432 MORTALITY MEASUREMENT AT ADVANCED AGES: ASTUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE Leonid A. Gavrilov* and Natalia S. Gavrilova † ABSTRACT Accurate estimates of mortality at advanced ages are essential to improving forecasts of mortality and the population size of the oldest old age group. However, estimation of hazard rates at extremely old ages poses serious challenges to researchers: (1) The observed mortality deceleration may be at least partially an artifact of mixing different birth cohorts with different mortality (het- erogeneity effect); (2) standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high at old ages and (3) ages of very old people may be exaggerated. One way of obtaining estimates of mortality at extreme ages is to pool together international records of persons surviving to extreme ages with subsequent efforts of strict age validation. This approach helps researchers to resolve the third of the above-mentioned problems but does not resolve the first two problems because of inevitable data heterogeneity when data for people belonging to different birth cohorts and countries are pooled together. In this paper we propose an alternative approach, which gives an opportunity to resolve the first two problems by compiling data for more homogeneous single-year birth cohorts with hazard rates measured at narrow (monthly) age intervals. Possible ways of resolving the third problem of hazard rate estimation are elaborated. This approach is based on data from the Social Security Administration Death Master File (DMF). Some birth cohorts covered by DMF could be studied by the method of extinct generations. Availability of month of birth and month of death information provides a unique opportunity to obtain hazard rate estimates for every month of age. Study of several single-year extinct birth cohorts shows that mortality trajectory at advanced ages follows the Gompertz law up to the ages 102–105 years without a noticeable deceleration. Earlier reports of mortality deceleration (devi- ation of mortality from the Gompertz law) at ages below 100 appear to be artifacts of mixing together several birth cohorts with different mortality levels and using cross-sectional instead of cohort data. Age exaggeration and crude assumptions applied to mortality estimates at advanced ages may also contribute to mortality underestimation at very advanced ages. 1. INTRODUCTION Accurate estimates of mortality at advanced ages are essential for improving forecasts of mortality and predicting the population size of the oldest old age group. It is believed that mortality at ad- * Leonid A. Gavrilov, PhD, is a Research Associate at the Center on Aging, NORC at the University of Chicago, 1155 E. 60th St., Chicago, IL 60637, [email protected]. † Natalia S. Gavrilova, PhD, is a Research Associate at the Center on Aging, NORC at the University of Chicago, 1155 E. 60th St., Chicago, IL 60637, [email protected]. vanced ages has a tendency to deviate from the Gompertz law (Gavrilov and Gavrilova 1991), so that the logistic model is suggested for fitting hu- man mortality after age 85 years (Horiuchi and Wilmoth 1998). The estimates of mortality force at extreme ages are difficult to make because of small numbers of survivors to these ages in most countries. Data for extremely long-lived individu- als are scarce and subjected to age exaggeration. Traditional demographic estimates of mortality based on period data encounter the well-known denominator problem. More accurate estimates are obtained using the method of extinct gener-
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432
MORTALITY MEASUREMENT AT ADVANCED AGES:A STUDY OF THE SOCIAL SECURITY
ADMINISTRATION DEATH MASTER FILELeonid A. Gavrilov* and Natalia S. Gavrilova†
ABSTRACT
Accurate estimates of mortality at advanced ages are essential to improving forecasts of mortalityand the population size of the oldest old age group. However, estimation of hazard rates atextremely old ages poses serious challenges to researchers: (1) The observed mortality decelerationmay be at least partially an artifact of mixing different birth cohorts with different mortality (het-erogeneity effect); (2) standard assumptions of hazard rate estimates may be invalid when risk ofdeath is extremely high at old ages and (3) ages of very old people may be exaggerated. Oneway of obtaining estimates of mortality at extreme ages is to pool together international recordsof persons surviving to extreme ages with subsequent efforts of strict age validation. This approachhelps researchers to resolve the third of the above-mentioned problems but does not resolve thefirst two problems because of inevitable data heterogeneity when data for people belonging todifferent birth cohorts and countries are pooled together. In this paper we propose an alternativeapproach, which gives an opportunity to resolve the first two problems by compiling data formore homogeneous single-year birth cohorts with hazard rates measured at narrow (monthly)age intervals. Possible ways of resolving the third problem of hazard rate estimation are elaborated.This approach is based on data from the Social Security Administration Death Master File (DMF).Some birth cohorts covered by DMF could be studied by the method of extinct generations.Availability of month of birth and month of death information provides a unique opportunity toobtain hazard rate estimates for every month of age. Study of several single-year extinct birthcohorts shows that mortality trajectory at advanced ages follows the Gompertz law up to the ages102–105 years without a noticeable deceleration. Earlier reports of mortality deceleration (devi-ation of mortality from the Gompertz law) at ages below 100 appear to be artifacts of mixingtogether several birth cohorts with different mortality levels and using cross-sectional instead ofcohort data. Age exaggeration and crude assumptions applied to mortality estimates at advancedages may also contribute to mortality underestimation at very advanced ages.
1. INTRODUCTION
Accurate estimates of mortality at advanced agesare essential for improving forecasts of mortalityand predicting the population size of the oldestold age group. It is believed that mortality at ad-
* Leonid A. Gavrilov, PhD, is a Research Associate at the Centeron Aging, NORC at the University of Chicago, 1155 E. 60th St.,Chicago, IL 60637, [email protected].† Natalia S. Gavrilova, PhD, is a Research Associate at the Centeron Aging, NORC at the University of Chicago, 1155 E. 60th St.,Chicago, IL 60637, [email protected].
vanced ages has a tendency to deviate from theGompertz law (Gavrilov and Gavrilova 1991), sothat the logistic model is suggested for fitting hu-man mortality after age 85 years (Horiuchi andWilmoth 1998). The estimates of mortality forceat extreme ages are difficult to make because ofsmall numbers of survivors to these ages in mostcountries. Data for extremely long-lived individu-als are scarce and subjected to age exaggeration.Traditional demographic estimates of mortalitybased on period data encounter the well-knowndenominator problem. More accurate estimatesare obtained using the method of extinct gener-
MORTALITY MEASUREMENT AT ADVANCED AGES: A STUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE 433
ations (Vincent 1951). In order to minimize sta-tistical noise in estimates of mortality at ad-vanced ages, researchers are forced to pool datafor several calendar periods (Depoid 1973;Thatcher 1999). Single-year life tables for manycountries have very small numbers of survivors toage 100, which makes estimates of mortality atadvanced ages unreliable. On the other hand, ag-gregation of deaths for several calendar periodscreates a heterogeneous mixture of cases fromdifferent birth cohorts. Theoretical models sug-gest that mortality deceleration at advanced agesmay be caused by population heterogeneity evenif individual risk of death follows the Gompertzlaw (Beard 1959, 1971; Vaupel et al. 1979). Inaddition, many standard assumptions used in haz-ard rate estimation are not valid for extreme oldages when mortality is particularly high.
Thus estimation of the hazard rate at extremelyold ages poses several serious challenges toresearchers:
(1) The observed mortality deceleration may beat least partially related to mixing differentbirth cohorts with different mortality (het-erogeneity effect)
(2) Standard assumptions of hazard rate esti-mates may be invalid when risk of death isextremely high at old ages
(3) Ages of very old people may be exaggerated.
One way of obtaining estimates of mortality atextreme ages is to pool international records ofpersons surviving to extreme ages with subse-quent efforts of strict age validation (Robine etal. 2005; Robine and Vaupel 2001). This approachhelps to resolve the third problem mentionedabove but does not allow researchers to resolvethe first two problems because of inevitable dataheterogeneity when data for people belonging todifferent birth cohorts and countries are pooled.
