns with age thatresult naturally from age overstatement alone. To understand better the combinedinfluence of defective coverage and age misreporting on observed mortality rate weneed to define more precisely the nature of the functions h(x) and g(x) , the natureof their dependence on patterns of age misreporting, and how they interact withdefective coverage. We investigate these issues in the section below.

2.4.2.3 Techniques to adjust for age misreporting

As indicated before, the main tool to adult detect age misreporting is highly sensitiveto relative completeness of census counts. Figure 2.2 displays the value of cmRx thatobtains when there is no age misreporting at all but there is differential completenessin census counts. It is plain that one cannot learn much about patterns of age misre-porting unless population census counts are first adjusted. This requires to identifymethods that provides robust estimates of completeness of one census relative tothe other. As we show below, the evaluation study confirms a result first reportedby Hill [Hill et al. (2009)] and show that the modified Brass technique (Brass-Hill)produces a robust estimate of C1/C2 .The ratio of completeness factor is sufficientto correct the observed values of cmRx.

Fig. 2.2 Behavior of index of age misstatement with differential censuses.

Age

Cum

ulat

ive

surv

ival

rat

io

0.0

0.2

0.4

0.6

0.8

1.0

45 50 55 60 65 70 75 80 85 90 95 100

Observed

0.9999990

0.9999995

1.0000000

1.0000005

1.0000010

45 50 55 60 65 70 75 80 85 90 95 100

Adjusted

Once the ratios are adjusted there remains the task of retrieving estimates of themagnitude of net adult age net overstatement. The model developed before basedon a known standard of age net overreporting includes two parameters, λ no and φ no


for the magnitude of population age over and understatement respectively. There arethree different methods to estimate these parameters.

i A brute force method: it is possible, but not advisable or even necessary (see (ii)below), to use the cumbersome but exact procedure that consists of computingall possible values for the vector [cmRx=45,100] that can be generated with infor-mation on population counts in two censuses and intercensal deaths, the stan-dard pattern of age over reporting and, finally, multiple combinations of pairs(λ no,φ no) One then chooses the (unique) pair of values that best reproduces theobserved vector [cmRx=45,100]

ii Parametric method I: this method is a short cut for Method I. We used the simu-lated data to estimate the following relation

(cmRx)−1 = α0x +α1xλ

no +α2xφno (2.10)

for all values of x≥ 45. The parameters of this relation, α0,α1 and α2, character-ize the space of solutions for the triplet (cmRx,λ

no,φ no) embedded in the simulateddata. As we show in section 2.4.3.2 the fit of the model is very good and the esti-mated values of the constant is always close to 1, as it should be. The tightness ofthe fit of the relation presents us with an unique opportunity. Indeed, if the observeddata is an element of the space of solutions, that is, if the observed data is generatedby one of the combinations of parameters spawned by the simulation, it might bepossible to invert the procedure in 2.10 and retrieve the unknown parameters for netage overstatement. 17. In fact, a constant-constrained regression, e.g. {α0x = 1}, ofthe vector of observed values {cmRx=45,100} on the vectors of estimates of the vectorof parameters {α1x=45,100} and {α2x=45,100} in 2.10 should produce two parameterestimates that correspond to the best fitting values of the unknowns λ no,φ no. Asshown below in section 2.4.3.2, this inverse procedure performs optimally18.

iii Parametric method II: the third method seeks to reproduce [cmRx=45,100] as afunction of age and then map parameters of the function onto the pairs (λ no,φ no) that generated the data. It consists of fitting a hyperbola to a range of valuesof cmRx

cmRx = β1/(ς −age)β2 (2.11)

where ς is set equal to 7619. We then use the estimated parameters of function2.11 to predict the pair of values (λ no,φ no). As we show below in section 2.4.3.2

17 The constrain imposed, namely, that the observed data must be in the space of populationsgenerated by the simulation is crucial for in the simulation we do not use all possible values of(λ no,φ no) but we limit them to a rather small range.18 Model 2.10 is best fitting in the sense that any interaction terms or higher order moments of theindependent variables do not reduce the mean squared error by a statistically significant amount.19 In cases when the values of the magnitude of age overstatement approaches the largest valuesallowed (close to 2 or 2.5), the function cmRx attains a point of discontinuity where the derivativeswith respect to age do not exist. In order to avoid such cases we used trial values for the parameterς and find that, in the space of simulated populations, ς = 76 is optimal as it always avoids points


the fit of the hyperbolic function to the distorted data is very tight but the retrievalof the hidden parameters governing net age overstatement is generally poor. Thisis due to under-identification: if one uses the entire range of values attainable byλ no and φ no , the function cmRx=45,100 can be mapped onto multiple pairs (λ no,φ no). The procedure works best when the pair of values (λ no, φ no) is withina limited range (approximately [0.10-1.5]). Because of this regularity one canuse method (ii) and (iii) jointly to seek consistency: if the observed values ofthe parameters λ no and φ no are within the permissible range devoid of points ofdiscontinuities, then both methods should produce the same results.

2.4.3 Results of the evaluation study

We now review results of applying candidate techniques for adjusting defective rela-tive completeness and age misreporting. We base our discussion on results from theset of simulated populations describe before, a space of fictitious populations anddeaths generated by five different demographic regimes combined with an exhaus-tive set of errors patterns. In section 2.4.3.1 we assess the techniques’ effectivenessto retrieve population parameters under several conditions: ignoring the error pat-terns embedded in the space of simulated populations, in subsets of populationsdefined by selected underlying conditions and, finally, isolating two types of errorsthat violate basic assumptions of all methods considered here, namely, age misre-porting and age dependent completeness. In section 2.4.3.2 we describe the behaviorof methods to adjust for age misreporting and, finally, in section 2.4.3.3 we reviewan empirical application.

2.4.3.1 Defective completeness and the average behavior of methods:evaluation using the pooled simulated populations

To facilitate assessment of techniques we create six different populations subsets: (a)total or pooled, (b) stable, (c) non-stable, (d) non-stable with no age misreporting,defective death and population coverage, (e) non-stable with age misreporting, in-complete death coverage and defective but identical population coverage in the twocensuses and (f) non-stable with age misreporting, incomplete death and defectivepopulation coverage. Each subpopulation with incomplete population and/or deathcoverage has three variants, one with constant relative completeness (of census anddeaths counts) and the others with age varying completeness.

Investigating the behavior of techniques isolating conditions that generate errorsis helpful when there is reliable external information about population stability, na-ture of age misreporting and/or patterns of age relative completeness. A techniquethat performs optimally in the pooled simulated population may not do so well un-

of discontinuity. This is equivalent to saying that one cannot reproduce the function for ages above76, a trait that is partially responsible for under identification.


der a specific set of conditions. The opposite situation is also possible: a techniquemay not behave well on average but could be optimal under some circumstances.Because the source of uncertainty matters for the final choice of method, our assess-ment is carried out across multiple subsets of simulated populations, each reflectingdifferent types of errors or conditions. We define the following six population sub-sets: a) pooled sample (n=31,500), b) stable populations (n=6,300), c) non-stablepopulations (n=25,200), d) non-stable populations with no age misreporting but de-fective completeness of death and population counts (n=700) e) non-stable popula-tions with age misreporting, defective coverage of death counts and equal (possibledefective) coverage of population counts (n=4,320) and, finally, f) non-stable pop-ulation with age misreporting, defective death registration, defective (but unequal)population counts (n=17,280). In each of these subsets we generate three variants,one assuming constant relative completeness and two variants imposing two differ-ent age-dependent patterns of relative completeness20.

We evaluate the following techniques: Brass technique [Brass (1975)] modifiedby Hill [Hill (1987)] (or BHill) to compute a robust estimate of relative complete-ness in two population censuses and 12 techniques to estimate relative complete-ness of death registration: a) original Brass method [Brass (1975)] modified by Hill[Hill (1987)] (BHill) and a variant by Martin [Martin (1980)] (BMartin), b) fourvariants of Bennett and Horiuchi (Bennett and Horiuchi, 1981;1984) (BH 1-BH 4),c) one method by Preston and Bennett [Preston and Bennett (1983)] (PB), d) twodifferent methods by Preston and Hill [Preston and Hill (1980)] (PH 1-PH 2), ande) two variants of a method proposed by Preston and Lahiri [Preston and Lahiri (1991)](PL 1 PL2). In addition, we employ a two-stage Bennet-Horiuchi method that con-sists of first estimating relative completeness of the two censuses using Brass-Hillmethod, adjusting rates of intercensal growth, and then applying any of the variantsof Bennett-Horiuchi method (2SBH 4)21. We include three flavors of the simulation,one with age-invariant relative completeness and two with different age patterns ofrelative completeness (see footnote to Tables 2.8A-2.8C).

The assessment focuses on the mean proportionate (absolute) errors for two pop-ulation parameters, the ratio of completeness of first to second census coverage,ρc=C1/C2 and the relative completeness of death registration, ρd =C3/(0.5∗(C1+C2)). The six panels of Tables 2.8A-2.8C display the mean of the proportionate ab-solute error for each of the six populations subsets defined above. Table 2.8A refersto simulations with constant relative completeness by age and Tables 2.8B and 2.8Creflect results using two different patterns of age varying relative relative complete-ness. The errors in each population subset s,s = 1,2...6, are Ξd

s = ∑j=Ksj=1 εd

s j and

Ξcs = ∑

j=Ksj=1 εc

s j, where εds j =| ρd

s j−ρds j | /ρd

s j, εcs j =| ρc

s j−ρcs j | /ρs j, ρc and ρd are

defined as before, ρds j and ρc

s j are estimates, and the summations are over all sim-

20 The two functions for age dependent census completeness are assumed to hold in both censusesand are defined as follows: (a) scenario 1: C1= 0.75 if age [15-34], C1= 0.85 elsewhere; C2= 0.85 ifage [15-34], C2= 0.95 elsewhere; C3= 0.80 if age [15-34], C1= 0.85 elsewhere; (b) scenario 2: C1=0.85 if age [15-34], C1= 0.75 elsewhere; C2= 0.95 if age [15-34], C2= 0.85 elsewhere; C3= 0.85 ifage [15-34], C3= 0.80 elsewhere.21 To avoid cluttering, we only show results of variant No 4


ulated populations j in each of six subsets. Naturally, different error metrics yielddifferent ranking of methods but the measure we use is the preferred one in mostapplications of this kind22.

22 We emphasize that the figures in Tables 2.8A-2.8C, are computed on a subset of rather benignpatterns of distortions as they exclude values of relative completeness lower than 0.7 and differ-ences between completeness of successive censuses higher than 0.10.


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006,

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Scen

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else

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],C

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else

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whe

re.


Tabl

e2.

8C

.Pro

port

iona

teab

solu

teer

rors

inea

chof

six

popu

latio

nssu

bset

sw

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ive

com

plet

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s(S

cena

rio

2).

A.S

tabl

ean

dN

onst

abe

B.S

tabl

eC

.Non

stab

leD

.Non

stab

le?

E.N

onst

able•

F.N

onst

able

‡

Indi

cato

rM

edia

nM

ean

Std.

Dev

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edia

nM

ean

Std.

Dev

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edia

nM

ean

Std.

Dev

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edia

nM

ean

Std.

Dev

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edia

nM

ean

Std.

Dev

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edia

nM

ean

Std.

Dev

.

Bra

ssH

illC

ensu

s(B

Hill

)10.

041

0.04

50.

031

0.04

20.

046

0.03

10.

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036

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8B

enne

tHor

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H1)

0.24

40.

311

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60.

249

0.32

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300

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307

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20.

218

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259

0.01

50.

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60.

234

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Ben

net-

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310

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285

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70.

218

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Ben

net-

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Ben

net-

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310

0.27

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247

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285

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225

0.23

30.

134

Ben

net-

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SBH

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081

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80.

276

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0.58

60.

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60.

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80.

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rass

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tin(B

Mar

tin)2

0.11

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154

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109

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118

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091

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40.

113

Bra

ssH

ill(B

Hill

)10.

094

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90.

083

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107

0.07

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023

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t(PB

)0.

724

0.70

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370

0.76

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739

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710

0.69

10.

369

0.53

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564

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90.

103

0.18

90.

235

0.76

10.

785

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7Pr

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0.41

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521

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0.48

90.

426

0.51

20.

473

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90.

180

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30.

343

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30.

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Pres

ton-

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508

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372

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Pres

ton

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o1

(PL

1)0.

486

35.5

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55.5

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310

1.11

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42.6

3052

04.8

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Pres

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100

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116

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N31

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700

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0• C

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and

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and

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dm

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s(C

1−

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<.1

01 V

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ofst

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orta

lity

decl

ine.

Scen

ario

2:C

1=

0.85

ifag

e[1

5-35

],C

1=

0.75

else

whe

re;C

2=

0.95

ifag

e[1

5-35

],C

1=

0.85

else

whe

re;C

3=

0.85

ifag

e[1

5-35

],C

1=

0.80

else

whe

re.


Search for an optimal estimate is carried out considering all prior informationavailable and the following are general rules:

i. In the absence of any knowledge whatsoever about errors or deviations fromstability, the search for best method should be concentrated in the first panel ofTables 2.8A-2.8C.

ii. When exogenous information suggests stability and not much else, the searchshould focus on the subset of stable populations or the second panel of Tables2.8A-2.8C. Instead, when there is prior empirical data confirming violation ofstability, for example past shifts in fertility regime, but one can be agnostic aboutcompleteness and age misreporting, the search of optimal method should con-centrate on the third panel of the tables.

iii. When, in addition to lack of stability, there is evidence of defective coverage ofpopulation and death counts but no suggestion of significant net age overstate-ment at adult ages, the search should shift to the subset in the fourth panel of thetables.

iv. When the researcher suspect a scenario like in (iii) above but, in addition, thereis evidence of age misreporting, identification of optimal method should be doneusing the fifth panel of Tables 2.8A-2.8C.

v. Finally, in cases scenario (iv) is most reasonable and one can establish that com-pleteness of two censuses is (possibly) defective but equal in both censuses, iden-tification of the optimal choice must be done with the sixth panel of the tables.

The results contained in these tables contain a number of salient characteristics.First, as already noted in the work by Hill and colleagues, Brass’s methods to es-timate relative completeness of the two censuses is uniformly good, regardless ofpopulation subset23. Second, with the exception of Brass methods, the magnitude oferrors are larger when census coverage is defective as long as completeness is NOTthe same in both censuses. This is because all methods (except Brass-Hill) rely ondirect computations of age specific growth rates from the observed data, a quantitythat will be in error when there is different coverage errors in two successive cen-suses. Indeed, the performance of these methods improves substantially when thereis accurate census coverage or, equivalently, when coverage is the same in both cen-suses (Table 2.8A, panel D). Fourth, age misreporting affects the accuracy of allestimates but more so in some cases (Brass-Hill and the second variant of Preston-Hill) than in others (Bennett-Horiuchi all variants). Fifth, the magnitude of errorsthat obtain when relative completeness is age dependent (panels in Tables 8A and8B 2.8A-2.8B) varies sharply by technique but, in general, are lowest in the methodby Bennett-Horiuchi.

23 To move beyond verification based purely on simulations only, Appendix 4 4 compares esti-mates of ratios of relative completeness with the ratios computed from CELADE’s estimates ofabsolute completeness in two successive censuses. Although the agreement of both sets of fig-ures is quite close, one should be cautious for a couple of reasons. First, agreement in ratios ofcompleteness may also be produced with incorrect census-specific estimates of completeness. Sec-ond, CELADE’s estimates are based on estimates and projection of populations and do not alwaysrely on post-enumeration assessments. Thus, the agreement we ascertain can be an agreement ofincorrect figures.


The most important inference from this evaluation exercise is as follows: if oneexcludes population subsets with defective census completeness, the optimal choiceis always one of the variants of Bennett-Horiuchi method followed by the two meth-ods proposed by Brass, irrespective of violations of stability assumptions or agemisreporting. This suggests the following strategies:

i. In the absence of exogenous information about the difference in completeness be-tween the two census and if the assumption of age invariant completeness holds,use Brass-Hill method;

ii. In the absence of exogenous information, whether or not age dependence of rel-ative completeness is suspected, use the two-stage procedure (2SBH 4): first es-timate relative completeness of census enumeration using Brass’ method, adjustintercensal rates of growth and then apply Bennett-Horiuchi method.

Throughout the book we use both strategies but when the difference between esti-mates is less than 0.05 we compute their average and when their difference exceed0.05 we chose strategy (ii)24.

2.4.3.2 Defective age reporting

Do the procedures to identify and adjust for age misreporting formulated beforeproduce robust estimates of the true population parameters? To answer this questionwe select the subset of simulated populations with age misreporting and defectivecompleteness, adjust for completeness following strategy (ii) above, identify theexistence of age misreporting, and then correct for it using techniques (ii) and (iii) insection 2.4.2.3. Tables 2.9A through 2.9C display the main results. First, Table 2.9Acontains parameters associated with expression 2.10 and reveals that the fit is almostperfect and that the estimated constant is close to one everywhere, irrespective ofage. Table 2.9B shows that when the procedure is inverted and we regress cmRx onthe known vectors of estimates of the parameters α1x=45,100 and α2x=45,100 we obtaine estimates of parameters of net age overstatement that are minimally affected byerrors. This suggests that if an observed population belongs to the space of simulatedpopulations, one can retrieve highly accurate estimates of the magnitude of age netover-reporting by simply using the estimated relation between the observed cmRxand estimates of parameters of α1x=45,100 and α2x=45,100 from Table 2.9A. Finally,as anticipated before, Table 2.9C reveals that the parametric technique based onthe hyperbola is of little help as the estimates of the underlying parameters of netoverstatement are subject to large errors even if the model fits the observables quitewell.

24 It is important to note that when relative completeness is age dependent, Bennett-Horiuchi ismean optimal, in the sense that the weighted average of relative death completeness of observeddata will be best estimated by Bennett-Horiuchi methods. It does not mean that, once applied, theadjusted mortality rates (and derived function of the life table) will also be best estimates. Noneof the methods we include in our evaluation can escape from the assumption of constant relativecompleteness and, therefore, we can only aspire to find a mean optimal candidate.


Table 2.9 A. Estimated regression models relating index of age misstatement and parameters ofage misreporting.