In this paper we propose an alternative ap-proach, which allows us to resolve partially theseproblems. This approach is based on using datafrom the Social Security Administration DeathMaster File (DMF), which allows compiling datafor large single-year birth cohorts. Some alreadyextinct birth cohorts covered by the DMF couldbe studied by the method of extinct generations.Availability of month-of-birth and month-of-deathinformation provides a unique opportunity to ob-tain hazard rate estimates for every month of age.
Possible ways of resolving the age exaggerationproblem in hazard rate estimation are alsoelaborated.
2. MORTALITY AT ADVANCED AGES:A HISTORICAL REVIEW
The history of mortality studies at extreme agesis very rich in ideas and findings. Early studiesstarting with Gompertz (1825) himself suggestedthat the Gompertz law of mortality is not appli-cable to extreme old ages, and that mortality de-celeration and leveling off takes place at ad-vanced ages (for a thoughtful historical review ofstudies on mortality deceleration at extreme oldages, see Olshansky 1998). In 1939 two Britishresearchers, Greenwood and Irwin, published anarticle, ‘‘Biostatistics of Senility,’’ with an intrigu-ing finding that mortality force stops increasingwith age at extreme old ages and becomes con-stant (Greenwood and Irwin 1939). This study,led by the famous British statistician and epide-miologist Major Greenwood, may be interestingto discuss here because it describes the mortalitypattern at advanced ages for humans reported byresearchers many years later.
The first important finding was formulated byGreenwood and Irwin in the following way: ‘‘theincrease of mortality rate with age advances at aslackening rate, that nearly all, perhaps all, meth-ods of graduation of the type of Gompertz’s for-mula over-state senile mortality’’ (Greenwoodand Irwin 1939, p. 14). This observation wasreported later by many authors (Depoid 1973;Gavrilov and Gavrilova 1991; Horiuchi and Wil-moth 1998; Thatcher 1999; Thatcher et al. 1998)and is now known as the ‘‘late-life mortalitydeceleration.’’
The authors also suggested ‘‘the possibilitythat with advancing age the rate of mortality as-ymptotes to a finite value’’ (Greenwood and Irwin1939, p. 14). Their conclusion that mortality atexceptionally high ages follows a first-order ki-netics (also known as the law of radioactive decaywith exponential decline in survival probabilities)was supported later by other researchers, includ-ing A. C. Economos (Economos 1979, 1980), whoreported that this law holds for humans and lab-oratory animals (linear decrease for the loga-rithm of the number of survivors). This observa-tion is known now as the ‘‘mortality leveling-off’’
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at advanced ages and as the ‘‘late-life mortalityplateau’’ (Curtsinger et al. 2006).
Moreover, Greenwood and Irwin made the firstestimates for the asymptotic value of human mor-tality (one-year probability of death, qx) at ex-treme ages using data from a life insurance com-pany. According to their estimates, ‘‘the limitingvalues of qx are 0.439 for women and 0.544 formen’’ (Greenwood and Irwin 1939, p. 21). It isinteresting that these first estimates are veryclose to estimates obtained later using more nu-merous and accurate human data, including re-cent data on supercentenarians (Robine and Vau-pel 2001). The authors also suggested anexplanation of this phenomenon. According toGreenwood and Irwin (1939), centenarians live ina more protected environment than younger agegroups and hence have a lower risk of death thanpredicted by the Gompertz formula.
The actuaries including Gompertz himself werethe first who noticed this phenomenon of mor-tality deceleration. They also proposed a logisticformula for fitting mortality growth with age inorder to account for mortality leveling off at ad-vanced ages (Perks 1932; Beard 1959, 1971).British actuary Robert Eric Beard introduced amodel of population heterogeneity with gamma-distributed individual risk in order to explainmortality deceleration at older ages (Beard1959). This explanation is now considered to bethe most common explanation of the mortalitydeceleration phenomenon (Horiuchi and Wil-moth 1998).
The same phenomenon of ‘‘almost nonaging’’survival dynamics at extreme old ages is detectedin many other biological species. In some speciesa mortality plateau can occupy a sizable part oftheir life (Carey et al. 1992; Gavrilov and Gavri-lova 2006). Biologists have been aware of mortal-ity leveling off since the 1960s. For example,Lindop (1961) applied Perks’ formula in order toaccount for mortality deceleration at older agesin mice. George Sacher believed that the ob-served mortality deceleration in mice and ratscan be explained by population heterogeneity:‘‘one effect of such residual heterogeneity is tobring about a decreased slope of the Gompertzianat advanced ages. This occurs because sub-populations with the higher injury levels die outmore rapidly, resulting in progressive selectionfor vigour in the surviving populations’’ (Sacher
1966, p. 435). Strehler and Mildvan (1960) con-sidered mortality deceleration at advanced agesas a prerequisite for all mathematical models ofaging to explain.
Later Economos published a series of articlesclaiming priority in the discovery of a ‘‘non-Gompertzian paradigm of mortality’’ (Economos1979, 1980, 1983, 1985). He found that mortal-ity leveling off is observed in rodents (guinea pigs,rats, mice) and invertebrates (nematodes,shrimps, bdelloid rotifers, fruit flies, and degen-erate medusae [Campanularia flexuosa]). In the1990s, the phenomenon of mortality decelerationand leveling off became widely known afterpublications that demonstrated mortality levelingoff in large samples of Drosophila melanogaster(Curtsinger et al. 1992, 2006) and medflies (Cer-atitis capitata) (Carey et al. 1992). Mortalityplateaus at advanced ages are observed forother insects: housefly (Musca vicina), blow-fly (Calliphora erythrocephala) (Gavrilov 1980),housefly (Musca domestica) (Gavrilov and Gavri-lova 2006), fruit flies (Anastrepha ludens, Anas-trepha obliqua, Anastrepha serpentine), parasitoidwasp (Diachasmimorpha longiacaudtis) (Vaupelet al. 1998), and bruchid beetle (Callosobruchusmaculates) (Tatar et al. 1993). Interestingly, thefailure kinetics of manufactured products (steelsamples, industrial relays, and motor heat insu-lators) also demonstrates the same ‘‘nonaging’’pattern at the end of their ‘‘lifespan’’ (Economos1979).
Population heterogeneity, first proposed byBeard in 1959, is by far the most common expla-nation of mortality deceleration (Horiuchi andWilmoth 1998). Another explanation of this phe-nomenon comes from the reliability theory of ag-ing, which explains mortality leveling off by anexhaustion of an organism’s redundancy (re-serves) at extremely old ages so that every ran-dom hit results in death (Gavrilov and Gavrilova1991, 2001, 2006). There is also an opinion thatlower risks of death for older people are due totheir less risky behavior (Greenwood and Irwin1939). Finally, some researchers suggest evolu-tionary explanations for the phenomenon of mor-tality leveling off (Charlesworth 2001; Muellerand Rose 1996).
The existence of mortality plateaus is well es-tablished for a number of lower organisms, mostlyinsects. In the case of mammals, data are much
MORTALITY MEASUREMENT AT ADVANCED AGES: A STUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE 435
more controversial. Lindop (1961) and Sacher(1966) reported short-term periods of mortalitydeceleration in mice at advanced ages and evenused Perks’ formula in their analyses. However,Austad later argued that rodents do not demon-strate mortality deceleration even in the case oflarge samples (Austad 2001). Study of baboonsfound no mortality deceleration at advanced ages(Bronikowski et al. 2002). In the case of humansthis problem is not yet resolved, because ofscarceness of data and/or their low reliability.Thus, more studies on larger human birth cohortsare required to establish with certainty the truemortality trajectory at advanced ages.
3. HAZARD RATE (MORTALITY FORCE)ESTIMATION AT ADVANCED AGES
A conventional way to obtain estimates of mor-tality at advanced ages is a construction of dem-ographic life table with probability of death (qx)as one of important life table functions. Althoughprobability of death is a useful indicator for mor-tality studies, it may be not the most convenientone for studies of mortality at advanced ages.First, the values of qx depend on the length of theage interval �x for which it is calculated. Thishampers both analyses and interpretation. For ex-ample, if one-year probability of death follows theGompertz law of mortality, probability of deathcalculated for another age interval does not fol-low this law (Gavrilov and Gavrilova 1991; Le Bras1976). Thus it turns out that the shape of agetrajectory for qx depends on the arbitrary choiceof age interval. Also, by definition qx is boundedby unity, which would inevitably produce apparentmortality deceleration when death rates are par-ticularly high.
A more useful indicator of mortality at ad-vanced age is the instantaneous mortality rate(mortality force) or hazard rate �x, which is de-fined as follows:
dN dln(N ) �ln(N )x x x� � � � � � � ,x N dx dx �xx
where Nx is a number of living individuals exposedto risk of death at age x. It follows from the def-inition of the hazard rate that it is equal to therate of decrease of the logarithmic survival func-tion with age. In actuarial practice, the hazardrate is often called mortality force as was done
in the original paper by Gompertz (Gompertz1825). The hazard rate does not depend on thelength of the age interval (it is measured at theinstant of time x), has no upper boundary, andhas a dimension of rate (time�1). It should alsobe noted that the famous law of mortality, theGompertz law, was first proposed for fitting theage-specific hazard rate function rather thanprobability of death (Gompertz 1825).
The empirical estimates of hazard rates are of-ten based on the suggestion that the age-specificmortality rate or death rate (number of deathsdivided by exposure) is a good estimate of thetheoretical hazard rate. One of the first empiricalestimates of the hazard rate was proposed byGeorge Sacher (Sacher 1956, 1966):
1� � [ln(l ) � ln(l )]x x��x/2 x��x/2�x
1 lx��x/2� ln .� ��x lx��x/2
This estimate is unbiased for slow changes in thehazard rate if �x��x �� 1 (Sacher 1966) andwas shown to be the maximum likelihood esti-mate (Gehan and Siddiqui 1973). A simplifiedversion of the Sacher estimate (for small age in-tervals equal to unity) is often used in biodemo-graphic studies of mortality: �x � �ln(1 � qx).This estimate was initially suggested by Gehan,who called it a ‘‘Sacher’’ estimate (Gehan 1969;Gehan and Siddiqui 1973). It is based on the as-sumption that hazard rate is constant over ageinterval and is shifted by one-half of a year toyounger ages compared to the original Sacherestimate.
Another empirical estimate of hazard rate, of-ten used in life table construction (Klein andMoesberger 1997), is the actuarial estimate,which is calculated in the following way (Kimball1960):
2q 2 l � lx x��x x� � � .x �x(2 � q ) �x l � lx x��x x
This estimate assumes uniform distribution ofdeaths over the age interval and is bounded by2/�x, so this is not the best estimate of the haz-ard rate at extreme old ages when death rates areparticularly high (Gavrilov and Gavrilova 1991).
At advanced ages, when death rates are veryhigh, the assumptions about small changes in the
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hazard rate or a constant hazard rate within theage interval become questionable. The same istrue for the assumption of uniform distributionof deaths within the age interval. Simulation stud-ies showed that the bias in hazard rate estimationincreases with the increase of the age interval(Kimball 1960). Narrowing the age interval forhazard rate estimation from one year to onemonth helps to improve the accuracy of hazardrate estimation.
4. SOCIAL SECURITY ADMINISTRATIONDEATH MASTER FILE AS A SOURCEOF MORTALITY DATA FORADVANCED AGES
The Social Security Administration Death MasterFile (DMF) is a publicly available data source thatallows a search for deceased individuals in theUnited States using various search criteria: birthdate, death date, first and last names, Social Se-curity number, place of last residence, etc. Thisresource covers deaths that occurred in the pe-riod 1937–2010 (Faig 2001) and captures about95% of deaths recorded by the National Death In-dex (Sesso et al. 2000). According to other esti-mates, the DMF covers about 90% of all deathsfor which death certificates are issued (Faig2001) and about 92–96% of deaths for personsolder than 65 years (Hill and Rosenwaike 2001).
The DMF was used in our study of age-relatedmortality dynamics after age 85 years. The advan-tage of this data source is that some already ex-tinct birth cohorts covered by the DMF could bestudied by the method of extinct generations(Kannisto 1988, 1994; Vincent 1951). Informa-tion available in the DMF includes names of thedeceased, his or her Social Security number, date,month, and year of birth, month and year ofdeath, state of Social Security number issuance,and place of the last residence.
In this study information from the DMF was col-lected for individuals who lived 88 years and moreand died before 2011. The DMF database isunique because it represents mortality experi-ence for very large birth cohorts of the oldest-oldpersons. In this study mortality measurementswere made for cohorts, which are more homoge-neous in respect to the year of birth and histori-cal life course experiences. Availability of month-of-birth and month-of-death information provides
a unique opportunity to obtain hazard rate esti-mates for every month of age, which is importantgiven extremely high mortality after age 100years. Despite certain limitations, this datasource allows researchers to obtain detailed esti-mates of mortality at advanced ages. We alreadyused this data resource for centenarians’ age val-idation in the study of centenarian family histo-ries (Gavrilova and Gavrilov 2007). This data re-source is also useful in mortality estimates forseveral extinct or almost extinct birth cohorts inthe United States.
5. HAZARD RATE ESTIMATES ATADVANCED AGES USING DATA FROMTHE DMF
In this study we collected information from theDMF publicly available at www.Rootsweb.com.The total number of records collected from thisresource was about 9 million, with more than900,000 records belonging to persons who lived100 years and more. We obtained data for personswho died before 2011 and were born in 1875–1895. Persons born in these years and alive in2010 should survive to at least age 115 years,which can be considered a very unlikely event.Thus, the 1875–1895 birth cohorts in this samplemay be considered practically extinct. Assumingthat the number of living persons belonging tothese birth cohorts in 2010 is close to zero, it ispossible to construct a cohort life table usingwell-known method of extinct generations, whichwas suggested and explained by Vincent (1951)and developed further by Kannisto (1994).
The DMF provides information about years andmonths of birth and death. However, informationon the exact day of death is not available for allrecords, so we were not able to calculate lifespanin days. Nevertheless we were able to calculateindividual lifespans with one-month accuracy,which is still higher than the accuracy of the tra-ditional yearly estimate of lifespan. In the firststage of our analyses we calculated an individuallifespan in completed months:
Lifespan in months � (death year � birth year)
� 12 � death month � birth month.