Age α0 α1 α2 R2

45 1.000 -0.027 -0.004 1.00046 1.000 -0.012 -0.005 1.00047 1.000 -0.006 -0.005 1.00048 1.000 -0.003 -0.006 1.00049 1.000 0.000 -0.007 1.00050 1.000 0.002 -0.008 1.00051 1.000 0.003 -0.009 1.00052 1.000 0.005 -0.010 1.00053 1.000 0.006 -0.011 1.00054 1.000 0.008 -0.013 1.00055 1.000 0.010 -0.014 1.00056 1.000 0.012 -0.016 0.99957 0.999 0.014 -0.019 0.99958 0.999 0.017 -0.022 0.99959 0.999 0.020 -0.025 0.99960 0.999 0.024 -0.030 0.99961 0.999 0.029 -0.035 0.99962 0.999 0.035 -0.041 0.99963 0.998 0.042 -0.048 0.99964 0.998 0.051 -0.057 0.99865 0.997 0.062 -0.069 0.99866 0.996 0.076 -0.082 0.99867 0.995 0.094 -0.099 0.99768 0.994 0.116 -0.121 0.99769 0.992 0.145 -0.148 0.99670 0.990 0.183 -0.183 0.99571 0.986 0.231 -0.228 0.99572 0.982 0.295 -0.285 0.99473 0.975 0.378 -0.360 0.99274 0.966 0.490 -0.458 0.99175 0.952 0.638 -0.586 0.989

2.4.3.3 An illustration: the case of Guatemala

Table 2.10 contains alternative estimates of parameters of conditional (age 5) lifetables for Guatemala during the period 1972-1982. Figures 2.3A and 2.3B displaythe corresponding age patterns of mortality above age 5 for males and females, re-spectively. The observed life expectancy at age 5 is 57 years for males and 60 forfemales. After adjusting for relative completeness the values drop to 55.1 for malesand 58.1 for females. And, once the age misreporting is corrected, the estimates de-scend even further to 54.3 for males and 57.3 for females. In proportional terms thedisparities between observed and adjusted values are largest for life expectancy atage 60: the observed values are between 15% and 20% too high for males and fe-males respectively. Since the errors are large for the period 1950-1972 and smaller


Table 2.9 B. Estimates and true values of parameters of net age overstatement from inversemethod.

run φ no φ no λ no λ no R2

1 0.000 0.061 0.350 0.370 1.0002 0.000 0.002 0.700 0.685 1.0003 0.000 -0.059 1.050 0.999 1.0004 0.000 -0.118 1.400 1.313 1.0005 0.000 -0.178 1.750 1.628 1.0006 0.000 -0.238 2.100 1.942 1.0007 0.000 -0.298 2.450 2.256 1.0008 0.000 -0.358 2.800 2.571 1.0009 0.350 0.393 0.700 0.727 1.00010 0.350 0.392 1.050 1.078 1.00011 0.350 0.391 1.400 1.429 1.00012 0.350 0.390 1.750 1.780 1.00013 0.350 0.388 2.100 2.130 1.00014 0.350 0.387 2.450 2.481 1.00015 0.350 0.386 2.800 2.832 1.00016 0.700 0.710 1.050 1.067 1.00017 0.700 0.755 1.400 1.445 1.00018 0.700 0.801 1.750 1.823 1.00019 0.700 0.846 2.100 2.201 1.00020 0.700 0.892 2.450 2.579 1.00021 0.700 0.938 2.800 2.957 1.00022 1.050 1.013 1.400 1.393 1.00023 1.050 1.096 1.750 1.791 1.00024 1.050 1.179 2.100 2.189 1.00025 1.050 1.262 2.450 2.587 1.00026 1.050 1.345 2.800 2.985 1.00027 1.400 1.303 1.750 1.704 1.00028 1.400 1.416 2.100 2.117 1.00029 1.400 1.530 2.450 2.530 1.00030 1.400 1.643 2.800 2.943 1.00031 1.750 1.582 2.100 2.004 0.99932 1.750 1.720 2.450 2.427 1.00033 1.750 1.859 2.800 2.851 1.00034 2.100 1.851 2.450 2.292 0.99935 2.100 2.009 2.800 2.723 1.00036 2.450 2.110 2.800 2.569 0.998

for the period after 1982, an unadjusted time trend would convey an highly mislead-ing picture of the time trajectory of mortality.

2.5 Estimation of adult mortality for the period 1850-1950

Availability of vital statistics for the period 1850-1950 is scarce (see Table 2.2).It is only in a few country-years that we can estimate life tables using techniques

2.5 Estimation of adult mortality for the period 1850-1950 41

Fig. 2.3 A. Observed and adjusted mortality rates for males in Guatemala.

Age

Mor

talit

y ra

tes

(log−

scal

e)

−6.0

−5.0

−4.0

−3.0

−2.0

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

−6.0

−5.0

−4.0

−3.0

−2.0●

●

●

●

●

●

●

●●

●●

●●

●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●

●

●

●

●

●

●

●

●●

●●

●●

●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●●●

●

●

●

●

●

●

●

●●

●●

●●

●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●●●●●

Adjusted for:Bennet−Horiuchi & age misstatementBennet−Horiuchi

Observed

●

●

●

Fig. 2.3 B. Observed and adjusted mortality rates for females in Guatemala.

Age

Mor

talit

y ra

tes

(log−

scal

e)

−6.0

−5.0

−4.0

−3.0

−2.0

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

−6.0

−5.0

−4.0

−3.0

−2.0●

●

●

●

●

●

●

●●

●●

●●

●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●

●

●

●

●

●

●

●

●●

●●

●●

●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●●●

●

●

●

●

●

●

●

●●

●●

●●

●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●

●●●●●●●●

Adjusted for:Bennet−Horiuchi & age misstatementBennet−Horiuchi

Observed

●

●

●


Table 2.9 C. Estimated and true values of parameters of net age overstatement using hyperbolicfunction.

run φ no φ no λ no λ no R2

1 0.000 1.243 0.350 0.071 1.0002 0.000 1.668 0.700 0.171 0.9983 0.000 2.662 1.050 0.339 0.9884 0.000 8.583 1.400 0.890 0.9365 0.000 11.785 1.750 0.952 0.9186 0.000 9.955 2.100 0.819 0.9377 0.000 20.000 2.450 4.240 0.9258 0.000 26.352 2.800 1.186 0.9059 0.350 1.244 0.700 0.070 1.00010 0.350 1.627 1.050 0.160 0.99811 0.350 2.428 1.400 0.303 0.99112 0.350 5.470 1.750 0.639 0.95813 0.350 7.273 2.100 0.721 0.94614 0.350 24.000 2.450 5.584 0.98615 0.350 43.669 2.800 1.533 0.88616 0.700 1.245 1.050 0.069 1.00017 0.700 1.593 1.400 0.152 0.99818 0.700 2.264 1.750 0.275 0.99419 0.700 4.228 2.100 0.519 0.97320 0.700 73.344 2.450 3.738 0.99321 0.700 45.485 2.800 1.833 0.90622 1.050 1.245 1.400 0.068 1.00023 1.050 1.565 1.750 0.144 0.99924 1.050 2.142 2.100 0.253 0.99525 1.050 3.562 2.450 0.445 0.98126 1.050 13.985 2.800 1.235 0.93927 1.400 1.246 1.750 0.067 1.00028 1.400 1.542 2.100 0.138 0.99929 1.400 2.047 2.450 0.236 0.99630 1.400 3.149 2.800 0.394 0.98631 1.750 1.246 2.100 0.066 1.00032 1.750 1.522 2.450 0.132 0.99933 1.750 1.972 2.800 0.221 0.99734 2.100 1.246 2.450 0.065 1.00035 2.100 1.504 2.800 0.127 0.99936 2.450 1.246 2.800 0.064 1.000

that rely on intercensal deaths. In some cases we use vital statistics for a periodof three years centered on a population census and the original method proposedby Brass (1975) requiring population stability to produce relative completeness ofdeath registration. In the remaining cases, where we can only access census counts,we use a generalized version of the ogive approach first formulated by Coale andDemeny (1967). In Sections 2.5.1-2.5.4 we describe the classic version of the ogiveapproach, a shortcut, a generalized version of the standard ogive method, and theextension of Brass classic method. In sections 2.5.5 and 2.5.6 we review methodsused to estimate adjusted rates of growth for completeness and for net international


Table 2.10 Observed and adjusted mortality indicators for Guatemala post 1950 for males andfemales.

Males FemalesObserved Adj RC1 Adj RC & Age2 Observed Adj RC1 Adj RC & Age2

E(5) 56.900 55.100 54.300 60.100 58.100 57.300E(40) 29.100 27.700 26.800 31.100 29.500 28.700E(60) 15.600 14.500 13.400 16.200 15.100 14.100C1/C2 1.062 1.073C3/0.5*(C1+C2) 0.893 0.873θ1 2.500 2.500θ3 0.000 0.000

1Relative completeness; 2Age misstatement.

migration in four countries heavily influenced by it at the turn of the century. Section2.5.7 summarizes results for the period.

2.5.1 Classic ogive method

Assume a stable population with a natural rate of increase r and mortality given bythe survival function S(y). The age distribution N(y),y = 0, . . . ,∞, of the populationis given by:

N(y) = N(0)exp(−ry)S(y) (2.12)

At a minimum, estimation of the full function S(x) requires knowledge of r andof N(x) in finite age groups, e.g single or five-year age groups. To reduce noiseand minimize effects of age heaping and systematic age misreporting, Coale andDemeny [Coale and Demeny (1967)] compute the cumulative age distribution from2.12, namely

cumN(x) = N(0)x∫

0

exp(−ry)S(y)dy (2.13)

a function that is insensitive to population transfers across ages older than x25.When the survival function S(y) is an element of a finite set within which there isvariation only due to mortality levels, expression 2.13 will have a unique solutionfor the unknown level of mortality for each age x. In most observed cases, dataerrors, inaccuracies in the observed value of r, or mild departures from stability,can yield different solutions for the unknown level of mortality associated with eachx. Coale and Demeny compute the median level estimated in a restricted range of

25 The label ‘ogive method’ is due to the shape of the cumulative age distribution.


ages that excludes very old and very young ages where errors are likely to be morepronounced.

A key difficulty remains unsolved, however. This is that S(y) may belong not toa unique but to one of M > 1 distinct families of mortality patterns within each ofwhich there is variability induced by differences due to levels of mortality only. Ifso, there will be as many as M distinct solutions for each x. Even in the absenceof multiple solutions due to inaccuracies above noted, this creates an identificationproblem that can only be resolved by a priori specifying the family of mortalitypattern to which S(y) belongs. Thus, the researcher requires exogenous informationabout the prevailing age pattern of mortality26.

2.5.2 Arriaga’s shortcut

We can re-express 2.12 as follows:

ln(

N(y)N(x)

)=−r(y− x)− I(x,y) (2.14)

where I(x,y) is the integrated force of mortality between ages x and y. When thesurvival function belongs to a known family can be used to recover the mortalitylevel. Arriaga’s suggestion [Arriaga (1968)] is to assume a given level, compute thedifference ln(N(y)/N(x))− I(x,y) and regress this variable (via OLS) on the agedifference (y− x). The procedure is repeated as many times as levels of mortalityone may want to consider. The life table consistent with the observed age distri-bution will yield a slope close to r. Admitting the possibility of multiple familiesof mortality patterns implies to repeat the search procedure with multiple mortality

26 Two remarks about these methods are important. First, the ogive method, or any of its variants,is designed to work only within a range of ages. In particular, it was never meant to include thepopulation below ages 5 or 10. This means that the observables only contain information about theforce of mortality above ages 5 or 10. Because the contrast between mortality patterns is rootedin differences in the relation between child and adult mortality, it is possible that multiple mortal-ity patterns may identify similar levels of mortality at adult ages thus shrinking considerably theidentification problem. We use this feature to our advantage in all applications of the generalizedogive method to LAC countries. The second remark is as follows: if we choose a parameterizationof mortality that returns an explicit expression for S(x) in terms of a handful of parameters, one formortality level and two or three ancillary ones to identify the shape of the mortality function at var-ious ages, it becomes possible to solve simultaneously for these parameters from the observables.For example, if S(x) is expressed as a Brass-type logit function of a standard survival functionand two parameters, one for the level of mortality and the other to determine the relation betweenchild and adult mortality, then expression 2.13 becomes a (highly non linear) function of theseparameters. Using all observations from a limited range of ages it should be possible to choose thepair of estimates that reproduces best (in squared error terms) the observables cumN(x). This wasnot the solution adopted in the original formulation of the ogive procedure because it was beingimplemented jointly with the Coale-Demeny mortality patterns that do not admit simple param-eterization. In this book we restrict searches in the Coale Demeny mortality patterns, the UnitedNations mortality patterns, and our own mortality models (see Chapter 3).


families and obtain multiple solutions consistent with observables. This procedureis a shortcut of the more general ogive method described before. Although is wasused quite successfully, it suffers from a number of problems that are less relevantin the Coale-Demeny formulation. The first is that it requires the population to be inone or five year age groups and is vulnerable to age misreporting. The second prob-lem is that the dependent variable is likely to be noisy and may produce outliersor observations with a disproportionate influence on the estimated slope, the targetparameter.

2.5.3 Generalized ogive method

When the assumption of stability is indefensible we use expressions from general-ized stable population [Preston and Coale (1982)] and write 2.12 as :

N(y, t) = N(0, t)exp

− y∫0

r(x, t)dx

exp

− y∫0

µ(x, t)dx

(2.15)

where the population distribution is observed at time t, r(x, t) is the age specific rateof growth at age x and time t and µ(x, t)is the instantaneous mortality rate at agex and time t. Expression 2.15 can be treated just as expression 2.12 to retrieve thelevel of mortality consistent with observables. There are two important differencesbetween the generalized ogive and the standard ogive method. First, unlike the stan-dard ogive method, the generalized procedure does not invoke the assumption ofstability but, instead, demands as input a complete age pattern of rates of growth.Second, unlike the standard procedure, significant migration flows do not underminethe procedure. Indeed, if available, age-specific rates of net migration can be usedto recalculate the rates of growth net of migration. This variant was implemented inArgentina, Brazil, Cuba and Uruguay, for the years 1850-1920, a period when thesecountries experienced substantial international migratory flows in both directions.

The core of the procedure based on expression 2.15 is analogous to the ap-plication of the standard ogive method. In a first stage we compute age spe-cific rates of growth to generate the exponent of the first exponential functionin 2.15. In a second stage we choose a pivotal age x and compute the quantityΛ(y) = ln(N(y, t)/N(x, t))+

∫ yx r(x, t)dx 27. We then search within a single family

of life tables the sequence −∫ y

x µ(x, t) that best approximates the value of Λ(y).Inevitably, different ages y ∈ [x, x] will identify different solutions. If so, we maychoose from two alternative strategies: (a) select the median solution among thoseobtained within the range of ages from x+1 to x or (b) select the level of mortalitythat minimizes

27 After some trial and error we settled for x = 15 and x = 70 as lower and upper upper bounds ofthe age ranger within which we search for solutions.


x

∑x[Λ(y)−

y∫x

µ(x, t)]2 (2.16)

where [x, x] is the range of ages selected for the computations. Throughout, wechose the first of these methods to estimate the optimal mortality level.

As happens with the standard ogive procedure, the application of the generalizedogive requires choosing ex ante a model within a family of mortality patterns. Inour application we choose from among five options: three mortality models from theCoale-Demeny families (West, South and North), one model contained in the UnitedNations family [United Nations (1982)] and the standard model from the new fam-ily patterns estimated with the adjusted post 1950 LAC life tables (see Chapter 3).To compare results and assess levels of uncertainty we compute life expectancy atages 45, 65, 75 associated with the unique solution in each of the five models. Wethen calculate mean and variances over five observations (e.g. five different mortal-ity patterns) for all three parameters and their coefficient of variation. In all cases,estimates of the three parameters associated with models South and West are verysimilar to each other but of lower magnitude than those from the other two mod-els. However, in the worst of cases differences never exceed 8% of the mean value.Both the UN and the new mortality models proposed in Chapter 5 reflect mortalitypatterns that emerge when the secular mortality is underway and the original rela-tion between early and adult mortality has already shifted as a consequence of themortality decline itself. Instead models West and South in the Coale-Demeny fam-ilies reflect mortality patterns in societies with fairly high mortality levels to beginwith, more closely approximating conditions likely to have prevailed at adult agesin LAC before the onset on mortality decline (See Chapters 4 and 5). Throughoutthis book we use the average mortality of rates associated with the unique solutionsfrom models South and West.

2.5.4 Brass’s method to estimate life tables with limited vitalstatistics

When the assumption of stability or quasi stability is defensible and when, in addi-tion, only one census and death counts for years centered on the census are available,we use Brass [Brass (1975)] original method and Martin’s variant [Martin (1980)]to estimate relative completeness of death registration. The variant by Martin relaxesthe assumption of stability and invokes instead relations estimated from quasi-stablepopulations. Application of both Brass original method and Martin’s variant pro-duces life tables for the following country years Argentina 1914, Costa Rica 1927,Mexico 1921 and Uruguay 1908.


2.5.5 Estimation of rates of growth for application of the ogivemethod

Rates of growth were computed in standard ways from two successive censuses. Incases when there is historical evidence of heavy international migration, these rateswere adjusted using estimates of net international migration (see 5.6). In all othercases the observed rates of growth (total and age specific) were further adjustedfor relative completeness of censuses. Three different methods were employed tocalculate adjusted rates of natural increase:

i. Splines: we fit cubic splines to observed rates of growth for the entire period1900-1970. Rates of population increase from 1950 on are all adjusted for differ-ential census completeness whereas, with some exceptions28, those for the periodbefore 1950 are not so (Chile and Costa Rica are exceptions).

ii. Third party estimates: when available, we used estimates adjusted by third par-ties provided the adjustments are based on exogenous information about accu-racy of census counts rather than estimated from models requiring unverifiableassumptions. We used Arriaga’s adjusted estimates [Arriaga (1968)]. Rates forArgentina, Brazil, Cuba and Uruguay for the earlier periods were estimated afteradjusting for net international migration (see below).

iii Inverse Brass’s method: as noted before, the extension of the classic Brassmethod to estimate relative completeness of death registration (Hill, 1987) yieldsremarkably robust estimates of relative completeness of two successive censuses,irrespective of relative completeness of death registration. Further, it does not as-sume the existence of stability though the population must be closed to migration(or, equivalently, adjusted for intercensal migration). Elsewhere [Palloni (2015)]we show that one can apply Brass-Hill method even in the absence of intercensaldeath counts. All that is required is a matrix of intercensal deaths computed froma standard mortality pattern with an approximate level of mortality suitable forthe intercensal period. One then applies the method, discards the estimate ofrelative completeness of death registrations but retains the estimate of relativecompleteness of censuses. This estimate is then used to adjust the observed inter-censal rates of growth during the period. The accuracy of this estimate dependson how closely the selected model pattern resembles the empirical mortality pat-tern

28 The exceptions are country years where we could apply Brass original method or Martin’s vari-ant (Chile, Costa Rica, Mexico) or for countries where we estimated adjustments for internationalmigration (Argentina, Brazil, Cuba, Uruguay).