Having this information it is possible to esti-mate hazard rates at each month of age by stan-
MORTALITY MEASUREMENT AT ADVANCED AGES: A STUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE 437
Figure 1Logarithm of Hazard Rate (per Year) asFunction of Age Calculated for Monthly
and Yearly Age Intervals
Note: Hazard rate estimates were obtained using the Nelson-Aalenestimate. Solid lines show polynomial fit with quadratic term. Datafor 1891 birth cohort from the DMF.
dard methods of survival analysis. All calculationswere done using Stata statistical software, release11 (StataCorp 2009). This software calculatesnonparametric estimates of major survival func-tions including the Nelson-Aalen estimator of thehazard rate (force of mortality). Note that thehazard rate �(x) in contrast to the probability ofdeath q(x) has a dimension of time frequency, be-cause of the time interval in the denominator (re-ciprocal time, time�1). Thus the values of hazardrates depend on the chosen units of time mea-surement (day�1, month�1, or year�1). In thisstudy, survival times were measured in months,so the estimates of hazard rates initially had adimension of month�1. For the purpose ofcomparability with other published studies, whichtypically use the year�1 time scale, we trans-formed the monthly hazard rates to the moreconventional units of year�1, by multiplying theseestimates by a factor of 12 (one month in thedenominator of hazard rate formula is equal to1⁄12 year). It should be noted that the hazard rate,in contrast to the probability of death, can begreater than 1, and therefore its logarithm canbe greater than 0 (and we indeed observed thesevalues at extreme old ages in some rare cases aswill be described later). In this paper we focus ouranalyses on 1881–1895 birth cohorts, because wefound that data quality for cohorts born before1881 is not particularly good.
In our study we used Nelson-Aalen hazard rateestimates provided by Stata (StataCorp 2009). Infact, the Nelson-Aalen method was initially pro-posed for cumulative hazard rate estimation (par-ticularly for right-censored survival data; Kleinand Moesberger 1997). In Stata hazard rate es-timation is made by taking the steps of theNelson-Aalen cumulative hazard function (Cleveset al. 2008), so that for each observed time ofdeath xj the estimated hazard contribution is
ˆ ˆ ˆ�H(x ) � H(x ) � H(x ),j j j�1
where is an estimate of cumulative hazardH(x)function.
The way of hazard rate estimation conducted inStata is similar to calculation of life-table prob-ability of death (StataCorp 2009); that is, thenumber of deaths in the studied age interval isdivided by the number alive at the beginning ofage interval. At advanced ages when mortality is
high and for relatively large age intervals, thenumber of persons exposed to risk of death in themiddle of age interval is substantially lower thanthe number alive at the beginning of the age in-terval. This would result in downward bias in haz-ard rate estimates at advanced ages, which weobserved in our study when the Nelson-Aalenestimates were applied to yearly age intervals(Fig. 1, lower line). However, for smaller monthlyage intervals, the problem described above is notso crucial, and the Nelson-Aalen method still canbe applied (Fig. 1, upper line).
Note that hazard rate values calculated on amonthly basis demonstrate steady exponentialgrowth up to age 105 years (Fig. 1). After age 105years mortality trajectories show a small tendencyfor deceleration with age. Study of hazard rateestimates for several other single-year birth co-horts (1886–1895) found a similar age patternfor mortality at advanced ages.
Another issue that makes monthly age intervalspreferable for hazard rate estimation is thepattern-of-deaths distribution within studied ageintervals at extremely old ages. This issue is im-portant because some estimates of the hazardrate (for example, the actuarial estimate of thehazard rate) are based on the assumption of ap-proximately uniform distribution of deaths within
438 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 3
one-year age intervals (Gehan and Siddiqui 1973;Kimball 1960; Watson and Leadbetter 1964). In-deed, this assumption is consistent with the datafor 90-year-olds, presented in Figure 2a. However,for 102-year-olds, the distribution of deaths is
Figure 2Deaths at Extreme Old Ages (102 Years) Are
Not Distributed Uniformly over One-YearAge Intervals in Contrast to Earlier
Age Groups (90 Years)
a)
b)
Note: 1891 birth cohort from the DMF. (a) Distribution for 90-year-olds. (b) Distribution for 102-year-olds.
obviously not uniform, with far more deaths oc-curring at the beginning of the age interval (Fig.2b). To alleviate this problem, it is preferable tomake mortality estimates at advanced ages forshorter (monthly) age intervals, rather than fortraditional one-year age intervals when the actu-arial estimate of the hazard rate is applied.
Using single-year birth cohort data from theDMF we were able to minimize the effects of pop-ulation heterogeneity on age-related mortality dy-namics. We were also able to calculate more ac-curate monthly estimates of hazard rates, whichare less prone to possible biases caused by viola-tion of typical assumptions used in mortality es-timation (Kimball 1960). However, we were notable to control for possible age misreportingamong the deceased. The most common type ofage misreporting among very old individuals isage exaggeration (Willcox et al. 2008). This typeof age misreporting results in underestimation ofmortality rates at advanced ages. Thus, we maysuggest that mortality deceleration after age 105years among DMF cohorts may be caused by poordata quality (age exaggeration) at very advancedages. If this hypothesis is correct, then mortalitydeceleration at advanced ages should be less ex-pressed for data with presumably better quality.In order to test this hypothesis we conducted adata quality study for DMF birth cohorts.
6. MEASURES TO IMPROVE DATAQUALITY
Mortality deceleration and even decline of mor-tality is observed usually for data of low quality.On the other hand, improvement of data qualityresults in a straighter mortality trajectory in asemilog scale (Kestenbaum and Ferguson 2002).Thus, we may suggest that mortality estimates atadvanced ages should be lower in populationswith less accurate age reporting compared topopulations with more accurate age reporting. Inorder to test this hypothesis, we compared mor-tality trajectories at advanced ages for popula-tions with a different quality of age reporting.
Study of age validation among individuals aged110 years or more (Rosenwaike and Stone 2003)demonstrated that age reporting among super-centenarians in the Social Security Administra-tion DMF is rather accurate with the exception ofpersons born in the Southern states. In order to
MORTALITY MEASUREMENT AT ADVANCED AGES: A STUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE 439
Figure 3Regional Mortality Data with Presumably
Different Quality
Note: Logarithm of hazard rate (per year) for the 1891 birth cohortas a function of age. Comparison of less reliable (‘‘Southern’’ group)and more reliable (‘‘Northern’’ group) data. Solid lines show a poly-nomial fit with a quadratic term. Data from the DMF.
Figure 4Historical Mortality Data with Presumably
Different Quality
Note: Logarithm of the hazard rate (per year) for two birth cohortsas a function of age. Comparison of less reliable (older 1881 birthcohort, lower line) and more reliable (more recent 1891 birth cohort,upper line) data. Solid lines show fit by a polynomial function witha quadratic term. Data from the DMF.
compare populations with presumably differentdata quality we split DMF records into twogroups. In the first (‘‘Southern’’) group of lessreliable data we included records for those per-sons who applied for a Social Security number inthe Southeast states (AL, AR, FL, GA, KY, LA, MS,NC, SC, TN, VA, WV), Southwest states (AZ, NM,OK, TX), Puerto Rico, and Hawaii. We also addedto this group all records for persons who appliedfor a Social Security number in New York and Cal-ifornia because of a very high proportion of im-migrants (with unknown quality of age report-ing) residing in these states. In the second(‘‘Northern’’) group with presumably better dataquality we included persons applying for a SocialSecurity number in all other states as well as re-tired railroad workers.
Figure 3 shows age trajectories of mortality forpersons born in 1891 and who had applied for aSocial Security number in either ‘‘Southern’’ or‘‘Northern’’ states. Note that the line correspond-ing to the polynomial fit of the logarithm of mor-tality with age is straighter for the ‘‘Northern’’group (upper line) compared with the line cor-responding to mortality of the ‘‘Southern’’ group(lower line). Thus, the estimates of age-specificmortality demonstrate a straighter trajectory ofmortality in the semilog scale for the populationof ‘‘Northern’’ states with presumably better qual-
ity of age reporting compared to the populationwith less reliable quality of age reporting (verifiedusing a polynomial fit of mortality data).