2.5.6 Adjusting rates of natural increase for net internationalmigration

Argentina, Brazil, Cuba and Uruguay experienced large flows of international mi-grants soon after 1850-1860 that lasted until about 1945, albeit with a sharp de-cline after World War I. Table 2.11 displays sources of total net international mi-gration available for all four countries. If the goal is to implement the generalizedogive method, total net migration rates are of only limited utility as they convey noage-specific information. To estimate age-specific net migration flows we use age-specific rates of net migration derived from a standard model pattern from amongthose proposed by Rogers (Rogers,19). Figure 2.4A displays a standard model pat-tern of net migration rates and, as an illustration, Figure 2.4B displays estimated agespecific male net migration rates consistent with total flows (from sources in Table2.11) for Argentina during the period 1870-1950. These estimates are then used toadjust rates of natural increase before applying generalized ogive29.

Table 2.11 Sources of estimates of international migration: 1860-1960.

Country Source 1870-1900 1900-1914 1914-1950 1950+

ArgentinaSomoza

√ √NA NA

Collver√ √ √ √

Sanchez-Albornoz√ √ √

NACollver

√ √ √ √

Sanchez-Albornoz√ √ √

NAMortara

√ √ √NA

Ferreira-Levy√ √ √

NACuba

Collver√ √ √

NASecretaria Hacienda

√ √ √NA

UruguayAnuarios estadisiticos

√ √ √NA

Sources by Periods:(a) [Somoza (1971)](b) [Collver (1965)](c) [Sanchez-Albornoz (1976)](d) [Mortara (1954)](e) [Ferreira-Levy (1974)](f) [Secretaria de Hacienda, Republica de Cuba (1976)](g) [Anuarios estadısticos, Republica del Uruguay (1976)].

29 There are, of course, multiple standard age patterns of net migration that could have been used.However, experimentation with alternatives that preserve a plausible shape (high rates at youngestages, increasing at ages of heavy labor force participation and decreasing at older ages) yield verysimilar estimates of mortality levels when GO is applied. The results are insensitive to alternativeage patterns of migration because GO rests on computations of cumulative quantities.


Fig. 2.4 A. Standard density function for net international migration.

Age

Sta

ndar

d ag

e−sp

ecifi

c pr

opor

tions

(su

m to

1)

of m

igra

nts

0.00

0.01

0.02

0.03

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100

0.00

0.01

0.02

0.03

●

●

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Source: Fitted model to Raymer-Rogers data (2006).

Fig. 2.4 B. Estimated net international migration rates. Argentina-Males

Age

Net

mig

ratio

n ra

tes

0.00

0.01

0.02

0.03

0.04

0.05

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

0.00

0.01

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0.05

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

Migration rates:1869−18951895−19141914−19471947−1963

●

●


2.5.7 Summary of life tables for the period 1850-1950

Table 2.12 displays country-years in the interval 1850-1950 and the methods used toestimate life tables in each case. The four methods are: generalized ogive (GO), GOwith migration-adjusted rates of growth (GOm), Martin’s variant of Brass’s methodfor quasi-stable populations (BMartin) and, finally, two-stage Bennett-Horiuchi(2SBH 4). After application of B, BMartin and 2SBH 4 we further adjust adultmortality for net age over-reporting using the earliest estimates of net overstatementfor the country in the post 1950 period30.

Table 2.12 Countries by Method of Life Table Estimation for 1850-1950.

Country Generalized Ogive (GO) GO with migration BMartin 2SBH 4

Argentina 1882, 1904 1914Bolivia 1925, 1950Brazil 1881, 1895, 1900, 1910Chile 1859, 1870, 1880, 1890, 1901, 1913 1925, 1935, 1946Colombia 1908, 1915, 1923, 1933 1944Costa Rica 1873, 1887, 1909 1927 1938Cuba 1851, 1869, 1882, 1893, 1903, 1913 1925, 1937, 1948Dominican Republic 1927 1942EcuadorEl Salvador 1940Guatemala 1886, 1907, 1930 1945Honduras 1932, 1937 1942Mexico 1897, 1905, 1915 1921 1925, 1935, 1945Nicaragua 1945Panama 1915, 1925, 1935 1945ParaguayPeru 1908Uruguay 1904 1908Venezuela 1931 1938, 1945

BMartin, Martin’s variant of Brass’s method for quasi-stable populations; 2SBH 4, two-stage Bennett-Horiuchi.

2.6 Estimation of child mortality

The life tables estimated so far are limited use as they only inform mortality ex-periences above age 5. Documentation of the history of mortality decline shouldinclude estimates of mortality below age 5, a task that demands different data andtechniques. In section 2.6.1 below we describe the production of estimates of childmortality below age 5 (Q(5)) for the period after 1950. Most of these estimates arebased on mixed methods, including indirect techniques with data from WFS, DHS

30 Not all estimates in the Table 2.12 are equally accurate and not all of them are used in analy-ses for this book. For example, the adjusted life tables for Argentina (1914) and Uruguay (1908)computed after applying BMartin are likely to contain biases due to violations of the stability as-sumption associated with international migration. However, they are included in the table as theyform part of the set of all possible estimates, albeit with a low probability of being unbiased.

2.6 Estimation of child mortality 51

and microcensus samples and direct techniques using adjusted vital statistics andbirth histories from surveys. Section 2.6.2 is about estimation of Q(5) for the pe-riod 1850-1950, when vital statistics are either unavailable, unreliable or the datato compute indirect estimates do no exist. Most of the estimates we suggest in thissection are derived from adjusted vital statistics and some invoke assumptions aboutmodel mortality patterns to establish consistency with adjusted adult mortality forthe same period.

2.6.1 Child mortality after 1950

The goal is to estimate ‘infant mortality’, or the probability of dying during thefirst year of life, Q(1) = 1− exp(−

∫ 10 µ(y)dy), the conditional probability of dy-

ing between the first and fifth birthday, Q(1,5) = 1− exp(−∫ 5

1 µ(y)dy) and ‘childmortality’ or Q(5) = 1− exp(−

∫ 50 µ(y)dy). We use three separate sources of infor-

mation: (i) vital registration including births and deaths as well as census counts ofpopulation below age 5 by single years of age, ii) survey data with birth historiesand reports of children ever born and surviving to mothers aged 15-49, and iii) mi-crocensus samples with requisite information on children ever born and survivingby maternal age. The first source of data is the basis for direct estimates computedwith adjustments whereas the second and third sources are the basis for indirectestimates.

These data will be combined to handle three classes of country-years observa-tions. The first are countries with no or highly erratic vital records but suitable dataon children ever born and surviving from either surveys or microcensuses. The sec-ond class contains country-years with (potentially adjustable) vital records and sur-vey/census information on children ever born and surviving. The third class includescountry-years with (potentially adjustable) vital records but with no survey or censusinformation on children born and surviving. The rules followed to compute alterna-tive estimates of parameters of interest are per force different in each of these threeclasses of countries and are dictated by the nature of indirect and direct estimates.We review these below.

2.6.1.1 Indirect estimates

Indirect methods were implemented in country-years with surveys (DHS, WFS,MICS) and census (microdata from census samples)31 on children ever born andsurviving by mother’s age. The list of country-years with suitable information isin Table 2.13. In addition, WFS and DHS data includes complete (WFS) or par-tial (DHS) maternal birth histories from which it is possible to compute estimates

31 We used census microdata directly released to us by CELADE. We did not use those archivedas part of IPUMS.


of single year of age probabilities of dying between birth and age 5. To facilitateidentification we consider these as “indirect” estimates also, even though their gen-esis is quite different from standard indirect estimates. There are then three types ofestimates:

Table 2.13 Availability of information to compute indirect estimates of child mortality.

Country Census WFS DHS MICS Vital

Argentina 1970, 1980, 1991, 2001 1966-2010Bolivia 1976, 1992, 2001 1989, 1994, 1998, 2003, 2008 2000Brazil 1970, 1980, 1991, 2000 1986, 1996Chile 1970, 1982, 1992, 2002 1955-2009Colombia 1973, 1985 1976 1986, 1990, 1995, 2000, 2005, 2009Costa Rica 1973, 1984, 2000 1976 1956-2010Cuba 1981 1964-2010Dominican Rep 1970, 1981, 2002 1975, 1980 1986, 1991, 1996, 1999, 2002, 2007 2006Ecuador 1974, 1982, 1990, 2001, 2010 1979 1987El Salvador 1971, 1992, 2007 1985 1992, 1993Guatemala 1973, 1981, 2002 1987, 1995, 1999Honduras 1974, 1988, 2001 2005Mexico 1980, 1990, 2000, 2005, 2010 1976 1987Nicaragua 1971, 1995, 2005 1998, 2001, 2006Panama 1980, 1990, 2000 1975, 1976 1955-2009Paraguay 1972, 1982, 1992, 2002 1977, 1979 1990Peru 1972, 1981, 1993, 2007 1978 1986, 1992, 1996, 2000, 2003, 2004, 2005Uruguay 1975, 1985, 1996 1955-2009Venezuela 1981, 1990, 2001, 2011 1977 1955-2007

i. Standard indirect estimates: These were computed following two different method-ologies. The first is the classic Brass technique augmented with estimates oftime reference associated the conditional probabilities of dying for women in theage groups 20-24, 25-29 and 30-34 [Brass and Coale (1968), Trussell (1975)].Throughout we assume models West and South from the Coale-Demeny familiesof mortality patterns. The second methodology is based on the United Nationslife tables also augmented with estimates of time references [Palloni and Heligman (1985)].Throughout we assume the Latin American model from the United Nations fam-ilies of mortality patterns [United Nations (1982)].

ii . Non standard indirect estimates from birth histories: These were computed di-rectly from mothers’ reports on children’s dates of birth and death within thefirst five years before the survey in WFS and DHS. We use conventional events-exposure ratios in single years of age as estimates of single year death risks andthe exponential of (negative) rates as estimates of conditional probabilities ofsurviving one year segments.

2.6.1.2 Direct estimates from adjusted vital statistics

In most cases, available information on births and child deaths during this period isdefective due to incomplete coverage and cannot be used without significant adjust-


ments. We compute adjusted estimates of Q(1), Q(1,5) or Q(1) and Q(5) from twosources. The first source is composed of third parties figures that include estimatesfrom different agencies using adjustments for completeness of death, birth and deathregistration that are fully documented and whose suitableness can be evaluated. Ina number of cases these estimates were computed by a country’s statistical officeand, in others, they are produced by international agencies An important fraction ofthe adjustments originate in contrasts between raw figures from vital statistics andindirect estimates from one or several methodologies identified in 6.1.1.

The second source of adjusted estimates is the product of reconciliation betweenestimates computed directly from vital statistics and estimates from indirect meth-ods. Following conventional rules we compute single year probabilities of dyingfrom vital statistics and then estimate adjustment factors to make them consistentwith estimates from standard and non-standard indirect estimates. To compute in-fant and child mortality from vital statistics we track registered births cohorts byyear of birth and then match them to registered deaths in single years of age. In allcases we used conventional Lexis diagrams and separation factors to allocate deathsin a calendar year to a birth cohort and obtain cohort based estimates. In addition,we compute period-based estimates of the same probabilities. We average periodand cohort estimates and compute adjustment factors that make them consistentwith standard and non standard indirect estimates. These factors are then used tocompute adjusted quantities during periods not longer than 10 years when only vitalstatistics are available. Furthermore, we estimate time trends of adjustment factors,a useful tool for the computation of parameters during 1900-1950 when indirectestimates are unavailable.

2.6.1.3 Estimation from pooled direct and indirect estimates

Pooling together the set of indirect and direct estimates for each country results ina time series, with possibly multiple observations for each time point, straddlingthe period 1950-2010. The country-specific pooled data contain non-independentobservations with some degree of redundancy, a property that is desirable whencomputations by different actors or agencies follow different conventions. Each es-timate, however, also provides different, possibly erroneous, information. The errorsare caused by inappropriate choice of mortality models, reporting errors in the sur-veys or censuses (of children born or dead, of maternal age), or inaccurate dating ofevents. Thus, despite redundancies, the pooled set for a single time point is richerand more informative about the true value of parameters that any single estimate forthe same time point. If anything, the pooled set reflects measurable uncertainty thatcan be informative in statistical analyses. A single point estimate conveys no suchinformation and conceals the uncertainty surrounding its genesis.

Figure 2.5 displays the pooled set of Q(1) in four countries.To demonstrate the detection of consistent time trends we also include estimates

from adjusted vital statistics or from third party sources for years before 1950. Thecloud of estimates are dense for years after 1950 and thin out before this period. The


estimates are tightly clustered around very narrow bands following smooth trajecto-ries with some critical points of acceleration and deceleration, suggesting unequalrates of change over time. Although for most analyses in this book we use the com-plete set of estimates, it is less cumbersome to identify only one point per year. Togenerate a single point estimate for each year and for the period after 1950 only wefit country-specific splines of the following form

Fig. 2.5 Pooled set of infant mortality rates, Q(1), in four countries.

Year

Infa

nt m

orta

lity

per

1,00

0 liv

e bi

rths

, Q(1

)

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50

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●

●● ●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●●●

● ●●●●●

●●

●●●●

● ●●

●●●

● ●●

●●●

●● ●

● ●●●●●

● ●●●

●●●

● ●

● ●

●

●●●●

●●

●

● ●

● ● ●

●

●●●

●●●

●●

●●●●

●●

●●●●●●●●●●●●●●●

●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●

●●

●●

●●

● ●● ●

D. Mexico

0

50

100

150

200

250

●

●●●●●

●●●

●●●●

●

●

●

●●●●●●

●

●●

● ●●

●●

●●

●●

●● ●●

●●

● ●●

●●

●

●●●●●●●

●●●●

●●

●●

●

●

●

●

●

●

●●●

●●●

●

●

●●

●●●

●●

●●●

●

●

●

●●

●●

●

●

●●

● ●●

●●

● ●●●●

● ●●●●● ●●●●

●●

●●

●

●

●

●

●●

●●

●

●

●●

●●

●●

●

●●

A. Bolivia

0

50

100

150

200

250●

●● ●

●● ●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●●●●●●●

● ●●●●●

● ●●

●●●

●●●

●●●●●

●●●

●●

●●●●●●●●

●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●

●●

●

● ●●

●

●

●

●● ● ● ● ●

B. Chile

logit(Q(i)) = β0 +n=k

∑n=1

βnTn (2.17)

where Q(i) stands for one of the three measures of child mortality defined before,Tn are nodes at predefined years from 1950 to 2010 in jumps of five and βn areassociated shifts of the spline at those nodes. Year-specific predicted values fromthe fitted spline will constitute our estimates for that year32.

32 In most cases we are able to compute all three indicators of child mortality. However, occa-sionally we can only compute Q(1) and Q(5). In these cases Q(1,5) was defined as Q(1,5) =(Q(5)−Q(1))/(1−Q(1)) .


2.6.2 Child mortality for the period 1850-1950

With only a few exceptions vital statistics for this period are either unavailable orhighly incomplete (with regional rather than national coverage). To estimate childmortality we employ a strategy that rests on three principles. First, when availablewe use adjusted adult mortality rates to determine values of Q(5) consistent witha handful of selected mortality patterns, e.g believed to fit well the mortality ex-perience during this period. Second, we include third party estimates if and whenthe methods that generated the estimates are thoroughly justified and one is ableto assess their suitableness. Third, for countries with vital statistics with national,but perhaps deficient, coverage we generate adjusted estimates using time trends ofestimated adjusted factors for the period 1950-2010.

Implementation of these three principle yields at least one and in most cases be-tween 2 and 5 different estimates of the target parameters33. In all cases the estimatesstretching back to 1905 form a cloud of points that either moves slowly upwards asone gets close to 1900 or tends to reach a ceiling earlier than in 1900. Only in thecases of Argentina, Brazil, Cuba and Uruguay are we able to compute estimates ofQ’s from adult mortality before 1900. In most other cases we can only verify timetrends up to 1900 (a handful of countries) or up to 1920 (most countries). Since invirtually all countries (with the exception of Argentina, Cuba and Uruguay) mor-tality decline could not have possibly started before 1900, we fit a country-specific,three-parameters Gompertz function to the cloud of point estimates. The function is

Qi = β1exp(−exp(−β2(ti−β3)))+ εi (2.18)

where Qi denotes one of the three parameters, ti is the year of estimation (withorigin in 1900) and β ’s are parameters. The function, a variant of a standard lo-gistic function, provides two pieces of information: a point estimate for each yearwithin the range of years where countries contribute with at least one estimate, andestimates of threshold values of the mortality parameters.

33 Because we chose two model mortality patterns, countries with estimates of adult mortality areinformed by two alternative estimates of the parameters.


Tabl

e2.

14C

lass

ifica

tion

ofco

untr

yye

ars

byso

urce

sus

edto

estim

ate

Q(5).