A similar phenomenon is observed when mor-tality data for earlier birth cohort are comparedto mortality data for later birth cohort. Recordsfor later-born persons are expected to be of betterquality because of the improvement of age re-porting over time. Taking into account that agereporting is improving over history, we may sug-gest that mortality for later birth cohorts wouldfollow the Gompertz law more closely comparedto the mortality of earlier birth cohorts with pre-sumably lower data quality. This suggestion isconfirmed by comparing the 1881 and 1891 birthcohorts (Fig. 4). The more recent 1891 birth co-hort demonstrates a straighter trajectory andlower statistical noise after age 105 than theolder 1881 one (Fig. 4). Thus, we may expect thatfuture extinct cohorts born after 1891 may dem-onstrate even better fit by the Gompertz modelthan the older ones because of continuously im-proving accuracy in age reporting.
Quantitative estimation of the degree of mor-tality deceleration at older ages was conducted bymodeling the logarithm of the hazard rate as apolynomial function of age. In the case of mor-tality deceleration, the hazard rate would be de-
440 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 3
Table 1Comparison of the Degree of MortalityDeceleration at Advanced Ages Using
1891 birth cohortYearly age intervals �48.9 (�49.1, �48.6)Monthly age intervals �25.5 (�25.8, �25.1)
Note: Logarithm of hazard rate after age 90 for a particular popu-lation was fitted by the polynomial regression model with a quad-ratic term.
scribed by a convex parabola in semilog coordi-nates. This is expressed as a negative parameterat a quadratic term, and the absolute value of thisparameter can be used as a measure of mortalitydeceleration. Table 1 presents values of the pa-rameter at a quadratic term for polynomial re-gression of the logarithm of the hazard rate as afunction of age (after age 90). Note that the ab-solute value of a parameter at the quadratic termis higher for older birth cohorts and the ‘‘South-ern’’ group of less reliable records (described ear-lier), confirming that data with lower qualitydemonstrate a higher degree of mortalitydeceleration.
Based on the results presented above we mayconclude that the better the quality of mortalitydata at advanced ages, the closer mortality tra-jectories are to the Gompertz function. However,we have to admit that the data at very old agesare noisy and have quality problems even for the‘‘Northern’’ group and the most recent birth co-horts. Thus, our next step in data quality controlis to identify the age interval with reasonablygood data quality and then to compare compet-ing mortality models in this age interval.
One approach to testing data quality at ad-vanced ages is to calculate the female-to-male ra-tio at advanced ages. Taking into account that
female mortality is always lower than male mor-tality, it is reasonable to expect that the female-to-male ratio should continuously increase withage. On the other hand, old men have a tendencyfor age exaggeration, and in populations withpoor age registration there is a relative excess ofmen at very advanced ages (Caselli et al. 2006;Willcox et al. 2008).
The DMF does not have information about sexof the deceased. To cope with this limitation ofthe data sample, we conducted a procedure of sexidentification using information about the 1,000most commonly used baby first names in the1900s provided by the Social Security Adminis-tration (http://www.ssa.gov/OACT/babynames).These data come from a sample of 5% of all SocialSecurity cards issued to individuals who wereborn during the 1900s in the United States. Fromthe lists of male and female names we removednames consisting of initials and potentially unisexnames (such as Jessie or Lonnie). It is interestingto note that the male list contains obviously fe-male names (Mary, Elizabeth), and the sameproblem was observed for the female list, whichindicates that the data apparently contain manysex misidentifications. These female names wereremoved from the male list, and the same pro-cedure was done for the female list. Additionalmale or female names found in DMF were alsoadded to our sex identification name lists. Usingthe final lists of male and female first names, weapplied the sex identification procedure to thesample of 1879–1895 birth cohorts. Eventuallywe were able to identify sex for 91–93% of recordsfor persons in our sample. The remaining 7–9%of persons with unknown sex had approximatelythe same mean lifespan as the remaining per-centage of individuals with identified sex pooled(checked with the t-test statistics), so the exis-tence of possible sex nonidentification bias inmortality seems unlikely. This data sample of774,926 individuals with known sex (out of846,982 individuals) was used for identifying theage interval of reasonably good data quality usinginformation on the female-male ratio at advancedages.
The result of hazard rate estimation for menand women born in 1891 is presented in Figure5. Note that male mortality continues to exceedfemale mortality up to very advanced ages, andthis mortality differential narrows very slowly with
MORTALITY MEASUREMENT AT ADVANCED AGES: A STUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE 441
Figure 5Age-Specific Hazard Rates (Log Scale) for
Men (Upper Line) and Women (Lower Line)Born in 1891
Note: Solid lines show fit with polynomial function with quadraticterm. Data from the DMF.
Figure 7Age of Maximum Female-to-Male Ratio, by
Birth Cohort
Figure 6Observed Female-to-Male Ratio at AdvancedAges for Combined 1887–1892 Birth Cohort
Note: If data are of good quality, then this ratio should grow withage.
age. At age 110 years the number of remainingmales (9) and females (61) is too small for ac-curate estimates of the hazard rate after this age.This pattern of sex differential in mortality at veryold ages confirms the validity of using the female-to-male ratio as a tool of data quality assessmentbecause higher male mortality at all observedages should theoretically lead to a continuouslygrowing female-male ratio.
We calculated the female-to-male ratio afterage 95 years for 1881–1895 U.S. birth cohortsfrom the DMF. Figure 6 demonstrates the age de-
pendency of this ratio for pooled sample of 1887–1892 birth cohorts (these cohorts have similarlevels of mortality). Note that the female-to-maleratio is growing steadily with age up to ages 106–107 years. After this age the female-to-male ratiostarts to decrease, indicating a declining qualityof age reporting. Thus, the estimates of hazardrates obtained from the DMF are of acceptablequality up to the age of 106 years. Figure 7 showsthe maximum age or tipping point when thefemale-to-male ratio starts to decline for differentbirth cohorts. It demonstrates that this age variesfrom 104 to 107, indicating the upper limit fordata of reasonably good quality in terms of agereporting.
Thus, our study of the female-to-male ratio fordifferent birth cohorts indicates that age report-ing has reasonably good quality up to an age ofabout 106 years. For this reason we used the ageinterval 88–106 years for mortality modeling.
7. MODELING MORTALITY ATADVANCED AGES
The next step of our study was to compare twocompeting models of mortality at advancedages—the Gompertz and the logistic models—using data of reasonably good quality. Study ofdata quality control at advanced ages described
442 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 3
Table 2Comparison of Goodness-of-Fit (BayesianInformation Criterion) for Gompertz and
Note: Estimates were made in the age interval 88–106 years for 10single-year U.S. birth cohorts and data of enhanced accuracy forindividuals who applied for Social Security numbers in the Northernstates (see explanation in the text). Cases when the Gompertz modelfits data better than the logistic model are highlighted in bold.
Figure 8Mortality Fitting Using Gompertz (Upper Line)
and Logistic (Lower Line) Models for 1891Birth Cohort of Persons Who Applied for
Social Security Numbers in the Northern States
Note: Data from the DMF.
earlier suggests that age reporting among theoldest-old in the United States is good until theage of 106 years. It means that comparing mor-tality models beyond this age is not feasible be-cause of poor quality of mortality data. For thisreason, we used a subsample of deaths for personswho applied for Social Security numbers in the‘‘Northern’’ states (described above) and born in1886–1895 because these data have reasonablygood quality. We applied Gompertz and logistic(Kannisto) models (Thatcher et al. 1998) to mor-tality modeling in the age interval 88–106 yearsusing the nonlinear regression method for param-eter estimation. Calculations were performed us-ing Stata.