Cou

ntry

Mod

elm

orta

lity

Vita

ladj

uste

dT

hird

part

y

Arg

entin

a18

69,1

882,

1895

,190

4,19

14,1

930,

1947

1914

,194

718

82,1

904,

1914

,194

7B

oliv

ia19

00,1

925

1900

,195

019

50B

razi

l18

72,1

881,

1890

,189

5,19

00,1

910,

1920

,193

0,19

40,1

945

1940

1940

Chi

le18

54,1

859,

1865

,187

0,18

75,1

880,

1885

,189

0,18

95,1

901,

1907

,191

3,19

20,1

925,

1930

,193

5,19

40,1

946

1920

,193

0,19

4019

20,1

930,

1935

,194

0,19

46C

olom

bia

1905

,190

8,19

12,1

915,

1918

,192

3,19

28,1

933,

1938

,194

419

3819

38,1

944

Cos

taR

ica

1864

,187

3,18

83,1

887,

1892

,190

9,19

27,1

938

1927

,193

819

27,1

938

Cub

a18

41,1

851,

1861

,186

9,18

77,1

882,

1887

,189

3,18

99,1

903,

1907

,191

3,19

19,1

925,

1931

,193

7,19

43,1

948

1927

,193

8.19

4319

25,1

937,

1948

Dom

inic

anR

ep19

20,1

927,

1935

,194

219

35,1

942

1935

,194

2E

cuad

orE

lSal

vado

r19

30,1

940

1930

,194

019

30,1

940

Gua

tem

ala

1880

,188

6,18

93,1

907,

1921

,193

0,19

40,1

945

1940

,194

519

40,1

945,

1949

Hon

dura

s19

30,1

932,

1935

,193

7,19

40,1

942,

1945

,194

719

40,1

945

1940

,194

5M

exic

o18

95,1

897,

1900

,190

5,19

10,1

915,

1921

,192

5,19

30,1

935,

1940

,194

519

21,1

925,

1930

,193

5,19

40,1

945

1940

,194

5N

icar

agua

1940

,194

519

40,1

945

1940

,194

5Pa

nam

a19

11,1

915,

1920

,192

5,19

30,1

935,

1940

,194

519

40,1

945

1940

,194

5Pa

ragu

ayPe

ru18

76,1

908,

1940

Uru

guay

1900

,190

4,19

08,1

935

1908

,193

519

08,1

935

Ven

ezue

la19

26,1

931,

1936

,193

8,19

41,1

945

1938

,194

1,19

4519

38,1

941,

1945

2.8 Causes of death and the problem if ill-defined deaths 57

2.7 The final step: complete life tables

In section 2.4.3.3 we illustrate the construction of conditional (after age 5) life ta-bles for Guatemala during in the post 1950 period. To complete the life table wenow estimate the probabilities of dying in the first year of life and between ages1 and 5 using procedures described in Section 2.5. Guatemala has abundant infor-mation to compute direct and indirect estimates of both Q(1) and Q(5) from 1970on. Estimates for the period 1950-1970 are obtained from adjusted vital statisticsand predicted values from fitted time trends. For the period of interest here we esti-mate Q(1) and (5) using the fitted values to the pool of available estimates, convertthese into mortality rates using standard separation factors, and then complete thesequence of mortality rates by merging the adjusted sequence of adult mortalityrates (see section 2.4.3.3) and the values of Q(1) and Q(5).

2.8 Causes of death and the problem if ill-defined deaths

As early as 1950, most LAC countries began establishing consisting reporting ofcause of death data. In most cases, death counts are available yearly after 1950by year of death, sex, and single years of age. These data is of varying quality inrelation to level of disaggregation by cause of death, owing to the use of differentinternational classification of diseases (ICD), and by age. Figure 2.6 shows availablecause of death data in LAC by country, year, ICD and last age group reported. Thefigure clearly shows differences in the use of ICD classification across countries andin the level of disaggregation of deaths by age. Before 1970 all countries used ICD-7to classified deaths with an open age group at 85+, except for Ecuador that reporteddata up to ages 65+ in 1960-65 and age 75+ in 1966-67. Between 1970 and 1995, allcountries used ICD-8 and ICD-9 and they all consistently reported deaths up to age85+. After 1995 all countries transitioned to ICD-10 and the level of disaggregationof deaths by age increased up to age 95+.

The most important deficiency of cause of death data in LAC is the large propor-tion of deaths classified as ill-defined. For some countries this proportion can be ashigh as 60% of all deaths at older ages (60 and older). This problem is somewhatexacerbated by the use of different coding scheme over the years in the ICD and byfurther disaggregation of deaths at older ages in recent times (Figure 2.6). Assigningan underlying cause of death at these ages is a challenging endeavour due to largenumber of comorbidities afflicting older adults.

Failure to correct for ill-defined deaths will lead to incorrect inferences about therole of causes of death on the timing and pace of the mortality decline. Our generalstrategy for adjustments is to start out with populations where the proportion of ill-defined death counts exceeds 10% by age, fit a linear regression model and obtainestimates of the fraction of deaths due to each cause that are ill-defined (see below).We then re-distribute ill-defined deaths across causes of death by sex, age, countryand year, and obtain adjusted cause-specific mortality rates. Finally, for populations


Fig. 2.6 Cause of death data in LAC by country, year, ICD-classification and last age group re-ported.

with small fraction of ill-defined deaths(< 10%), we re-allocate these deaths pro-portionally according to the cause of death distribution by sex, age, country, andyear.

Figure 2.7 shows LAC countries where the proportion of ill-defined deaths ex-ceeded 10%. This figure shows that ill-defined deaths are very permissive amongcountries that experienced a late (top) and an intermediate demographic transition(middle), but less so in countries with early demographic transition (bottom). More-over, ill-defined deaths are mostly concentrated at younger (triangles) and olderadult ages (stars).

2.8.1 Methods for correcting ill-defined deaths

Let PO(ill) be the probability of observing a death classified as ‘ill-defined’; letP(ill| j) be the probability that death due to cause j will be classified as ill-definedcategory; let PT ( j) and PO( j) be the true and observed probabilities, respectively,of a death being due to cause j. Then

PO(ill) =k

∑j=1

PT ( j)P(ill| j) and


Fig. 2.7 LAC countries where the proportion of ill-defined deaths exceeds 10% for Males andFemales.

Years

1950 1960 1970 1980 1990 2000 2010

Uruguay

Cuba

Costa Rica

Chile

Argentina

Venezuela

Paraguay

Mexico

Ecuador

Colombia

Brazil

Peru

Panama

Nicaragua

Honduras

Guatemala

El Salvador

Dominican Rep

1950 1960 1970 1980 1990 2000 2010

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●● ●●● ● ●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●

●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●● ●●●●●●● ●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●●●

●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●

● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ●

●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●● ● ●●●●●●●●●● ●●●● ●●●● ●●●● ●● ●●●● ●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●

Male

1950 1960 1970 1980 1990 2000 2010

1950 1960 1970 1980 1990 2000 2010

Uruguay

Cuba

Costa Rica

Chile

Argentina

Venezuela

Paraguay

Mexico

Ecuador

Colombia

Brazil

Peru

Panama

Nicaragua

Honduras

Guatemala

El Salvador

Dominican Rep

●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

● ●● ●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●

●●●●●●●●●● ● ●●●● ● ●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●

●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●

●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●

●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●● ●● ●●● ● ● ●●●● ●●● ● ●●●● ●● ● ●●● ●● ●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●

Female

Ages:0−1415−4950+

●

Shaded areas indicate countries with ‘late’ (top), ‘intermediate’ (middle), and ‘early’ (bottom)demographic transition, respectively.

PO( j) = PT ( j)(1−P(ill| j)) (2.19)

It then follows that

PO(ill) =k

∑j=1

PO( j)P(ill| j)

1−P(ill| j)

PO(ill) =k

∑j=1

PO( j)β j (2.20)

where β j = P(ill| j)/(1−P(ill| j)) represents the odds of a death due to causej being classified as ill-defined. Expression 2.20 illustrates a regression equationrelating observed values of P(ill) and of probabilities of death due to each cause,PO( j) for j = 1, ...k. We can assume that the β j’s are functions of (a) country, (b)time, and then convert the expression into a GLS model that can be estimated in apooled cross section time series with random or fixed effects. The estimation needsto be done with one constraint, namely, constant =0. Also, one needs to select care-fully the groups of causes to include. Perhaps instead of country we can includedummies for groups of countries and instead of years we can use dummies for peri-ods. Finally, we could also enter a term associated with total levels of mortality, thatis:


β j = β0 j +∑∀r

βr jCr +∑∀s

βs jTs +θ j ln(DT )

where C and T are dummies for group of countries and years, and DT is the totalmortality rate. Then the regression equation will look as follows:

PO(ill) =k

∑j=1

β0 jPO( j)+k

∑j=1

∑∀r

βr jPO( j)Cr +k

∑j=1

∑∀s

βs jPO( j)Ts +k

∑j=1

θ jPO( j) ln(DT )

Once we have estimates of β j, say β j, we estimate the fraction of deaths due tocause j that are ill-defined using a 4 step process: (1) estimate P(ill| j), (2) computePT ( j), (3) use (1) and (2) to estimate P(ill∩ j), and (4) from here the final pro-portions can be inferred as the ratio P(ill∩ j)/PO(ill). More specifically, we followthese steps:

1. Probability that deaths due to cause j will be classified as ill-defined category,P(ill| j). These values are estimated as P(ill| j) = exp(β j)/(1+ exp(β j))

2. True probability of cause of death j, PT ( j). From 2.19, this value is estimated as

PT ( j) =PO( j)

1− P(ill| j)=

PO( j)

1− P(ill| j)∗P(ill| j)P(ill| j)

=PO( j)∗ OddsP(ill| j)

=PO( j)∗ β jP(ill| j)

3. Probability of being ill-defined and cause j, P(ill∩ j). By definition, P(ill| j) =P(ill ∩ j)/PT ( j). It thus follows that P(ill ∩ j) can be estimated as P(ill∩ j) =P(ill| j)∗ PT ( j).

4. Fraction of deaths due to cause j that are ill-defined. These fractions are esti-mated as P( j|ill) = P(ill∩ j)/PO(ill).

Important caveat: because we will estimate the equation via GLS, the predictedvalues will not be equal to the observed values. In other words, the following in-equality will prevail:

P(ill) =k

∑j=0

PO( j)β j 6= PO(ill)

To adjust for this we simply normalized the estimated proportions so the vectorP( j|ill) has length 1. That is, we compute: P( j|ill)/∑kj=1

P( j|ill).


2.8.2 Empirical estimates

All models were estimated using group of causes, groups of countries, and timeperiods to increase the robustness in the estimates. Data was classified as shown inTable 2.15.

Table 2.15 Cause of death classification, country groups, and time periods for Cause-of-deathmodel fitting.

Group Causes of death

Cancers Neoplasms, Respiratory Neoplasms, Digestive Neoplasms, Breast NeoplasmsCardiovascular Circulatory Diseases, Heart, Hypertension, Cerebrovascular, ArteriosclerosisRespiratory Respiratory Diseases, Acute Upper Respiratory Infections, Influenza

Pneumonia, Acute Bronchitis, Chronic Bronchitis, Emphysema, AsthmaDigestive/diabetes Digestive Diseases, Cirrhosis, Ulcers, Diabetes MellitusInfections Infectious DiseasesAccidents Accidents, Homicides, SuicidesResidual All other causes

Time periods Years

1950-1969 1950-19691970-1989 1970-19891990-2010 1990-2010

Stage dem tran Countries

Late Dominican Republic, El Salvador, Guatemala, Honduras, NicaraguaPanama and Peru.

Intermediate Brazil, Colombia, Ecuador, Mexico, Paraguay, VenezuelaEarly Argentina, Chile, Costa Rica, Cuba, Uruguay

Note: Stage dem trans, stage of the demographic transition.

2.8.3 Statistical modeling

We fitted 3 models as follows:

1. Basic:

Milld = β0 jM j +∑∀k

βkCr ∗M j +∑∀m

βmTs ∗M j (2.21)

2. Basic+ln(CMRT ):

Milld = Basic+θ jM j ∗ ln(CMRT ) (2.22)

3. Saturated:

Milld = Basic+θ jM j ∗ ln(CMRT )+allinteractions (2.23)


where Milld and M j are mortality rates for ill-defined and for cause of death j, re-spectively; Cr and Ts are a set of dummy variables for country-region and time-period, respectively; and CMRT is the crude mortality rate at time T .

2.8.4 Results

The figure below shows R2 adjusted from the models shown above estimated sepa-rately by age and sex.

Age

R2

adju

sted

0.75

0.80

0.85

0.90

0.95

0−14 15−49 50−+

Female

0−14 15−49 50−+0.75

0.80

0.85

0.90

0.95Male

Models:BasicBasic+ln(CMR)Saturated

In order to improve model fitting, we estimated models of the form Basic+ln(CMRT ) shown in equation 2.22. Table 2.16 shows coefficient estimates. Thisleads to at least 80% variance explained among women at all ages and for men<50 (red bars in the above figure). Using these coefficient estimates we estimatedconditional probabilities and then estimated the fraction of deaths due to cause jthat are ill-defined by country-year-sex-age.

For comparison purposes, we also computed similar fractions using a standardapproach based on proportional allocation of ill-defined deaths according to the dis-tribution of well-defined causes of death. To simplify comparability between the twoapproaches, we plotted these proportions by cause of death, country, year and agefor males. Figures 2.8–2.14 show these comparisons for cancers, cardiovascular, res-piratory, digestive/diabetes, infections, accidents and residual causes, respectively.

The general pattern among males indicates that our approach assigns a greaterfraction of ill-defined deaths in most causes of death, but this pattern varies by age.At ages < 15 our approach assigns more ill-defined deaths to cancers, CVD, and res-piratory diseases, but at ages 15-49 we assign greater fraction to digestive/diabetesconditions and at ages >50, our approach assigns more deaths to respiratory dis-eases and about the same fraction as the traditional approach to CVD and residualcauses.


Table 2.16 Cause-of-death coefficient estimates from Model 2.22 by age and sex.

Men WomenAge 0-14 Age 15-49 Age 50+ Age 0-14 Age 15-49 Age 50+

variable coef p-value coef p-value coef p-value coef p-value coef p-value coef p-value

cancer 40.02 0.29 2.12 0.06 -3.50 0.10 55.69 0.13 -1.29 0.03 0.41 0.85cvd 12.76 0.20 0.83 0.26 2.02 0.01 23.35 0.02 7.69 0.00 1.15 0.09resp 3.70 0.00 -1.77 0.06 2.84 0.03 2.57 0.00 -0.98 0.27 6.39 0.00dig diab 0.38 0.96 1.57 0.01 -5.86 0.00 19.37 0.00 -2.09 0.01 -4.35 0.02infe -3.33 0.00 0.13 0.80 -10.88 0.00 -3.15 0.00 0.32 0.59 -20.08 0.00acc -1.41 0.80 0.01 0.94 0.10 0.97 -4.22 0.53 2.83 0.00 24.77 0.00res -0.97 0.00 -0.22 0.19 1.89 0.00 -1.99 0.00 0.01 0.96 -1.04 0.03time50Xcancer 3.68 0.57 -0.54 0.01 -1.95 0.00 8.68 0.20 0.65 0.00 -1.35 0.01time50Xcvd -7.30 0.01 0.07 0.65 -0.27 0.04 -8.32 0.00 -1.61 0.00 -0.17 0.19time50Xresp -2.43 0.00 -0.21 0.60 0.79 0.00 -1.18 0.00 -0.33 0.31 0.18 0.54time50Xdig diab 1.02 0.88 0.76 0.00 2.99 0.00 -17.01 0.01 1.43 0.00 1.26 0.02time50Xinfe 1.53 0.00 -0.21 0.29 0.10 0.88 1.22 0.00 -0.21 0.34 0.53 0.41time50Xacc 1.51 0.21 0.04 0.38 0.60 0.43 0.42 0.77 0.05 0.76 1.58 0.36time50Xres 0.19 0.02 0.34 0.00 1.84 0.00 0.38 0.00 0.38 0.00 2.61 0.00time70Xcancer 10.97 0.07 -0.70 0.00 -1.19 0.00 4.90 0.41 -0.10 0.46 -0.69 0.09time70Xcvd 0.79 0.74 0.19 0.12 -0.05 0.64 -1.60 0.51 -0.11 0.43 0.16 0.13time70Xresp -2.26 0.00 -0.44 0.28 0.24 0.22 -0.80 0.00 -0.56 0.07 -0.26 0.23time70Xdig diab 1.89 0.78 0.60 0.00 2.38 0.00 -16.13 0.01 0.50 0.15 0.79 0.13time70Xinfe 1.48 0.00 0.12 0.54 0.56 0.36 1.16 0.00 0.18 0.39 0.54 0.35time70Xacc -1.16 0.25 -0.03 0.46 -0.65 0.29 -0.75 0.49 -0.16 0.20 0.44 0.75time70Xres 0.02 0.76 0.07 0.09 0.07 0.28 0.03 0.68 0.12 0.03 0.04 0.51earlyXcancer -6.93 0.20 -0.54 0.04 1.93 0.00 -6.06 0.25 0.06 0.83 2.65 0.00earlyXcvd -0.35 0.90 0.07 0.75 -0.24 0.02 0.41 0.89 -0.39 0.34 -0.20 0.04earlyXresp 0.66 0.04 0.59 0.40 0.07 0.71 0.30 0.32 1.14 0.20 0.48 0.01earlyXdig diab -0.60 0.00 -0.77 0.23 -2.92 0.00 -0.48 0.00 -0.75 0.15 -2.77 0.00earlyXinfe -0.74 0.00 -0.20 0.51 -0.90 0.33 -0.78 0.00 -1.17 0.02 -3.45 0.00earlyXacc 0.44 0.71 -0.06 0.56 2.48 0.02 1.08 0.43 0.21 0.47 -0.39 0.77earlyXres 0.03 0.74 0.13 0.25 -0.49 0.00 0.14 0.08 -0.29 0.06 -0.03 0.72interXcancer 5.88 0.27 0.07 0.69 1.36 0.00 10.02 0.03 0.91 0.00 3.49 0.00interXcvd -4.92 0.02 0.07 0.43 0.14 0.08 -5.37 0.00 -1.04 0.00 -0.19 0.01interXresp -0.62 0.00 1.00 0.00 0.66 0.00 -0.89 0.00 0.96 0.00 2.26 0.00interXdig diab 0.42 0.00 -1.24 0.00 -1.13 0.00 0.62 0.00 -1.64 0.00 -2.14 0.00interXinfe 0.24 0.01 0.24 0.01 -1.80 0.00 0.51 0.00 0.15 0.14 -5.53 0.00interXacc -0.74 0.35 -0.09 0.00 -0.64 0.24 -1.40 0.13 0.20 0.03 -4.49 0.00interXres 0.01 0.81 0.09 0.00 -0.21 0.00 -0.07 0.06 -0.31 0.00 0.02 0.74ln cmrXcancer 7.80 0.29 0.27 0.21 -0.54 0.18 11.07 0.10 -0.22 0.03 0.57 0.14ln cmrXcvd 2.85 0.12 0.14 0.32 0.31 0.02 4.17 0.01 1.27 0.00 0.15 0.21ln cmrXresp 0.31 0.00 -0.28 0.10 0.58 0.03 0.33 0.00 -0.02 0.89 1.30 0.00ln cmrXdig diab 0.15 0.09 0.26 0.02 -1.00 0.00 0.36 0.00 -0.50 0.00 -1.11 0.00ln cmrXinfe -0.51 0.00 -0.04 0.64 -2.54 0.00 -0.51 0.00 -0.04 0.72 -4.75 0.00ln cmrXacc -0.21 0.84 -0.02 0.28 -0.04 0.94 -0.73 0.54 0.53 0.00 4.23 0.00ln cmrXres -0.22 0.00 -0.07 0.03 0.27 0.00 -0.40 0.00 -0.06 0.15 -0.23 0.01

Note: resp, respiratory; dig diab, digestive/diabetes; infe, infectious diseases; acc, accidents; res,residual; time50, period 1950-1969; time70, period 1970-1989; inter, intermediate demographictransition.Source: Data from LAMBdA.


Fig.

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Proportion of Ill−defined deaths due to Cancer 0.2

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Gua

1−4

Gua

5−9

Gua

10−

14

Gua

15−

19

Gua

20−

24

Gua

25−

29

Gua

30−

34

Gua

35−

39

Gua

40−

44

Gua

45−

49

Gua

50−

54

Gua

55−

59

Gua

60−

64

Gua

65−

69

Gua

70−

74

Gua

75−

79

Gua

80−

84

Gua

0.2

0.4

0.6

0.8

185

−+

Gua

0.2

0.4

0.6

0.81

0−1

ElSal

1−4

ElSal

5−9

ElSal

10−

14

ElSal

15−

19

ElSal

20−

24

ElSal

25−

29

ElSal

30−

34

ElSal

35−

39

ElSal

40−

44

ElSal

45−

49

ElSal

50−

54

ElSal

55−

59

ElSal

60−

64

ElSal

65−

69

ElSal

70−

74

ElSal

75−

79

ElSal

80−

84

ElSal

0.2

0.4

0.6

0.8

185

−+

ElSal

0.2

0.4

0.6

0.81

0−1

DomRep

1−4

DomRep

5−9

DomRep

10−

14

DomRep

15−

19

DomRep

20−

24

DomRep

25−

29

DomRep

30−

34

DomRep

35−

39

DomRep

40−

44

DomRep

45−

49

DomRep

50−

54

DomRep

55−

59

DomRep

60−

64

DomRep

65−

69

DomRep

70−

74

DomRep

75−

79

DomRep

80−

84

DomRep

0.2

0.4

0.6

0.8

185

−+

DomRep


Fig.

2.9

Est

imat

edpr

opor

tion

ofill

-defi

ned

deat

hsdu

eto

card

iova

scul

ardi

seas

esfo

rM

ales

byco

untr

y,ag

e,an

dye

ar.

Red

valu

esar

ees

timat

edus

ing

our

appr

oach

and

blac

kva

lues

com

efr

oma

prop

ortio

nald

istr

ibut

ion

Year

Proportion of Ill−defined deaths due to Cardiovascular disease 0.2

0.4

0.6

0.8

19601970198019902000

0−1

Uru

19601970198019902000

1−4

Uru

19601970198019902000

5−9

Uru

19601970198019902000

10−

14

Uru

19601970198019902000

15−

19

Uru

19601970198019902000

20−

24

Uru

19601970198019902000

25−

29

Uru

19601970198019902000

30−

34

Uru

19601970198019902000

35−

39

Uru

19601970198019902000

40−

44

Uru

19601970198019902000

45−

49Uru

19601970198019902000

50−

54

Uru

19601970198019902000

55−

59

Uru

19601970198019902000

60−

64

Uru

19601970198019902000

65−

69

Uru

19601970198019902000

70−

74

Uru

19601970198019902000

75−

79

Uru

19601970198019902000

80−

84

Uru

19601970198019902000

0.2

0.4

0.6

0.8

85−

+

Uru

0.2

0.4

0.6

0.8

0−1

CRica

1−4

CRica

5−9

CRica

10−

14

CRica

15−

19

CRica

20−

24

CRica

25−

29

CRica

30−

34

CRica

35−

39

CRica

40−

44CRica

45−

49

CRica

50−

54

CRica

55−

59

CRica

60−

64

CRica

65−

69

CRica

70−

74

CRica

75−

79

CRica

80−

84

CRica

0.2

0.4

0.6

0.8

85−

+

CRica

0.2

0.4

0.6

0.8

0−1

Chi

1−4

Chi

5−9

Chi

10−

14

Chi

15−

19

Chi

20−

24

Chi

25−

29

Chi

30−

34

Chi

35−

39Chi

40−

44

Chi

45−

49

Chi

50−

54

Chi

55−

59

Chi

60−

64

Chi

65−

69

Chi

70−

74

Chi

75−

79

Chi

80−

84

Chi

0.2

0.4

0.6

0.8

85−

+

Chi

0.2

0.4

0.6

0.8

0−1

Arg

1−4

Arg

5−9

Arg

10−

14

Arg

15−

19

Arg

20−

24

Arg

25−

29

Arg

30−

34

Arg

35−

39

Arg

40−

44

Arg

45−

49

Arg

50−

54

Arg

55−

59

Arg

60−

64

Arg

65−

69

Arg

70−

74

Arg

75−

79

Arg

80−

84

Arg

0.2

0.4

0.6

0.8

85−

+

Arg

0.2

0.4

0.6

0.8

0−1

Ven

1−4

Ven

5−9

Ven

10−

14

Ven

15−

19

Ven

20−

24

Ven

25−

29

Ven

30−

34

Ven

35−

39

Ven

40−

44

Ven

45−

49

Ven

50−

54

Ven

55−

59

Ven

60−

64

Ven

65−

69

Ven

70−

74

Ven

75−

79

Ven

80−

84

Ven

0.2

0.4

0.6

0.8

85−

+

Ven

0.2

0.4

0.6

0.8

0−1

Par

1−4

Par

5−9

Par

10−

14

Par

15−

19

Par

20−

24

Par

25−

29

Par

30−

34

Par

35−

39

Par

40−

44

Par

45−

49

Par

50−

54

Par

55−

59

Par

60−

64

Par

65−

69

Par

70−

74

Par

75−

79

Par

80−

84

Par

0.2

0.4

0.6

0.8

85−

+

Par

0.2

0.4

0.6

0.8

0−1

Mex

1−4

Mex

5−9

Mex

10−

14

Mex

15−

19

Mex

20−

24

Mex

25−

29

Mex

30−

34

Mex

35−

39

Mex

40−

44

Mex

45−

49

Mex

50−

54

Mex

55−

59

Mex

60−

64

Mex

65−

69

Mex

70−

74

Mex

75−

79

Mex

80−

84

Mex

0.2

0.4

0.6

0.8

85−

+

Mex

0.2

0.4

0.6

0.8

0−1

Ecu

1−4

Ecu

5−9

Ecu

10−

14

Ecu

15−

19Ecu

20−

24

Ecu

25−

29

Ecu

30−

34

Ecu

35−

39

Ecu

40−

44

Ecu

45−

49

Ecu

50−

54

Ecu

55−

59

Ecu

60−

64

Ecu

65−

69

Ecu

70−

74

Ecu

75−

79

Ecu

80−

84

Ecu

0.2

0.4

0.6

0.8

85−

+

Ecu

0.2

0.4

0.6

0.8

0−1

Col

1−4

Col

5−9

Col

10−

14Col

15−

19

Col

20−

24

Col

25−

29

Col

30−

34

Col

35−

39

Col

40−

44

Col

45−

49

Col

50−

54

Col

55−

59

Col

60−

64

Col

65−

69

Col

70−

74

Col

75−

79

Col

80−

84

Col

0.2

0.4

0.6

0.8

85−

+

Col

0.2

0.4

0.6

0.8

0−1

Bra

1−4

Bra

5−9

Bra10

−14

Bra

15−

19

Bra

20−

24

Bra

25−

29

Bra

30−

34

Bra

35−

39

Bra

40−

44

Bra

45−

49

Bra

50−

54

Bra

55−

59

Bra

60−

64

Bra

65−

69

Bra

70−

74

Bra

75−

79

Bra

80−

84

Bra

0.2

0.4

0.6

0.8

85−

+

Bra

0.2

0.4

0.6

0.8

0−1

Per

1−4

Per5−

9

Per

10−

14

Per

15−

19

Per

20−

24

Per

25−

29

Per

30−

34

Per

35−

39

Per

40−

44

Per

45−

49

Per

50−

54

Per

55−

59

Per

60−

64

Per

65−

69

Per

70−

74

Per

75−

79

Per

80−

84

Per

0.2

0.4

0.6

0.8

85−

+

Per

0.2

0.4

0.6

0.8

0−1

Pan1−

4

Pan

5−9

Pan

10−

14

Pan

15−

19

Pan

20−

24

Pan

25−

29

Pan

30−

34

Pan

35−

39

Pan

40−

44

Pan

45−

49

Pan

50−

54

Pan

55−

59

Pan

60−

64

Pan

65−

69

Pan

70−

74

Pan

75−

79

Pan

80−

84

Pan

0.2

0.4

0.6

0.8

85−

+

Pan

0.2

0.4

0.6

0.8

0−1

Nic

1−4

Nic

5−9

Nic

10−

14

Nic

15−

19

Nic

20−

24

Nic

25−

29

Nic

30−

34

Nic

35−

39

Nic

40−

44

Nic

45−

49

Nic

50−

54

Nic

55−

59

Nic

60−

64

Nic

65−

69

Nic

70−

74

Nic

75−

79

Nic

80−

84

Nic

0.2

0.4

0.6

0.8

85−

+

Nic

0.2

0.4

0.6

0.8

0−1

Hon

1−4

Hon

5−9

Hon

10−

14

Hon

15−

19

Hon

20−

24

Hon

25−

29

Hon

30−

34

Hon

35−

39

Hon

40−

44

Hon

45−

49

Hon

50−

54

Hon

55−

59

Hon

60−

64

Hon

65−

69

Hon

70−

74

Hon

75−

79

Hon

80−

84

Hon

0.2

0.4

0.6

0.8

85−

+

Hon

0.2

0.4

0.6

0.8

0−1

Gua

1−4

Gua

5−9

Gua

10−

14

Gua

15−

19

Gua

20−

24

Gua

25−

29

Gua

30−

34

Gua

35−

39

Gua

40−

44

Gua

45−

49

Gua

50−

54

Gua

55−

59

Gua

60−

64

Gua

65−

69

Gua

70−

74

Gua

75−

79

Gua

80−

84

Gua

0.2

0.4

0.6

0.8

85−

+

Gua

0.2

0.4

0.6

0.8

0−1

ElSal

1−4

ElSal

5−9

ElSal

10−

14

ElSal

15−

19

ElSal

20−

24

ElSal

25−

29

ElSal

30−

34

ElSal

35−

39

ElSal

40−

44

ElSal

45−

49

ElSal

50−

54

ElSal

55−

59

ElSal

60−

64

ElSal

65−

69

ElSal

70−

74

ElSal

75−

79

ElSal

80−

84

ElSal

0.2

0.4

0.6

0.8

85−

+

ElSal

0.2

0.4

0.6

0.8

0−1

DomRep

1−4

DomRep

5−9

DomRep

10−

14

DomRep

15−

19

DomRep

20−

24

DomRep

25−

29

DomRep

30−

34

DomRep

35−

39

DomRep

40−

44

DomRep

45−

49

DomRep

50−

54

DomRep

55−

59

DomRep

60−

64

DomRep

65−

69

DomRep

70−

74

DomRep

75−

79

DomRep

80−

84

DomRep

0.2

0.4

0.6

0.8

85−

+

DomRep


Fig.