Figure 8 presents the results of mortality fit-ting after age 88 for Gompertz and Kannistomodels using data for ‘‘Northern’’ states and the1891 birth cohort. The visual impression is thatthe Gompertz model fits data better than thelogistic model, although for objective compari-sons we need to use conventional measures ofgoodness-of-fit. In this study, goodness-of-fit foreach model was estimated using the Bayesian in-formation criterion (BIC). Table 2 shows valuesof the BIC for both the Gompertz and the logisticmodel for 10 studied birth cohorts. Note that in8 of 10 cases (studied birth cohorts), the Gom-pertz model demonstrates better fit (lower BIC)
than the logistic model for age interval 88–106years.
At this time, we cannot make a conclusion thatthe Gompertz model fits mortality data betterthan the logistic model after age 106, because ofthe low quality of age reporting beyond this age.At the same time, the data indicate that theGompertz model fits mortality data well until age106 years. Taking into account that survival be-yond age 106 years is a rather rare event, it wouldbe reasonable to suggest the Gompertz modelrather than the logistic model for closing cohortlife tables in actuarial practice. In this case, mor-tality modeling could be done first for the hazardrate (mortality force) function, and then all lifetable functions (including probability of death qx)could be derived on the basis of modeled valuesof the hazard rate.
8. COMPARISON OF DMF OLD-AGEMORTALITY DATA WITH THE 1900ACTUARIAL LIFE TABLE
Because of limitations of the DMF data (incom-pleteness of death registration before the 1970s),we were unable to estimate mortality before theage 85–88 years. There is also a question whetherour estimates of the Gompertz parameters (slopeparameter in particular) are applicable to a wider
MORTALITY MEASUREMENT AT ADVANCED AGES: A STUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE 443
Figure 9Actuarial 1900 U.S. Cohort Life Table and
1894 Birth Cohort
Age
50 60 70 80 90 100
log
(haz
ard
rate
)
-2
-1
0
1894 birth cohort, DMF1900 cohort, U.S. actuarial life table
Note: Source for actuarial life table: Bell and Miller (2002).
age interval or whether they are specific only foradvanced ages (the so-called two-stage Gompertzmodel). In this study we compared our empiricalestimates of cohort mortality at advanced ages,based on DMF data, with the 1900 actuarial co-hort life table (Bell et al. 1992). Unfortunately,no reliable U.S. cohort life tables exist for cohortsolder than 1900, so we compared our mortalitydata for earlier birth cohorts to the later 1900birth cohort. It should be noted that the adultU.S. population did not experience substantialmortality changes between 1890 and 1900. Ac-cording to the estimates made by historical de-mographers, life expectancy at age 40 in theUnited States in 1890 (27.61 years) was practi-cally the same as in 1900 (27.52 years) (Haines1998). Thus, we believe that mortality at ad-vanced ages for the 1900 birth cohort in the ac-tuarial study and the 1894 birth cohort we usedfor comparison should not be substantiallydifferent.
We initially compared our DMF data withprobability-of-death estimates from the actuariallife table (qx) and found that life table values ofprobability of death show deceleration after age100. It should be noted, however, that the yearlyestimates of probability of death (qx) and mortal-ity force (�x) diverge at advanced ages. This hap-pens because probability of death cannot con-tinue its rapid growth when it approaches itsmathematical upper limit equal to one, whereasmortality force theoretically can increase indefi-nitely with age. Therefore, estimates of probabil-ity of death at advanced ages are biased downwardcompared to the hazard rate estimates. Hence,probability-of-death estimates beyond age 100show deceleration in semilog coordinates even ifvalues of hazard rates follow the Gompertz model.
A formula proposed by George Sacher (Sacher1956, 1966) gives more accurate estimates of thehazard rate for yearly age intervals, and so we ap-plied a simplified Sacher formula for the age-specific hazard rate conversion from qx values re-ported in the 1900 actuarial cohort life table:
� � �ln(1 � q ),x x
where qx is a probability-of-death value from thelife table.
Figure 9 shows the trajectories of age-specifichazard rates for the 1900 birth cohort from the
actuarial study and 1894 birth cohort from DMFover a broad age interval starting at age 50 years.Note that hazard rate estimates for the 1894birth cohort are practically identical to the haz-ard rate estimates calculated on the basis of theactuarial life table. Also note that mortality of the1894 birth cohort has the same slope in semilogcoordinates as the mortality of the actuarial birthcohort calculated for a much wider age interval,which does not support the suggestion about thetwo-stage Gompertz model of mortality at ad-vanced ages. Indeed, the maximum likelihood es-timator of the Gompertz slope parameter formortality in the 1894 cohort measured in the in-terval 88–106 years (0.0786 year�1, 95% CI:0.0786–0.0787) does not differ from the slopeparameter calculated over the age interval 40–104 years in the 1900 life table: 0.0785 year�1,95% CI: 0.0772–0.0797. Thus, we may concludethat more accurate estimates of hazard rates atadvanced ages based on individual mortality datapractically coincide with hazard rate estimatescalculated on the basis of the 1900 actuarial co-hort life table (Bell et al. 1992). It is remarkablethat estimates in the actuarial life table obtainedby extrapolation of annual probabilities of deathafter age 95 (Bell et al. 1992) are practically iden-tical to the observed estimates of hazard rates
444 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 3
based on mortality experience of a very large co-hort of old individuals.
9. DISCUSSION
This study of large single-year U.S. birth cohortsfound that mortality deceleration at advancedages is negligible up to the age of 106 years. Be-low the age of 107 years and for data of reason-ably good quality the Gompertz model fits mor-tality better than the logistic model (no mortalitydeceleration). We also found that deceleration ofmortality in later life is more expressed for dataof lower quality. Quality of age reporting in DMFbecomes particularly poor after the age of 107years. It is also interesting that the DMF mortal-ity data agree remarkably well with mortality es-timates reported by the 1900 actuarial cohort lifetable (if hazard rate values are compared).
There are several reasons why earlier studies,including our own research (Gavrilov 1984; Gav-rilov and Gavrilova 1991), reported mortality de-celeration and mortality leveling off at advancedages (Horiuchi and Wilmoth 1998; Kannisto1994; Robine and Vaupel 2001; Thatcher 1999;Thatcher et al. 1998). First, many studies presentinformation for age-specific probability of deathrather than the hazard rate (Gallop and Macdon-ald 2005; Robine and Vaupel 2001). It is not sur-prising that probability of death has a tendencyof deceleration at advanced ages when mortalityis high, taking into account that this mortalityindicator has a theoretical upper limit equal toone. For example, a study of mortality among su-percentenarians demonstrated that probability ofdeath for this group does not increase with age(Robine and Vaupel 2001). The authors do notprovide estimates of the hazard rate for this smallheterogeneous population, so at this moment itis difficult to make a conclusion about the realmortality trajectory in this sample of very oldindividuals. In the study of validated French-Canadian centenarians born in 1870–1894 theauthors found no growth of mortality with age forlife table death rates rather than hazard rates(Beaudry-Godin et al. 2008). It is true that foryoung and middle ages (when mortality is rela-tively low), probability-of-death and hazard ratevalues are numerically close. As a result, some au-thors do not distinguish between probability of
death and hazard rate in their calculations (LeBras 2005).