2.10

Est

imat

edpr

opor

tion

ofill

-defi

ned

deat

hsdu

eto

resp

irat

ory

dise

ases

forM

ales

byco

untr

y,ag

e,an

dye

ar.R

edva

lues

are

estim

ated

usin

gou

rapp

roac

han

dbl

ack

valu

esco

me

from

apr

opor

tiona

ldis

trib

utio

n

Year

Proportion of Ill−defined deaths due to Respiratory diseases 0.2

0.4

0.6

0.81

19601970198019902000

0−1

Uru

19601970198019902000

1−4

Uru

19601970198019902000

5−9

Uru

19601970198019902000

10−

14

Uru

19601970198019902000

15−

19

Uru

19601970198019902000

20−

24

Uru

19601970198019902000

25−

29

Uru

19601970198019902000

30−

34

Uru

19601970198019902000

35−

39

Uru

19601970198019902000

40−

44

Uru

19601970198019902000

45−

49Uru

19601970198019902000

50−

54

Uru

19601970198019902000

55−

59

Uru

19601970198019902000

60−

64

Uru

19601970198019902000

65−

69

Uru

19601970198019902000

70−

74

Uru

19601970198019902000

75−

79

Uru

19601970198019902000

80−

84

Uru

19601970198019902000

0.2

0.4

0.6

0.8

185

−+

Uru

0.2

0.4

0.6

0.81

0−1

CRica

1−4

CRica

5−9

CRica

10−

14

CRica

15−

19

CRica

20−

24

CRica

25−

29

CRica

30−

34

CRica

35−

39

CRica

40−

44CRica

45−

49

CRica

50−

54

CRica

55−

59

CRica

60−

64

CRica

65−

69

CRica

70−

74

CRica

75−

79

CRica

80−

84

CRica

0.2

0.4

0.6

0.8

185

−+

CRica

0.2

0.4

0.6

0.81

0−1

Chi

1−4

Chi

5−9

Chi

10−

14

Chi

15−

19

Chi

20−

24

Chi

25−

29

Chi

30−

34

Chi

35−

39Chi

40−

44

Chi

45−

49

Chi

50−

54

Chi

55−

59

Chi

60−

64

Chi

65−

69

Chi

70−

74

Chi

75−

79

Chi

80−

84

Chi

0.2

0.4

0.6

0.8

185

−+

Chi

0.2

0.4

0.6

0.81

0−1

Arg

1−4

Arg

5−9

Arg

10−

14

Arg

15−

19

Arg

20−

24

Arg

25−

29

Arg

30−

34

Arg

35−

39

Arg

40−

44

Arg

45−

49

Arg

50−

54

Arg

55−

59

Arg

60−

64

Arg

65−

69

Arg

70−

74

Arg

75−

79

Arg

80−

84

Arg

0.2

0.4

0.6

0.8

185

−+

Arg

0.2

0.4

0.6

0.81

0−1

Ven

1−4

Ven

5−9

Ven

10−

14

Ven

15−

19

Ven

20−

24

Ven

25−

29

Ven

30−

34

Ven

35−

39

Ven

40−

44

Ven

45−

49

Ven

50−

54

Ven

55−

59

Ven

60−

64

Ven

65−

69

Ven

70−

74

Ven

75−

79

Ven

80−

84

Ven

0.2

0.4

0.6

0.8

185

−+

Ven

0.2

0.4

0.6

0.81

0−1

Par

1−4

Par

5−9

Par

10−

14

Par

15−

19

Par

20−

24

Par

25−

29

Par

30−

34

Par

35−

39

Par

40−

44

Par

45−

49

Par

50−

54

Par

55−

59

Par

60−

64

Par

65−

69

Par

70−

74

Par

75−

79

Par

80−

84

Par

0.2

0.4

0.6

0.8

185

−+

Par

0.2

0.4

0.6

0.81

0−1

Mex

1−4

Mex

5−9

Mex

10−

14

Mex

15−

19

Mex

20−

24

Mex

25−

29

Mex

30−

34

Mex

35−

39

Mex

40−

44

Mex

45−

49

Mex

50−

54

Mex

55−

59

Mex

60−

64

Mex

65−

69

Mex

70−

74

Mex

75−

79

Mex

80−

84

Mex

0.2

0.4

0.6

0.8

185

−+

Mex

0.2

0.4

0.6

0.81

0−1

Ecu

1−4

Ecu

5−9

Ecu

10−

14

Ecu

15−

19Ecu

20−

24

Ecu

25−

29

Ecu

30−

34

Ecu

35−

39

Ecu

40−

44

Ecu

45−

49

Ecu

50−

54

Ecu

55−

59

Ecu

60−

64

Ecu

65−

69

Ecu

70−

74

Ecu

75−

79

Ecu

80−

84

Ecu

0.2

0.4

0.6

0.8

185

−+

Ecu

0.2

0.4

0.6

0.81

0−1

Col

1−4

Col

5−9

Col

10−

14Col

15−

19

Col

20−

24

Col

25−

29

Col

30−

34

Col

35−

39

Col

40−

44

Col

45−

49

Col

50−

54

Col

55−

59

Col

60−

64

Col

65−

69

Col

70−

74

Col

75−

79

Col

80−

84

Col

0.2

0.4

0.6

0.8

185

−+

Col

0.2

0.4

0.6

0.81

0−1

Bra

1−4

Bra

5−9

Bra10

−14

Bra

15−

19

Bra

20−

24

Bra

25−

29

Bra

30−

34

Bra

35−

39

Bra

40−

44

Bra

45−

49

Bra

50−

54

Bra

55−

59

Bra

60−

64

Bra

65−

69

Bra

70−

74

Bra

75−

79

Bra

80−

84

Bra

0.2

0.4

0.6

0.8

185

−+

Bra

0.2

0.4

0.6

0.81

0−1

Per

1−4

Per5−

9

Per

10−

14

Per

15−

19

Per

20−

24

Per

25−

29

Per

30−

34

Per

35−

39

Per

40−

44

Per

45−

49

Per

50−

54

Per

55−

59

Per

60−

64

Per

65−

69

Per

70−

74

Per

75−

79

Per

80−

84

Per

0.2

0.4

0.6

0.8

185

−+

Per

0.2

0.4

0.6

0.81

0−1

Pan1−

4

Pan

5−9

Pan

10−

14

Pan

15−

19

Pan

20−

24

Pan

25−

29

Pan

30−

34

Pan

35−

39

Pan

40−

44

Pan

45−

49

Pan

50−

54

Pan

55−

59

Pan

60−

64

Pan

65−

69

Pan

70−

74

Pan

75−

79

Pan

80−

84

Pan

0.2

0.4

0.6

0.8

185

−+

Pan

0.2

0.4

0.6

0.81

0−1

Nic

1−4

Nic

5−9

Nic

10−

14

Nic

15−

19

Nic

20−

24

Nic

25−

29

Nic

30−

34

Nic

35−

39

Nic

40−

44

Nic

45−

49

Nic

50−

54

Nic

55−

59

Nic

60−

64

Nic

65−

69

Nic

70−

74

Nic

75−

79

Nic

80−

84

Nic

0.2

0.4

0.6

0.8

185

−+

Nic

0.2

0.4

0.6

0.81

0−1

Hon

1−4

Hon

5−9

Hon

10−

14

Hon

15−

19

Hon

20−

24

Hon

25−

29

Hon

30−

34

Hon

35−

39

Hon

40−

44

Hon

45−

49

Hon

50−

54

Hon

55−

59

Hon

60−

64

Hon

65−

69

Hon

70−

74

Hon

75−

79

Hon

80−

84

Hon

0.2

0.4

0.6

0.8

185

−+

Hon

0.2

0.4

0.6

0.81

0−1

Gua

1−4

Gua

5−9

Gua

10−

14

Gua

15−

19

Gua

20−

24

Gua

25−

29

Gua

30−

34

Gua

35−

39

Gua

40−

44

Gua

45−

49

Gua

50−

54

Gua

55−

59

Gua

60−

64

Gua

65−

69

Gua

70−

74

Gua

75−

79

Gua

80−

84

Gua

0.2

0.4

0.6

0.8

185

−+

Gua

0.2

0.4

0.6

0.81

0−1

ElSal

1−4

ElSal

5−9

ElSal

10−

14

ElSal

15−

19

ElSal

20−

24

ElSal

25−

29

ElSal

30−

34

ElSal

35−

39

ElSal

40−

44

ElSal

45−

49

ElSal

50−

54

ElSal

55−

59

ElSal

60−

64

ElSal

65−

69

ElSal

70−

74

ElSal

75−

79

ElSal

80−

84

ElSal

0.2

0.4

0.6

0.8

185

−+

ElSal

0.2

0.4

0.6

0.81

0−1

DomRep

1−4

DomRep

5−9

DomRep

10−

14

DomRep

15−

19

DomRep

20−

24

DomRep

25−

29

DomRep

30−

34

DomRep

35−

39

DomRep

40−

44

DomRep

45−

49

DomRep

50−

54

DomRep

55−

59

DomRep

60−

64

DomRep

65−

69

DomRep

70−

74

DomRep

75−

79

DomRep

80−

84

DomRep

0.2

0.4

0.6

0.8

185

−+

DomRep


Fig.

2.11

Est

imat

edpr

opor

tion

ofill

-defi

ned

deat

hsdu

eto

dige

stiv

e/di

abet

esdi

seas

esfo

rM

ales

byco

untr

y,ag

e,an

dye

ar.R

edva

lues

are

estim

ated

usin

gou

rap

proa

chan

dbl

ack

valu

esco

me

from

apr

opor

tiona

ldis

trib

utio

n

Year

Proportion of Ill−defined deaths due to Digestive/Diabetes diseases 0.2

0.4

0.6

0.81

19601970198019902000

0−1

Uru

19601970198019902000

1−4

Uru

19601970198019902000

5−9

Uru

19601970198019902000

10−

14

Uru

19601970198019902000

15−

19

Uru

19601970198019902000

20−

24

Uru

19601970198019902000

25−

29

Uru

19601970198019902000

30−

34

Uru

19601970198019902000

35−

39

Uru

19601970198019902000

40−

44

Uru

19601970198019902000

45−

49Uru

19601970198019902000

50−

54

Uru

19601970198019902000

55−

59

Uru

19601970198019902000

60−

64

Uru

19601970198019902000

65−

69

Uru

19601970198019902000

70−

74

Uru

19601970198019902000

75−

79

Uru

19601970198019902000

80−

84

Uru

19601970198019902000

0.2

0.4

0.6

0.8

185

−+

Uru

0.2

0.4

0.6

0.81

0−1

CRica

1−4

CRica

5−9

CRica

10−

14

CRica

15−

19

CRica

20−

24

CRica

25−

29

CRica

30−

34

CRica

35−

39

CRica

40−

44CRica

45−

49

CRica

50−

54

CRica

55−

59

CRica

60−

64

CRica

65−

69

CRica

70−

74

CRica

75−

79

CRica

80−

84

CRica

0.2

0.4

0.6

0.8

185

−+

CRica

0.2

0.4

0.6

0.81

0−1

Chi

1−4

Chi

5−9

Chi

10−

14

Chi

15−

19

Chi

20−

24

Chi

25−

29

Chi

30−

34

Chi

35−

39Chi

40−

44

Chi

45−

49

Chi

50−

54

Chi

55−

59

Chi

60−

64

Chi

65−

69

Chi

70−

74

Chi

75−

79

Chi

80−

84

Chi

0.2

0.4

0.6

0.8

185

−+

Chi

0.2

0.4

0.6

0.81

0−1

Arg

1−4

Arg

5−9

Arg

10−

14

Arg

15−

19

Arg

20−

24

Arg

25−

29

Arg

30−

34

Arg

35−

39

Arg

40−

44

Arg

45−

49

Arg

50−

54

Arg

55−

59

Arg

60−

64

Arg

65−

69

Arg

70−

74

Arg

75−

79

Arg

80−

84

Arg

0.2

0.4

0.6

0.8

185

−+

Arg

0.2

0.4

0.6

0.81

0−1

Ven

1−4

Ven

5−9

Ven

10−

14

Ven

15−

19

Ven

20−

24

Ven

25−

29

Ven

30−

34

Ven

35−

39

Ven

40−

44

Ven

45−

49

Ven

50−

54

Ven

55−

59

Ven

60−

64

Ven

65−

69

Ven

70−

74

Ven

75−

79

Ven

80−

84

Ven

0.2

0.4

0.6

0.8

185

−+

Ven

0.2

0.4

0.6

0.81

0−1

Par

1−4

Par

5−9

Par

10−

14

Par

15−

19

Par

20−

24

Par

25−

29

Par

30−

34

Par

35−

39

Par

40−

44

Par

45−

49

Par

50−

54

Par

55−

59

Par

60−

64

Par

65−

69

Par

70−

74

Par

75−

79

Par

80−

84

Par

0.2

0.4

0.6

0.8

185

−+

Par

0.2

0.4

0.6

0.81

0−1

Mex

1−4

Mex

5−9

Mex

10−

14

Mex

15−

19

Mex

20−

24

Mex

25−

29

Mex

30−

34

Mex

35−

39

Mex

40−

44

Mex

45−

49

Mex

50−

54

Mex

55−

59

Mex

60−

64

Mex

65−

69

Mex

70−

74

Mex

75−

79

Mex

80−

84

Mex

0.2

0.4

0.6

0.8

185

−+

Mex

0.2

0.4

0.6

0.81

0−1

Ecu

1−4

Ecu

5−9

Ecu

10−

14

Ecu

15−

19Ecu

20−

24

Ecu

25−

29

Ecu

30−

34

Ecu

35−

39

Ecu

40−

44

Ecu

45−

49

Ecu

50−

54

Ecu

55−

59

Ecu

60−

64

Ecu

65−

69

Ecu

70−

74

Ecu

75−

79

Ecu

80−

84

Ecu

0.2

0.4

0.6

0.8

185

−+

Ecu

0.2

0.4

0.6

0.81

0−1

Col

1−4

Col

5−9

Col

10−

14Col

15−

19

Col

20−

24

Col

25−

29

Col

30−

34

Col

35−

39

Col

40−

44

Col

45−

49

Col

50−

54

Col

55−

59

Col

60−

64

Col

65−

69

Col

70−

74

Col

75−

79

Col

80−

84

Col

0.2

0.4

0.6

0.8

185

−+

Col

0.2

0.4

0.6

0.81

0−1

Bra

1−4

Bra

5−9

Bra10

−14

Bra

15−

19

Bra

20−

24

Bra

25−

29

Bra

30−

34

Bra

35−

39

Bra

40−

44

Bra

45−

49

Bra

50−

54

Bra

55−

59

Bra

60−

64

Bra

65−

69

Bra

70−

74

Bra

75−

79

Bra

80−

84

Bra

0.2

0.4

0.6

0.8

185

−+

Bra

0.2

0.4

0.6

0.81

0−1

Per

1−4

Per5−

9

Per

10−

14

Per

15−

19

Per

20−

24

Per

25−

29

Per

30−

34

Per

35−

39

Per

40−

44

Per

45−

49

Per

50−

54

Per

55−

59

Per

60−

64

Per

65−

69

Per

70−

74

Per

75−

79

Per

80−

84

Per

0.2

0.4

0.6

0.8

185

−+

Per

0.2

0.4

0.6

0.81

0−1

Pan1−

4

Pan

5−9

Pan

10−

14

Pan

15−

19

Pan

20−

24

Pan

25−

29

Pan

30−

34

Pan

35−

39

Pan

40−

44

Pan

45−

49

Pan

50−

54

Pan

55−

59

Pan

60−

64

Pan

65−

69

Pan

70−

74

Pan

75−

79

Pan

80−

84

Pan

0.2

0.4

0.6

0.8

185

−+

Pan

0.2

0.4

0.6

0.81

0−1

Nic

1−4

Nic

5−9

Nic

10−

14

Nic

15−

19

Nic

20−

24

Nic

25−

29

Nic

30−

34

Nic

35−

39

Nic

40−

44

Nic

45−

49

Nic

50−

54

Nic

55−

59

Nic

60−

64

Nic

65−

69

Nic

70−

74

Nic

75−

79

Nic

80−

84

Nic

0.2

0.4

0.6

0.8

185

−+

Nic

0.2

0.4

0.6

0.81

0−1

Hon

1−4

Hon

5−9

Hon

10−

14

Hon

15−

19

Hon

20−

24

Hon

25−

29

Hon

30−

34

Hon

35−

39

Hon

40−

44

Hon

45−

49

Hon

50−

54

Hon

55−

59

Hon

60−

64

Hon

65−

69

Hon

70−

74

Hon

75−

79

Hon

80−

84

Hon

0.2

0.4

0.6

0.8

185

−+

Hon

0.2

0.4

0.6

0.81

0−1

Gua

1−4

Gua

5−9

Gua

10−

14

Gua

15−

19

Gua

20−

24

Gua

25−

29

Gua

30−

34

Gua

35−

39

Gua

40−

44

Gua

45−

49

Gua

50−

54

Gua

55−

59

Gua

60−

64

Gua

65−

69

Gua

70−

74

Gua

75−

79

Gua

80−

84

Gua

0.2

0.4

0.6

0.8

185

−+

Gua

0.2

0.4

0.6

0.81

0−1

ElSal

1−4

ElSal

5−9

ElSal

10−

14

ElSal

15−

19

ElSal

20−

24

ElSal

25−

29

ElSal

30−

34

ElSal

35−

39

ElSal

40−

44

ElSal

45−

49

ElSal

50−

54

ElSal

55−

59

ElSal

60−

64

ElSal

65−

69

ElSal

70−

74

ElSal

75−

79

ElSal

80−

84

ElSal

0.2

0.4

0.6

0.8

185

−+

ElSal

0.2

0.4

0.6

0.81

0−1

DomRep

1−4

DomRep

5−9

DomRep

10−

14

DomRep

15−

19

DomRep

20−

24

DomRep

25−

29

DomRep

30−

34

DomRep

35−

39

DomRep

40−

44

DomRep

45−

49

DomRep

50−

54

DomRep

55−

59

DomRep

60−

64

DomRep

65−

69

DomRep

70−

74

DomRep

75−

79

DomRep

80−

84

DomRep

0.2

0.4

0.6

0.8

185

−+

DomRep


Fig.