In the studies of period mortality, decelerationat advanced ages is probably caused by mixingdata for several birth cohorts with different mor-tality levels (heterogeneity effect) and usingcross-sectional instead of cohort data (Horiuchiand Wilmoth 1998; Kannisto 1994; Thatcher etal. 1998). In addition to that, if informationabout the population at risk is taken from cen-suses, there is always a possibility of a mismatchin the accuracy of age reporting between deathsand population at risk due to a higher proportionof age misreporting in censuses (the so-called de-nominator problem). It should be noted that mor-tality deceleration was also reported for cohortmortality data (Thatcher et al. 1998). In thiscase, mortality deceleration may be caused by agemisreporting in death data for older persons, aswe already found in our study. Earlier studies,conducted more than 10 years ago, used data forolder birth cohorts when age reporting was notparticularly accurate even for such countries asthe United Kingdom (Gallop and Macdonald2005). In addition, most developed countrieshave much smaller populations compared to theUnited States, and hence studies of mortality atadvanced ages for these countries have to com-bine many single-year birth cohorts, thereby in-creasing the heterogeneity of the sample. Thus,age exaggeration, use of probabilities of death in-stead of hazard rates, and perhaps data hetero-geneity could lead to downward biases in mortal-ity estimates at older ages reported in previousstudies.
In our study we found no significant mortalitydeceleration at advanced age for humans, whilemany less complex organisms demonstrate verylong late-age mortality plateaus (Carey et al.1992; Curtsinger et al. 1992; Gavrilov and Gav-rilova 1991; Vaupel et al. 1998). Earlier studiesshowed that the period of mortality decelerationin mammalian species is very short (Lindop 1961;Sacher 1966) compared to less complex organ-isms (Gavrilov and Gavrilova 2006; Vaupel et al.1998). It appears to be relatively short if not neg-ligible in humans too. This observation agreeswith the prediction of the reliability theory of ag-ing that more complex living systems/organismswith many vital subsystems (such as mammals)
MORTALITY MEASUREMENT AT ADVANCED AGES: A STUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE 445
may experience a very short period of mortalityplateau at advanced ages in contrast to less com-plex organisms (Gavrilov and Gavrilova 1991,2001, 2006). Thus, human mortality at advancedages may show a very short or even negligiblemortality plateau despite the evidence of mortal-ity leveling off among other organisms such asinsects.
The results obtained in this study may be im-portant for actuarial practice, particularly if mor-tality is analyzed for birth cohorts. For cohortdata, hazard rates can be extrapolated with theGompertz formula up to 107 years of age, andthen probabilities of death can be reconstructedwith the Sacher formula if necessary. Our studyalso suggests that improvements in age reportingshould result in mortality trajectories that betterfollow the Gompertz function until very advancedages. DMF data confirm that the 1900 actuarialcohort life table (Bell et al. 1992) provides a gooddescription of mortality at advanced ages, sug-gesting that cohort life tables obtained in thisstudy may indeed be useful in actuarial practice.
10. ACKNOWLEDGMENTS
We would like to thank three anonymous review-ers for their constructive criticism and useful sug-gestions. This study was made possible thanks togenerous support from the Society of Actuariesand the National Institute on Aging (NIA grantR01 AG028620). We are grateful to Thomas Ed-walds, Kenneth Faig, and Ward Kinkade for help-ful comments and suggestions on an earlierversion of this manuscript as well as to theparticipants of the 2008 and 2010 InternationalSymposiums ‘‘Living to 100’’ in Orlando, FL, or-ganized by the Society of Actuaries for stimulat-ing discussion.
REFERENCES
AUSTAD, S. N. 2001. Concepts and Theories of Aging. In Hand-book of the Biology of Aging, edited by E. J. Masoro and S.N. Austad, pp. 3–22. San Diego: Academic Press.
BEARD, R. E. 1959. Note on Some Mathematical Mortality Mod-els. In The Lifespan of Animals, edited by E. W. Wolsten-holme and M. O. O’Connor, pp. 302–311. Boston: Little,Brown.
BEARD, R. E. 1971. Some Aspects of Theories of Mortality, Causeof Death Analysis, Forecasting and Stochastic Processes. InBiological Aspects of Demography, edited by W. Brass, pp.57–68. London: Taylor and Francis.
BEAUDRY-GODIN, M., R. BOURBEAU, AND B. DESJARDINS. 2008. DataValidation and Measurement of Cohort Mortality amongCentenarians in Quebec (Canada) according to Ethnic Or-igin. In Living to 100 and Beyond [online monograph].Schaumburg, IL: Society of Actuaries.
BELL, F. C., AND M. L. MILLER. 2002. Life Tables for the UnitedStates Social Security Area 1900–2100. Actuarial Study No.116. Baltimore, MD: U.S. Department of Health and Hu-man Services.
BELL, F. C., A. H. WADE, AND S. C. GOSS. 1992. Life Tables forthe United States Social Security Area 1900–2080. Actuar-ial Study No. 107. Baltimore, MD: U.S. Department ofHealth and Human Services.
BRONIKOWSKI, A. M., S. C. ALBERTS, J. ALTMANN, C. PACKER, K. D.CAREY, AND M. TATAR. 2002. The Aging Baboon: Compara-tive Demography in a Non-Human Primate. Proceedings ofthe National Academy of Sciences of the USA 99: 9591–9595.
CAREY, J. R., P. LIEDO, D. OROZCO, AND J. W. VAUPEL. 1992. Slow-ing of Mortality-Rates at Older Ages in Large Medfly Co-horts. Science 258(5081): 457–461.
CASELLI, G., L. POZZI, J. W. VAUPEL, L. DEIANA, G. PES, C. CARRU,C. FRANCESCHI, AND G. BAGGIO. 2006. Family Clustering inSardinian Longevity: A Genealogical Approach. Experimen-tal Gerontology 41(8): 727–736.
CHARLESWORTH, B. 2001. Patterns of Age-Specific Means and Ge-netic Variances of Mortality Rates Predicted by Mutation-Accumulation Theory of Ageing. Journal of Theoretical Bi-ology 210: 47–65.
CLEVES, M. A., R. G. GUTIERREZ, W. W. GOULD, AND Y. MAR-CHENKO. 2008. An Introduction to Survival Analysis UsingStata. College Station, TX: Stata Press.
CURTSINGER, J. W., H. FUKUI, D. TOWNSEND, AND J. W. VAUPEL.1992. Demography of Genotypes: Failure of the LimitedLife-Span Paradigm in Drosophila melanogaster. Science258: 461–463.
CURTSINGER, J. W., N. S. GAVRILOVA, AND L. A. GAVRILOV. 2006.Biodemography of Aging and Age-Specific Mortality in Dro-sophila melanogaster. In Handbook of the Biology of Aging,edited by E. J. Masoro and S. N. Austad, pp. 261–288. SanDiego: Academic Press.
DEPOID, F. 1973. Mortality of Old People over 85. Population28(4–5): 755–792.
ECONOMOS, A. C. 1979. A Non-Gompertzian Paradigm for Mor-tality Kinetics of Metazoan Animals and Failure Kinetics ofManufactured Products. AGE 2: 74–76.
ECONOMOS, A. C. 1980. Kinetics of Metazoan Mortality. Journalof Social and Biological Structures 3(4): 317–329.
ECONOMOS, A. C. 1983. Rate of Aging, Rate of Dying and theMechanism of Mortality. Archives in Gerontology and Ger-iatrics 1: 3–27.
ECONOMOS, A. C. 1985. Rate of Aging, Rate of Dying and Non-Gompertzian Mortality—Encore. Gerontology 31(2): 106–111.
FAIG, K. 2001. Reported Deaths of Centenarians and Near-Centenarians in the U.S. Social Security Administration’sDeath Master File. Paper presented at the Society of Ac-
446 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 3
tuaries Living to 100 and Beyond International Sympo-sium, Orlando, FL.