2.12

Est

imat

edpr

opor

tion

ofill

-defi

ned

deat

hsdu

eto

infe

ctio

usdi

seas

esfo

rMal

esby

coun

try,

age,

and

year

.Red

valu

esar

ees

timat

edus

ing

oura

ppro

ach

and

blac

kva

lues

com

efr

oma

prop

ortio

nald

istr

ibut

ion

Year

Proportion of Ill−defined deaths due to Infectious diseases 0.2

0.4

0.6

0.8

19601970198019902000

0−1

Uru

19601970198019902000

1−4

Uru

19601970198019902000

5−9

Uru

19601970198019902000

10−

14

Uru

19601970198019902000

15−

19

Uru

19601970198019902000

20−

24

Uru

19601970198019902000

25−

29

Uru

19601970198019902000

30−

34

Uru

19601970198019902000

35−

39

Uru

19601970198019902000

40−

44

Uru

19601970198019902000

45−

49Uru

19601970198019902000

50−

54

Uru

19601970198019902000

55−

59

Uru

19601970198019902000

60−

64

Uru

19601970198019902000

65−

69

Uru

19601970198019902000

70−

74

Uru

19601970198019902000

75−

79

Uru

19601970198019902000

80−

84

Uru

19601970198019902000

0.2

0.4

0.6

0.8

85−

+

Uru

0.2

0.4

0.6

0.8

0−1

CRica

1−4

CRica

5−9

CRica

10−

14

CRica

15−

19

CRica

20−

24

CRica

25−

29

CRica

30−

34

CRica

35−

39

CRica

40−

44CRica

45−

49

CRica

50−

54

CRica

55−

59

CRica

60−

64

CRica

65−

69

CRica

70−

74

CRica

75−

79

CRica

80−

84

CRica

0.2

0.4

0.6

0.8

85−

+

CRica

0.2

0.4

0.6

0.8

0−1

Chi

1−4

Chi

5−9

Chi

10−

14

Chi

15−

19

Chi

20−

24

Chi

25−

29

Chi

30−

34

Chi

35−

39Chi

40−

44

Chi

45−

49

Chi

50−

54

Chi

55−

59

Chi

60−

64

Chi

65−

69

Chi

70−

74

Chi

75−

79

Chi

80−

84

Chi

0.2

0.4

0.6

0.8

85−

+

Chi

0.2

0.4

0.6

0.8

0−1

Arg

1−4

Arg

5−9

Arg

10−

14

Arg

15−

19

Arg

20−

24

Arg

25−

29

Arg

30−

34

Arg

35−

39

Arg

40−

44

Arg

45−

49

Arg

50−

54

Arg

55−

59

Arg

60−

64

Arg

65−

69

Arg

70−

74

Arg

75−

79

Arg

80−

84

Arg

0.2

0.4

0.6

0.8

85−

+

Arg

0.2

0.4

0.6

0.8

0−1

Ven

1−4

Ven

5−9

Ven

10−

14

Ven

15−

19

Ven

20−

24

Ven

25−

29

Ven

30−

34

Ven

35−

39

Ven

40−

44

Ven

45−

49

Ven

50−

54

Ven

55−

59

Ven

60−

64

Ven

65−

69

Ven

70−

74

Ven

75−

79

Ven

80−

84

Ven

0.2

0.4

0.6

0.8

85−

+

Ven

0.2

0.4

0.6

0.8

0−1

Par

1−4

Par

5−9

Par

10−

14

Par

15−

19

Par

20−

24

Par

25−

29

Par

30−

34

Par

35−

39

Par

40−

44

Par

45−

49

Par

50−

54

Par

55−

59

Par

60−

64

Par

65−

69

Par

70−

74

Par

75−

79

Par

80−

84

Par

0.2

0.4

0.6

0.8

85−

+

Par

0.2

0.4

0.6

0.8

0−1

Mex

1−4

Mex

5−9

Mex

10−

14

Mex

15−

19

Mex

20−

24

Mex

25−

29

Mex

30−

34

Mex

35−

39

Mex

40−

44

Mex

45−

49

Mex

50−

54

Mex

55−

59

Mex

60−

64

Mex

65−

69

Mex

70−

74

Mex

75−

79

Mex

80−

84

Mex

0.2

0.4

0.6

0.8

85−

+

Mex

0.2

0.4

0.6

0.8

0−1

Ecu

1−4

Ecu

5−9

Ecu

10−

14

Ecu

15−

19Ecu

20−

24

Ecu

25−

29

Ecu

30−

34

Ecu

35−

39

Ecu

40−

44

Ecu

45−

49

Ecu

50−

54

Ecu

55−

59

Ecu

60−

64

Ecu

65−

69

Ecu

70−

74

Ecu

75−

79

Ecu

80−

84

Ecu

0.2

0.4

0.6

0.8

85−

+

Ecu

0.2

0.4

0.6

0.8

0−1

Col

1−4

Col

5−9

Col

10−

14Col

15−

19

Col

20−

24

Col

25−

29

Col

30−

34

Col

35−

39

Col

40−

44

Col

45−

49

Col

50−

54

Col

55−

59

Col

60−

64

Col

65−

69

Col

70−

74

Col

75−

79

Col

80−

84

Col

0.2

0.4

0.6

0.8

85−

+

Col

0.2

0.4

0.6

0.8

0−1

Bra

1−4

Bra

5−9

Bra10

−14

Bra

15−

19

Bra

20−

24

Bra

25−

29

Bra

30−

34

Bra

35−

39

Bra

40−

44

Bra

45−

49

Bra

50−

54

Bra

55−

59

Bra

60−

64

Bra

65−

69

Bra

70−

74

Bra

75−

79

Bra

80−

84

Bra

0.2

0.4

0.6

0.8

85−

+

Bra

0.2

0.4

0.6

0.8

0−1

Per

1−4

Per5−

9

Per

10−

14

Per

15−

19

Per

20−

24

Per

25−

29

Per

30−

34

Per

35−

39

Per

40−

44

Per

45−

49

Per

50−

54

Per

55−

59

Per

60−

64

Per

65−

69

Per

70−

74

Per

75−

79

Per

80−

84

Per

0.2

0.4

0.6

0.8

85−

+

Per

0.2

0.4

0.6

0.8

0−1

Pan1−

4

Pan

5−9

Pan

10−

14

Pan

15−

19

Pan

20−

24

Pan

25−

29

Pan

30−

34

Pan

35−

39

Pan

40−

44

Pan

45−

49

Pan

50−

54

Pan

55−

59

Pan

60−

64

Pan

65−

69

Pan

70−

74

Pan

75−

79

Pan

80−

84

Pan

0.2

0.4

0.6

0.8

85−

+

Pan

0.2

0.4

0.6

0.8

0−1

Nic

1−4

Nic

5−9

Nic

10−

14

Nic

15−

19

Nic

20−

24

Nic

25−

29

Nic

30−

34

Nic

35−

39

Nic

40−

44

Nic

45−

49

Nic

50−

54

Nic

55−

59

Nic

60−

64

Nic

65−

69

Nic

70−

74

Nic

75−

79

Nic

80−

84

Nic

0.2

0.4

0.6

0.8

85−

+

Nic

0.2

0.4

0.6

0.8

0−1

Hon

1−4

Hon

5−9

Hon

10−

14

Hon

15−

19

Hon

20−

24

Hon

25−

29

Hon

30−

34

Hon

35−

39

Hon

40−

44

Hon

45−

49

Hon

50−

54

Hon

55−

59

Hon

60−

64

Hon

65−

69

Hon

70−

74

Hon

75−

79

Hon

80−

84

Hon

0.2

0.4

0.6

0.8

85−

+

Hon

0.2

0.4

0.6

0.8

0−1

Gua

1−4

Gua

5−9

Gua

10−

14

Gua

15−

19

Gua

20−

24

Gua

25−

29

Gua

30−

34

Gua

35−

39

Gua

40−

44

Gua

45−

49

Gua

50−

54

Gua

55−

59

Gua

60−

64

Gua

65−

69

Gua

70−

74

Gua

75−

79

Gua

80−

84

Gua

0.2

0.4

0.6

0.8

85−

+

Gua

0.2

0.4

0.6

0.8

0−1

ElSal

1−4

ElSal

5−9

ElSal

10−

14

ElSal

15−

19

ElSal

20−

24

ElSal

25−

29

ElSal

30−

34

ElSal

35−

39

ElSal

40−

44

ElSal

45−

49

ElSal

50−

54

ElSal

55−

59

ElSal

60−

64

ElSal

65−

69

ElSal

70−

74

ElSal

75−

79

ElSal

80−

84

ElSal

0.2

0.4

0.6

0.8

85−

+

ElSal

0.2

0.4

0.6

0.8

0−1

DomRep

1−4

DomRep

5−9

DomRep

10−

14

DomRep

15−

19

DomRep

20−

24

DomRep

25−

29

DomRep

30−

34

DomRep

35−

39

DomRep

40−

44

DomRep

45−

49

DomRep

50−

54

DomRep

55−

59

DomRep

60−

64

DomRep

65−

69

DomRep

70−

74

DomRep

75−

79

DomRep

80−

84

DomRep

0.2

0.4

0.6

0.8

85−

+

DomRep


Fig.

2.13

Est

imat

edpr

opor

tion

ofill

-defi

ned

deat

hsdu

eto

acci

dent

sfo

rMal

esby

coun

try,

age,

and

year

.Red

valu

esar

ees

timat

edus

ing

oura

ppro

ach

and

blac

kva

lues

com

efr

oma

prop

ortio

nald

istr

ibut

ion

Year

Proportion of Ill−defined deaths due to Accidents 0.2

0.4

0.6

0.8

19601970198019902000

0−1

Uru

19601970198019902000

1−4

Uru

19601970198019902000

5−9

Uru

19601970198019902000

10−

14

Uru

19601970198019902000

15−

19

Uru

19601970198019902000

20−

24

Uru

19601970198019902000

25−

29

Uru

19601970198019902000

30−

34

Uru

19601970198019902000

35−

39

Uru

19601970198019902000

40−

44

Uru

19601970198019902000

45−

49Uru

19601970198019902000

50−

54

Uru

19601970198019902000

55−

59

Uru

19601970198019902000

60−

64

Uru

19601970198019902000

65−

69

Uru

19601970198019902000

70−

74

Uru

19601970198019902000

75−

79

Uru

19601970198019902000

80−

84

Uru

19601970198019902000

0.2

0.4

0.6

0.8

85−

+

Uru

0.2

0.4

0.6

0.8

0−1

CRica

1−4

CRica

5−9

CRica

10−

14

CRica

15−

19

CRica

20−

24

CRica

25−

29

CRica

30−

34

CRica

35−

39

CRica

40−

44CRica

45−

49

CRica

50−

54

CRica

55−

59

CRica

60−

64

CRica

65−

69

CRica

70−

74

CRica

75−

79

CRica

80−

84

CRica

0.2

0.4

0.6

0.8

85−

+

CRica

0.2

0.4

0.6

0.8

0−1

Chi

1−4

Chi

5−9

Chi

10−

14

Chi

15−

19

Chi

20−

24

Chi

25−

29

Chi

30−

34

Chi

35−

39Chi

40−

44

Chi

45−

49

Chi

50−

54

Chi

55−

59

Chi

60−

64

Chi

65−

69

Chi

70−

74

Chi

75−

79

Chi

80−

84

Chi

0.2

0.4

0.6

0.8

85−

+

Chi

0.2

0.4

0.6

0.8

0−1

Arg

1−4

Arg

5−9

Arg

10−

14

Arg

15−

19

Arg

20−

24

Arg

25−

29

Arg

30−

34

Arg

35−

39

Arg

40−

44

Arg

45−

49

Arg

50−

54

Arg

55−

59

Arg

60−

64

Arg

65−

69

Arg

70−

74

Arg

75−

79

Arg

80−

84

Arg

0.2

0.4

0.6

0.8

85−

+

Arg

0.2

0.4

0.6

0.8

0−1

Ven

1−4

Ven

5−9

Ven

10−

14

Ven

15−

19

Ven

20−

24

Ven

25−

29

Ven

30−

34

Ven

35−

39

Ven

40−

44

Ven

45−

49

Ven

50−

54

Ven

55−

59

Ven

60−

64

Ven

65−

69

Ven

70−

74

Ven

75−

79

Ven

80−

84

Ven

0.2

0.4

0.6

0.8

85−

+

Ven

0.2

0.4

0.6

0.8

0−1

Par

1−4

Par

5−9

Par

10−

14

Par

15−

19

Par

20−

24

Par

25−

29

Par

30−

34

Par

35−

39

Par

40−

44

Par

45−

49

Par

50−

54

Par

55−

59

Par

60−

64

Par

65−

69

Par

70−

74

Par

75−

79

Par

80−

84

Par

0.2

0.4

0.6

0.8

85−

+

Par

0.2

0.4

0.6

0.8

0−1

Mex

1−4

Mex

5−9

Mex

10−

14

Mex

15−

19

Mex

20−

24

Mex

25−

29

Mex

30−

34

Mex

35−

39

Mex

40−

44

Mex

45−

49

Mex

50−

54

Mex

55−

59

Mex

60−

64

Mex

65−

69

Mex

70−

74

Mex

75−

79

Mex

80−

84

Mex

0.2

0.4

0.6

0.8

85−

+

Mex

0.2

0.4

0.6

0.8

0−1

Ecu

1−4

Ecu

5−9

Ecu

10−

14

Ecu

15−

19Ecu

20−

24

Ecu

25−

29

Ecu

30−

34

Ecu

35−

39

Ecu

40−

44

Ecu

45−

49

Ecu

50−

54

Ecu

55−

59

Ecu

60−

64

Ecu

65−

69

Ecu

70−

74

Ecu

75−

79

Ecu

80−

84

Ecu

0.2

0.4

0.6

0.8

85−

+

Ecu

0.2

0.4

0.6

0.8

0−1

Col

1−4

Col

5−9

Col

10−

14Col

15−

19

Col

20−

24

Col

25−

29

Col

30−

34

Col

35−

39

Col

40−

44

Col

45−

49

Col

50−

54

Col

55−

59

Col

60−

64

Col

65−

69

Col

70−

74

Col

75−

79

Col

80−

84

Col

0.2

0.4

0.6

0.8

85−

+

Col

0.2

0.4

0.6

0.8

0−1

Bra

1−4

Bra

5−9

Bra10

−14

Bra

15−

19

Bra

20−

24

Bra

25−

29

Bra

30−

34

Bra

35−

39

Bra

40−

44

Bra

45−

49

Bra

50−

54

Bra

55−

59

Bra

60−

64

Bra

65−

69

Bra

70−

74

Bra

75−

79

Bra

80−

84

Bra

0.2

0.4

0.6

0.8

85−

+

Bra

0.2

0.4

0.6

0.8

0−1

Per

1−4

Per5−

9

Per

10−

14

Per

15−

19

Per

20−

24

Per

25−

29

Per

30−

34

Per

35−

39

Per

40−

44

Per

45−

49

Per

50−

54

Per

55−

59

Per

60−

64

Per

65−

69

Per

70−

74

Per

75−

79

Per

80−

84

Per

0.2

0.4

0.6

0.8

85−

+

Per

0.2

0.4

0.6

0.8

0−1

Pan1−

4

Pan

5−9

Pan

10−

14

Pan

15−

19

Pan

20−

24

Pan

25−

29

Pan

30−

34

Pan

35−

39

Pan

40−

44

Pan

45−

49

Pan

50−

54

Pan

55−

59

Pan

60−

64

Pan

65−

69

Pan

70−

74

Pan

75−

79

Pan

80−

84

Pan

0.2

0.4

0.6

0.8

85−

+

Pan

0.2

0.4

0.6

0.8

0−1

Nic

1−4

Nic

5−9

Nic

10−

14

Nic

15−

19

Nic

20−

24

Nic

25−

29

Nic

30−

34

Nic

35−

39

Nic

40−

44

Nic

45−

49

Nic

50−

54

Nic

55−

59

Nic

60−

64

Nic

65−

69

Nic

70−

74

Nic

75−

79

Nic

80−

84

Nic

0.2

0.4

0.6

0.8

85−

+

Nic

0.2

0.4

0.6

0.8

0−1

Hon

1−4

Hon

5−9

Hon

10−

14

Hon

15−

19

Hon

20−

24

Hon

25−

29

Hon

30−

34

Hon

35−

39

Hon

40−

44

Hon

45−

49

Hon

50−

54

Hon

55−

59

Hon

60−

64

Hon

65−

69

Hon

70−

74

Hon

75−

79

Hon

80−

84

Hon

0.2

0.4

0.6

0.8

85−

+

Hon

0.2

0.4

0.6

0.8

0−1

Gua

1−4

Gua

5−9

Gua

10−

14

Gua

15−

19

Gua

20−

24

Gua

25−

29

Gua

30−

34

Gua

35−

39

Gua

40−

44

Gua

45−

49

Gua

50−

54

Gua

55−

59

Gua

60−

64

Gua

65−

69

Gua

70−

74

Gua

75−

79

Gua

80−

84

Gua

0.2

0.4

0.6

0.8

85−

+

Gua

0.2

0.4

0.6

0.8

0−1

ElSal

1−4

ElSal

5−9

ElSal

10−

14

ElSal

15−

19

ElSal

20−

24

ElSal

25−

29

ElSal

30−

34

ElSal

35−

39

ElSal

40−

44

ElSal

45−

49

ElSal

50−

54

ElSal

55−

59

ElSal

60−

64

ElSal

65−

69

ElSal

70−

74

ElSal

75−

79

ElSal

80−

84

ElSal

0.2

0.4

0.6

0.8

85−

+

ElSal

0.2

0.4

0.6

0.8

0−1

DomRep

1−4

DomRep

5−9

DomRep

10−

14

DomRep

15−

19

DomRep

20−

24

DomRep

25−

29

DomRep

30−

34

DomRep

35−

39

DomRep

40−

44

DomRep

45−

49

DomRep

50−

54

DomRep

55−

59

DomRep

60−

64

DomRep

65−

69

DomRep

70−

74

DomRep

75−

79

DomRep

80−

84

DomRep

0.2

0.4

0.6

0.8

85−

+

DomRep


Fig.

2.14

Est

imat

edpr

opor

tion

ofill

-defi

ned

deat

hsdu

eto

resi

dual

caus

esfo

rMal

esby

coun

try,

age,

and

year

.Red

valu

esar

ees

timat

edus

ing

oura

ppro

ach

and

blac

kva

lues

com

efr

oma

prop

ortio

nald

istr

ibut

ion.

Year

Proportion of Ill−defined deaths due to Residual deaths 0.2

0.4

0.6

0.81

19601970198019902000

0−1

Uru

19601970198019902000

1−4

Uru

19601970198019902000

5−9

Uru

19601970198019902000

10−

14

Uru

19601970198019902000

15−

19

Uru

19601970198019902000

20−

24

Uru

19601970198019902000

25−

29

Uru

19601970198019902000

30−

34

Uru

19601970198019902000

35−

39

Uru

19601970198019902000

40−

44

Uru

19601970198019902000

45−

49Uru

19601970198019902000

50−

54

Uru

19601970198019902000

55−

59

Uru

19601970198019902000

60−

64

Uru

19601970198019902000

65−

69

Uru

19601970198019902000

70−

74

Uru

19601970198019902000

75−

79

Uru

19601970198019902000

80−

84

Uru

19601970198019902000

0.2

0.4

0.6

0.8

185

−+

Uru

0.2

0.4

0.6

0.81

0−1

CRica

1−4

CRica

5−9

CRica

10−

14

CRica

15−

19

CRica

20−

24

CRica

25−

29

CRica

30−

34

CRica

35−

39

CRica

40−

44CRica

45−

49

CRica

50−

54

CRica

55−

59

CRica

60−

64

CRica

65−

69

CRica

70−

74

CRica

75−

79

CRica

80−

84

CRica

0.2

0.4

0.6

0.8

185

−+

CRica

0.2

0.4

0.6

0.81

0−1

Chi

1−4

Chi

5−9

Chi

10−

14

Chi

15−

19

Chi

20−

24

Chi

25−

29

Chi

30−

34

Chi

35−

39Chi

40−

44

Chi

45−

49

Chi

50−

54

Chi

55−

59

Chi

60−

64

Chi

65−

69

Chi

70−

74

Chi

75−

79

Chi

80−

84

Chi

0.2

0.4

0.6

0.8

185

−+

Chi

0.2

0.4

0.6

0.81

0−1

Arg

1−4

Arg

5−9

Arg

10−

14

Arg

15−

19

Arg

20−

24

Arg

25−

29

Arg

30−

34

Arg

35−

39

Arg

40−

44

Arg

45−

49

Arg

50−

54

Arg

55−

59

Arg

60−

64

Arg

65−

69

Arg

70−

74

Arg

75−

79

Arg

80−

84

Arg

0.2

0.4

0.6

0.8

185

−+

Arg

0.2

0.4

0.6

0.81

0−1

Ven

1−4

Ven

5−9

Ven

10−

14

Ven

15−

19

Ven

20−

24

Ven

25−

29

Ven

30−

34

Ven

35−

39

Ven

40−

44

Ven

45−

49

Ven

50−

54

Ven

55−

59

Ven

60−

64

Ven

65−

69

Ven

70−

74

Ven

75−

79

Ven

80−

84

Ven

0.2

0.4

0.6

0.8

185

−+

Ven

0.2

0.4

0.6

0.81

0−1

Par

1−4

Par

5−9

Par

10−

14

Par

15−

19

Par

20−

24

Par

25−

29

Par

30−

34

Par

35−

39

Par

40−

44

Par

45−

49

Par

50−

54

Par

55−

59

Par

60−

64

Par

65−

69

Par

70−

74

Par

75−

79

Par

80−

84

Par

0.2

0.4

0.6

0.8

185

−+

Par

0.2

0.4

0.6

0.81

0−1

Mex

1−4

Mex

5−9

Mex

10−

14

Mex

15−

19

Mex

20−

24

Mex

25−

29

Mex

30−

34

Mex

35−

39

Mex

40−

44

Mex

45−

49

Mex

50−

54

Mex

55−

59

Mex

60−

64

Mex

65−

69

Mex

70−

74

Mex

75−

79

Mex

80−

84

Mex

0.2

0.4

0.6

0.8

185

−+

Mex

0.2

0.4

0.6

0.81

0−1

Ecu

1−4

Ecu

5−9

Ecu

10−

14

Ecu

15−

19Ecu

20−

24

Ecu

25−

29

Ecu

30−

34

Ecu

35−

39

Ecu

40−

44

Ecu

45−

49

Ecu

50−

54

Ecu

55−

59

Ecu

60−

64

Ecu

65−

69

Ecu

70−

74

Ecu

75−

79

Ecu

80−

84

Ecu

0.2

0.4

0.6

0.8

185

−+

Ecu

0.2

0.4

0.6

0.81

0−1

Col

1−4

Col

5−9

Col

10−

14Col

15−

19

Col

20−

24

Col

25−

29

Col

30−

34

Col

35−

39

Col

40−

44

Col

45−

49

Col

50−

54

Col

55−

59

Col

60−

64

Col

65−

69

Col

70−

74

Col

75−

79

Col

80−

84

Col

0.2

0.4

0.6

0.8

185

−+

Col

0.2

0.4

0.6

0.81

0−1

Bra

1−4

Bra

5−9

Bra10

−14

Bra

15−

19

Bra

20−

24

Bra

25−

29

Bra

30−

34

Bra

35−

39

Bra

40−

44

Bra

45−

49

Bra

50−

54

Bra

55−

59

Bra

60−

64

Bra

65−

69

Bra

70−

74

Bra

75−

79

Bra

80−

84

Bra

0.2

0.4

0.6

0.8

185

−+

Bra

0.2

0.4

0.6

0.81

0−1

Per

1−4

Per5−

9

Per

10−

14

Per

15−

19

Per

20−

24

Per

25−

29

Per

30−

34

Per

35−

39

Per

40−

44

Per

45−

49

Per

50−

54

Per

55−

59

Per

60−

64

Per

65−

69

Per

70−

74

Per

75−

79

Per

80−

84

Per

0.2

0.4

0.6

0.8

185

−+

Per

0.2

0.4

0.6

0.81

0−1

Pan1−

4

Pan

5−9

Pan

10−

14

Pan

15−

19

Pan

20−

24

Pan

25−

29

Pan

30−

34

Pan

35−

39

Pan

40−

44

Pan

45−

49

Pan

50−

54

Pan

55−

59

Pan

60−

64

Pan

65−

69

Pan

70−

74

Pan

75−

79

Pan

80−

84

Pan

0.2

0.4

0.6

0.8

185

−+

Pan

0.2

0.4

0.6

0.81

0−1

Nic

1−4

Nic

5−9

Nic

10−

14

Nic

15−

19

Nic

20−

24

Nic

25−

29

Nic

30−

34

Nic

35−

39

Nic

40−

44

Nic

45−

49

Nic

50−

54

Nic

55−

59

Nic

60−

64

Nic

65−

69

Nic

70−

74

Nic

75−

79

Nic

80−

84

Nic

0.2

0.4

0.6

0.8

185

−+

Nic

0.2

0.4

0.6

0.81

0−1

Hon

1−4

Hon

5−9

Hon

10−

14

Hon

15−

19

Hon

20−

24

Hon

25−

29

Hon

30−

34

Hon

35−

39

Hon

40−

44

Hon

45−

49

Hon

50−

54

Hon

55−

59

Hon

60−

64

Hon

65−

69

Hon

70−

74

Hon

75−

79

Hon

80−

84

Hon

0.2

0.4

0.6

0.8

185

−+

Hon

0.2

0.4

0.6

0.81

0−1

Gua

1−4

Gua

5−9

Gua

10−

14

Gua

15−

19

Gua

20−

24

Gua

25−

29

Gua

30−

34

Gua

35−

39

Gua

40−

44

Gua

45−

49

Gua

50−

54

Gua

55−

59

Gua

60−

64

Gua

65−

69

Gua

70−

74

Gua

75−

79

Gua

80−

84

Gua

0.2

0.4

0.6

0.8

185

−+

Gua

0.2

0.4

0.6

0.81

0−1

ElSal

1−4

ElSal

5−9

ElSal

10−

14

ElSal

15−

19

ElSal

20−

24

ElSal

25−

29

ElSal

30−

34

ElSal

35−

39

ElSal

40−

44

ElSal

45−

49

ElSal

50−

54

ElSal

55−

59

ElSal

60−

64

ElSal

65−

69

ElSal

70−

74

ElSal

75−

79

ElSal

80−

84

ElSal

0.2

0.4

0.6

0.8

185

−+

ElSal

0.2

0.4

0.6

0.81

0−1

DomRep

1−4

DomRep

5−9

DomRep

10−

14

DomRep

15−

19

DomRep

20−

24

DomRep

25−

29

DomRep

30−

34

DomRep

35−

39

DomRep

40−

44

DomRep

45−

49

DomRep

50−

54

DomRep

55−

59

DomRep

60−

64

DomRep

65−

69

DomRep

70−

74

DomRep

75−

79

DomRep

80−

84

DomRep

0.2

0.4

0.6

0.8

185

−+

DomRep

References 71

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[Bhat (1987)] Bhat, M. P. (1987). Mortality in India: Levels, Trends and Patterns. Ph.d., Univer-sity of Pennsylvania.

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[Brass and Coale (1968)] Brass, W. and Coale, A. (1968). Methods of analysis and estimation. InThe Demography of Tropical Africa, book section 3, 88–150. Princeton University Press.

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1 Appendix 1. Definition of demographic profiles for thesimulation

Five different master populations were created, one stable and four nonstable pop-ulations. In each case we start with a stable population in 1900 and we computeyearly populations until the year 2000. The age distribution is in single years of agebut for totals (not by gender).

The four non-stable populations were generated following approximately themortality and fertility schedules for Costa Rica, Mexico, Guatemala and Argentina,Uruguay for the period 1900-2000.

1.1 Stable population

The stable population is generated using constant values for GRR= 3.03 and E(0)=45 for the period 1900 and 2000 with a natural rate of increase r = 0.025.

1.2 Non-stable populations (a)(b)(c)

I II III IVYear E(0) GRR r E(0) GRR r E(0) GRR r E(0) GRR r

1900 34.70 3.60 0.05 26.30 6.20 0.04 22.10 5.80 0.03 45.40 1.80 0.021910 35.10 3.40 0.05 29.60 5.70 0.04 25.40 5.70 0.03 48.90 1.70 0.021920 35.10 3.20 0.05 32.90 5.20 0.04 28.70 5.20 0.03 51.30 1.60 0.021930 42.20 2.60 0.05 36.20 4.70 0.04 32.00 4.70 0.03 54.40 1.50 0.021940 46.90 2.50 0.05 41.80 4.20 0.04 37.40 3.80 0.03 59.60 1.40 0.021950 55.60 2.40 0.05 50.70 3.40 0.04 40.20 3.50 0.03 66.30 1.30 0.021960 62.60 2.30 0.05 58.50 3.30 0.04 47.00 3.30 0.03 68.40 1.40 0.021970 65.40 2.10 0.05 62.60 3.20 0.04 53.90 3.10 0.03 68.80 1.50 0.021980 72.60 1.70 0.05 67.70 2.10 0.04 58.20 3.00 0.03 71.00 1.30 0.021990 75.70 1.50 0.05 71.50 1.50 0.04 62.60 2.60 0.03 72.80 1.20 0.022000 77.30 1.30 0.05 73.40 1.20 0.04 65.90 2.20 0.03 75.20 1.10 0.02

(a) Non Stable population I, II, III and IV follow the patterns of mortality andfertility between 1900 and 2000 assessed with current (Adjusted data) for CostaRica, Mexico, Guatemala and Argentina/Uruguay respectively.

(b) Population parameters were directly estimated for each decade and then in-terpolated linearly within each decade to obtain yearly values.

(c) The initial population age distribution for I, II and III correspond to the stablepopulation associated with parameter values in 1900. In case IV the initial popula-tion corresponded to the average of census populations closest to 1900.

3 Appendix 3. Behavior of the age misreporting index cmRox,[t1,t2]

75

2 Appendix 2. Proof of lack of identification of parameters of netage overstatement

Using the same notation as in the text we have

ΠT = (1/φ

no) ˆ[ΘS]−1

ΠO

and

∆T = (1/λ

no)[Θ S]−1∆

O

In a closed population the relation between the vectors for populations in twosuccessive censuses and the vector of intercensal deaths is:

ΠTt+k = Π

Tt +∆

T[t,t+k] (.1)

Using the first two expressions in .1 yields:

(1/φno) ˆ[Θ

S]−1

ΠOt+k = (1/φ

no) ˆ[ΘS]−1

ΠOt − (1/λ

no)[Θ S]−1∆

O[t,t+k] (.2)

From .2 we see that only (φ no/λ no) is identifiable with the available information.


The expression of the age misreporting index is

cmRox,[t1,t2]

=cmPo

x+k,t2/cmPo

x,t1

1− (cmDox,[t1,t2]

/cmPox,t1)

a ratio of two different estimators of the same quantity, namely the cumulativeprobability of survival of the population aged x and over at time t1 to age (x+ k)and over at time t2. Use of cumulative quantities in the index is an important pre-requisite since it minimizes the impact of age misreporting within the bounds of thecumulative quantities. Thus, erroneous transfers over age x do not affect populationcounts at ages x and over. These quantities are influenced only by transfers fromages younger than x into ages x and above or by transfers from ages x and above toages younger than x. Admittedly, however, use of cumulative quantities complicatesthe algebra and muddles interpretation. To circumvent this difficulty and preserving


the same set up and assumptions defined in the text, we redefine the expression forsingle years of age to obtain:

Rox,[t1,t2]

=Po

x+k,t2/Po

x,t1

1− (Dx,[t1,t2]/Pox,t1)

or the ratio of a conventional survival ratio computed from two successive pop-ulation counts to the survival ratio computed from the complement of a measureof the conditional probability of dying between the two censuses. If the populationis stationary, the numerator is simply the ratio Lx+k/Lx in a life table and the de-nominator is the complement of the probability of dying in the intercensal period,namely, 1− (1−Lx+k/Lx). From this it follows that,

ln(

Rox,[t1,t2]

)∼−IN

x,x+k− ln(1−[1− exp

(−ID

x,x+k)])

(.3)

where IDx,x+k and IN

x,x+k are estimators of the integrated hazards between x and x+k consistent with the survival ratios in the denominator and numerator respectively.When the population is closed to migration, there is perfect coverage and no net ageoverstatement, expression .3 equals 0 as both estimators of the integrated hazardsare identical. When there is age overstatement expression .3 becomes

ln(

Rox,[t1,t2]

)∼ ln

(h(x+ k)

h(x)

)− IN

x,x+k− ln(

1− g(x)h(x)

[1− exp(−ID

x,x+k)])

(.4)

where h(.) and g(.) are defined in the text and refer to increasing functions ofage that reflect age overstatement of ages of population and deaths respectively.When these functions are equal to 1, there is neither population nor death age over-statement or, if there is, their effects cancel each other out. Expression .4 can besimplified if we expand the inner log expression in a Taylor series around a value off (x) = g(x)/h(x) = 1:

ln(

Rox,[t1,t2]

)∼ ln

(h(x+ k)

h(x)

)− IN

x,x+k +

(g(x)h(x)−1)(1+ ID

x,x+k)+ IDx,x+k (.5)

an expression that reduces to 0 when h(x+ k)/h(x) =1 and f (x) = 1.Expression .5 is the analytic support for inferences regarding the effects of age

overstatement on the index of age misstatement cmRx,[t1,t2] (see text). Deviationsfrom the assumption of population stationarity introduce only minor changes in thealgebra but leave the implications of expression .5 intact. However,when, as requiredby the original index, we restore the cumulative functions, the algebra becomes in-tractable even in the case of a stationary population. The way out of this conundrumis to think of the cumulative ratios as functions not of the exact integrated hazards, as


77

in expressions .3-.5 but rather as expressions of mean values of corresponding inte-grated hazards. Thus, in a stationary population, the survival ratio of the cumulativepopulations at ages x and x+ k is the ratio T (x+ k)/T (x) which can be written as∫

∞

x+k[exp(−∫ y

0 µ(s)ds)]dx/∫

∞

x [exp(−∫ y

0 µ(s)ds)]dx. Using the mean value theorem

in numerator and denominator leads to the approximation exp(−∫ x+k+i′

x+i µ(s)ds) or,more generally, exp(−

∫ x∗∗x∗ µ(s)ds) where x∗ > x and x∗∗ > x+ k. Upon taking logs

in this expression we retrieve an integrated hazard that expresses integration of theforce of mortality over two ages that are not fixed ex ante (such as x and x+ k) but,rather, between limits (ages) that are a function of the underlying force of mortal-ity. For this reason, in the text, we use the symbols IN

x,x+k and IDx,x+k associated with

cumulative quantities as “integrated hazard analogues”.


4 Appendix 4. Comparison of relative census completeness(C1/C2) estimates from Brass-Hill method and ratios computedfrom CELADE’s completeness estimates in two consecutivecensuses

Country, Year Males Females Country, Year Males FemalesOurs CELADE∗ Ours CELADE∗ Ours CELADE∗ Ours CELADE∗

Argentina Guatemala1947-1960 0.960 0.967 0.940 0.944 1950-1964 0.912 0.965 0.924 0.9521960-1970 0.981 0.989 0.995 0.999 1964-1973 1.047 1.073 1.105 1.0581970-1980 0.984 0.974 0.987 0.988 1973-1981 1.062 1.057 1.073 1.0541980-1991 0.997 1.012 0.986 1.010 1981-1994 0.974 0.999 0.984 1.0131991-2001 1.033 1.017 1.022 1.012 1994-2002 0.966 0.898 1.011 0.904Brazil Honduras1980-1991 0.998 1.015 1.000 1.010 1950-1961 0.944 1.016 0.951 1.0031991-2000 0.975 0.990 0.950 0.991 1961-1974 0.992 1.038 0.973 1.0312000-2010 1.003 1.011 0.999 1.001 1974-1988 0.964 0.958 0.977 0.964Chile Mexico1952-1960 0.950 0.967 0.937 0.964 1950-1960 0.976 0.995 0.926 1.0171960-1970 1.035 1.034 1.050 1.031 1960-1970 0.963 0.987 0.936 0.9861970-1982 0.938 0.949 0.949 0.952 1970-1980 0.981 0.969 0.956 0.9541982-1992 1.002 0.997 0.994 0.998 1980-1990 0.904 1.017 0.916 1.0091992-2002 1.011 1.011 1.002 1.016 1990-2000 0.952 0.990 0.968 0.981Colombia 2000-2010 0.954 1.006 0.958 1.0011951-1964 0.947 0.951 0.930 0.931 Nicaragua1964-1973 1.060 1.085 1.029 1.070 1950-1963 0.971 1.035 0.945 1.0301973-1985 0.915 0.962 0.955 0.983 1963-1971 0.941 1.048 0.955 1.0331985-1993 0.929 0.984 0.920 0.979 1971-1995 0.883 0.793 0.879 0.8151993-2005 1.005 0.954 1.008 0.955 1995-2005 0.935 0.985 0.940 0.992Costa Rica Panama1950-1963 0.936 0.919 0.919 0.931 1950-1960 1.031 0.981 1.019 0.9881963-1973 0.979 0.959 0.959 0.967 1960-1970 0.975 0.952 0.949 0.9511973-1984 0.995 1.020 1.028 1.031 1970-1980 0.958 1.014 0.958 1.0241984-2000 0.961 0.935 0.935 0.944 1980-1990 1.023 0.979 1.016 0.9782000-2011 0.974 1.057 0.976 1.021 1990-2000 1.000 1.005 0.988 1.006Cuba 2000-2010 1.001 0.997 1.002 0.9961953-1970 0.941 0.927 0.970 0.955 Paraguay1970-1981 0.965 1.008 0.992 0.992 1950-1962 0.967 0.991 0.980 0.9971981-2002 0.932 0.987 0.956 0.983 1962-1972 0.990 1.004 1.025 0.9942002-2012 0.995 1.016 0.990 1.016 1972-1982 0.994 1.006 0.968 1.012Dominican Republic 1982-1992 1.017 0.961 1.008 0.9591950-1960 0.986 0.971 0.914 0.981 1992-2002 0.988 0.997 0.981 1.0021960-1970 0.905 1.047 0.979 1.031 Peru1970-1981 0.930 0.943 0.962 0.966 1961-1972 0.995 0.986 0.967 1.0041981-1993 0.980 1.004 0.985 0.944 1972-1981 1.035 1.007 1.025 1.0051993-2002 0.948 0.963 0.988 1.015 1981-1993 1.006 1.008 1.020 1.0022002-2010 1.038 1.005 1.047 1.026 1993-2007 0.990 0.982 0.977 0.982Ecuador Uruguay1950-1962 0.993 0.996 0.998 0.989 1963-1975 0.960 1.003 0.939 0.9961962-1974 0.951 0.973 0.988 0.965 1975-1985 0.961 0.997 0.953 1.0111974-1982 0.912 1.001 0.990 0.997 1985-1996 0.977 1.005 0.981 1.0081982-1990 0.900 1.016 0.985 1.024 1996-2004 0.998 0.999 0.991 0.9991990-2001 0.970 1.000 0.975 0.997 2004-2011 0.996 1.019 0.998 1.0112001-2010 0.937 0.975 0.950 0.978 VenezuelaEl Salvador 1950-1961 1.041 1.030 1.004 1.0331950-1961 0.944 0.959 0.951 0.949 1961-1971 0.958 0.998 0.985 0.9701961-1971 0.992 0.939 0.973 0.937 1971-1981 1.016 0.973 1.005 0.9861971-1992 1.038 1.002 1.030 0.999 1981-1990 0.943 1.009 0.953 1.0071992-2007 0.958 1.003 0.977 0.999 1990-2001 0.928 0.982 0.943 0.976

2001-2011 1.035 1.013 1.005 1.031

* CELADE’s estimates of completeness provided by Guiomar Bay

5 Appendix 5. Short explanation of methods Bennet-Horiuchi 1-4 and Preston-Lahiri 1-2 79

5 Appendix 5. Short explanation of methods Bennet-Horiuchi1-4 and Preston-Lahiri 1-2

Bennet-Horiuchi No 1, (BH 1)Bennett-Horiuchi (1981) completeness factor can be estimated in two differentways. First (cumulated from bottom to top), as the ratio of the estimated numberof persons-years in the age group “a” to “a+5” (10Na−5) to the observed averagenumber of persons-years in the age group “a” to “a+5” over the ten-year period(10Na−5). This was labeled as C5 because the computation of cumulated numbersstarted at age 5. Second (cumulated from top to bottom), a more robust measure-ment was created by cumulating 10Na−5 and 10Na−5 from bottom to top; since theaccumulation started at age 75, the ratio was tagged as C75.

• Bennet-Horiuchi No 1 is the average of these two estimates.

Bennet-Horiuchi No 2, (BH 2)Bennett-Horiuchi (1984) is a variation of the previous estimates that introduces aslight correction in the calculation of 10Na−5 for ages above 60. This new indicatorcan also be computed in two different ways: first, as the ratio of 10Na−5 to 10Na−5(C5), and as the ratio of the accumulated values of 10Na−5 to 10Na−5 (C75). We onlyuse the latter and labeled it Bennet-Horiuchi No 2.

• Bennet-Horiuchi No 2 is the average of these two estimates.

Bennet-Horiuchi No 3 (BH 3) & Bennet-Horiuchi No 4 (BH 4)These estimates are computed as Bennet-Horiuchi No 1 & Bennet-Horiuchi No2 but now we use adjusted age-specific rate of population growth. The adjustmentfactors are estimated used Brass classic method.

Preston-Lahiri 1-2Preston-Lahiri method can also estimate two completeness measurements depend-ing on the age at which the functions are calculated (birth rates, deaths, rates, meanage, etc.) We computed two variants: using age 5+ (labeled C5) and 10+ (labeledC10).

• These are respectively called Preston-Lahiri No 1 (PL 1) and Preston-Lahiri No2 (PL 2).

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