GALLOP, A., AND A. S. MACDONALD. 2005. Mortality at AdvancedAges in the United Kingdom. In Living to 100 and BeyondMonograph [online edition]. Schaumburg, IL: Society ofActuaries.
GAVRILOV, L. A. 1980. Study of Life Span Genetics Using theKinetic Analysis. PhD thesis. Moscow: Moscow StateUniversity.
GAVRILOV, L. A. 1984. Does the Limit of the Life-Span ReallyExist? Biofizika 29(5): 908–909.
GAVRILOV, L. A., AND N. S. GAVRILOVA. 1991. The Biology of LifeSpan: A Quantitative Approach. New York: HarwoodAcademic.
GAVRILOV, L. A., AND N. S. GAVRILOVA. 2001. The Reliability The-ory of Aging and Longevity. Journal of Theoretical Biology213(4): 527–545.
GAVRILOV, L. A., AND N. S. GAVRILOVA. 2006. Reliability Theoryof Aging and Longevity. In Handbook of the Biology of Ag-ing, edited by E. J. Masoro and S. N. Austad, pp. 3–42. SanDiego: Academic Press.
GAVRILOVA, N. S., AND L. A. GAVRILOV. 2007. Search for Predic-tors of Exceptional Human Longevity: Using ComputerizedGenealogies and Internet Resources for Human LongevityStudies. North American Actuarial Journal 11(1): 49–67.
GEHAN, E. A. 1969. Estimating Survival Functions from Life Ta-ble. Journal of Chronic Diseases 21(9–10): 629–644.
GEHAN, E. A., AND M. M. SIDDIQUI. 1973. Simple Regression Meth-ods for Survival Time Studies. Journal of the American Sta-tistical Association 68(344): 848–856.
GOMPERTZ, B. 1825. On the Nature of the Function Expressiveof the Law of Human Mortality and on a New Mode of De-termining Life Contingencies. Philosophical Transactionsof the Royal Society of London A 115: 513–585.
GREENWOOD, M., AND J. O. IRWIN. 1939. The Biostatistics of Se-nility. Human Biology 11: 1–23.
HAINES, M. R. 1998. Estimated Life Tables for the United States,1850–1910. Historical Methods 31(4): 149–169.
HORIUCHI, S., AND J. R. WILMOTH. 1998. Deceleration in the AgePattern of Mortality at Older Ages. Demography 35: 391–412.
KANNISTO, V. 1988. On the Survival of Centenarians and the Spanof Life. Population Studies 42: 389–406.
KANNISTO, V. 1994. Development of Oldest-Old Mortality, 1950–1990: Evidence from 28 Developed Countries. Odense:Odense University Press.
KESTENBAUM, B., AND B. R. FERGUSON. 2002. Mortality of the Ex-treme Aged in the United States in the 1990s, Based onImproved Medicare Data. In Living to 100 and Beyond In-ternational Symposium [online edition] Schaumburg, IL:Society of Actuaries.
KIMBALL, A. W. 1960. Estimation of Mortality Intensities in An-imal Experiments. Biometrics 16(4): 505–521.
KLEIN, J. P., AND M. L. MOESBERGER. 1997. Survival AnalysisTechniques for Censored and Truncated Data. New York:Springer.
LE BRAS, H. 1976. Lois de mortalite et age limite. Population31: 655–692.
LE BRAS, H. 2005. Mortality Tempo versus Removal of Causes ofMortality: Opposite Views Leading to Different Estimationsof Life Expectancy. Demographic Research 13: 615–640.
LINDOP, P. J. 1961. Growth Rate, Lifespan and Causes of Deathin SAS/4 Mice. Gerontologia 5: 193–208.
MUELLER, L. D., AND M. R. ROSE. 1996. Evolutionary Theory Pre-dicts Late-Life Mortality Plateaus. Proceedings of the Na-tional Academy of Sciences USA 93: 15249–15253.
OLSHANSKY, S. J. 1998. On the Biodemography of Aging: A Re-view Essay. Population and Development Review 24: 381–393.
ROBINE, J.-M., A. COURNIL, J. GAMPE, AND J. W. VAUPEL. 2005.IDL, the International Database on Longevity. In Living to100 and Beyond Monograph [online edition]. Schaumburg,IL: Society of Actuaries.
ROBINE, J. M., AND J. W. VAUPEL. 2001. Supercentenarians:Slower Ageing Individuals or Senile Elderly? ExperimentalGerontology 36(4–6): 915–930.
ROSENWAIKE, I., AND L. F. STONE. 2003. Verification of the Agesof Supercentenarians in the United States: Results of aMatching Study. Demography 40(4): 727–739.
SACHER, G. A. 1956. On the Statistical Nature of Mortality, withEspecial Reference to Chronic Radiation Mortality. Radi-ology 67: 250–257.
SACHER, G. A. 1966. The Gompertz Transformation in the Studyof the Injury-Mortality Relationship: Application to LateRadiation Effects and Ageing. In Radiation and Aging, ed-ited by P. J. Lindop and G. A. Sacher, pp. 411–441. Lon-don: Taylor and Francis.
SESSO, H. D., R. S. PAFFENBARGER, AND I. M. LEE. 2000. Compar-ison of National Death Index and World Wide Web DeathSearches. American Journal of Epidemiology 152(2): 107–111.
STATACORP. 2009. Stata Statistical Software: Release 11. CollegeStation, TX: StataCorp.
STREHLER, B. L., AND A. S. MILDVAN. 1960. General Theory ofMortality and Aging. Science 132(3418): 14–21.
TATAR, M., J. R. CAREY, AND J. W. VAUPEL. 1993. Long-Term Costof Reproduction with and without Accelerated Senescencein Callosobruchus maculatus: Analysis of Age-Specific Mor-tality. Evolution 47: 1302–1312.
THATCHER, A. R. 1999. The Long-Term Pattern of Adult Mortalityand the Highest Attained Age. Journal of the Royal Statis-tical Society Series A–Statistics in Society 162: 5–30.
THATCHER, A. R., V. KANNISTO, AND J. W. VAUPEL. 1998. The Forceof Mortality at Ages 80 to 120. Odense: Odense UniversityPress.
VAUPEL, J. W., J. R. CAREY, K. CHRISTENSEN, T. E. JOHNSON, A. I.YASHIN, N. V. HOLM, I. A. IACHINE, V. KANNISTO, A. A. KHA-ZAELI, P. LIEDO, V. D. LONGO, Y. ZENG, K. G. MANTON, AND
J. W. CURTSINGER. 1998. Biodemographic Trajectories ofLongevity. Science 280(5365): 855–860.
VAUPEL, J. W., K. G. MANTON, AND E. STALLARD. 1979. Impact ofHeterogeneity in Individual Frailty on the Dynamics of Mor-tality. Demography 16(3): 439–454.
VINCENT, P. 1951. La Mortalite des vieillards. Population 6: 181–204.
MORTALITY MEASUREMENT AT ADVANCED AGES: A STUDY OF THE SOCIAL SECURITY ADMINISTRATION DEATH MASTER FILE 447
WATSON, G. S., AND M. R. LEADBETTER. 1964. Hazard Analysis. I.Biometrika 51(1–2): 175–184.
WILLCOX, D. C., B. J. WILLCOX, Q. HE, N. C. WANG, AND M. SUZUKI.2008. They Really Are That Old: A Validation Study of Cen-tenarian Prevalence in Okinawa. Journals of GerontologySeries A–Biological Sciences and Medical Sciences 63(4):338–349.
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