The Long-Term Pattern of Adult Mortality and the Highest Attained Age Author(S): A

The Long-Term Pattern of Adult Mortality and the Highest Attained Age Author(s): A. R. Thatcher Source: Journal of the Royal Statistical Society. Series A (Statistics in Society), Vol. 162, No. 1 (1999), pp. 5-43 Published by: Blackwell Publishing for the Royal Statistical Society Stable URL: http://www.jstor.org/stable/2680465 Accessed: 05/10/2010 21:57

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http://www.jstor.org J. R. Statist.Soc. A (1999) 162, Part 1,pp. 5-43

The long-termpattern of adult mortalityand the highest attained age

A. R. Thatcher New Malden, UK

[Read beforeThe Royal StatisticalSociety on Wednesday, June 17th, 1998, the President, Professor R. N. Curnow,in the Chair]

Summary. Recerrtnew data on old age mortalitypoint to a particularmodel for the way in whichthe probabilityof dying increases withage. The model is foundto fitnot only modern data butalso some widelyspaced historicaldata forthe 19thand 17thcenturies, and even some estimatesfor the early mediaeval period.The resultsshow a patternwhich calls forexplanation. The model can also be used to predicta probabilitydistribution for the highest age which will be attained in given circumstances.The resultsare relevantto the currentdebate about whetherthere is a fixedupper limitto the lengthof human life. Keywords: Aging;Extreme values; Lifespan;Logistic model; Longevity;Mortality

1. Introduction 1. 1. Background At everyage, thereis a probabilityof dyingwithin 12 months.From soon after30 yearsof age thisprobability starts to rise by about 10% witheach successiveyear of age. In round numbers,the probability of dying(for modern males) increasesfrom 0.001 at age 30 to 0.1 at age 80 years.This inexorableincrease follows quite closelythe 'law of mortality'which was discoveredby Gompertz(1825). By 80 years the rate of increaseis noticeablyslower and thereis controversyabout what happens at higherages still.There are alternativemodels, whichwe shall discuss. In the 19thcentury, the probabilitiesof dyingat ages 30-80 yearswere much higherthan theyare today. However,until relatively recently it was widelybelieved that, although there had been spectacularimprovements at lowerages, therehad been littlechange in the probabilitiesof dyingabove 80 yearsof age. Many believedthat, somewhere above 100 yearsold, thereis a maximumlifespan which has remainedunchanged since ancient times. In a veryinfluential paper, Fries (1980) expressedhis beliefas a physicianthat thereis a naturallimit to thelength of human life, even in theabsence of disease,and thatthe maximum lifespandoes not change.However, the average length of lifehas risendramatically since the mid-19th century. This combinationof a risingaverage and a fixedupper limit means that the shape of thesurvival curve has becomemore rectangular. Fries thought that the expectation of life at birthcan be expectedto increaseto an 'ideal average life span' of about 85 years. Chronicillnesses can be postponedby changesin lifestyle. Adult vigourcan be extendedto higherages, but stillwithin the same maximumlifespan. As a result,morbidity will become compressedinto a period of senescencebefore the end of life.A researchstrategy on aging should be fundamentallyshifted, towards postponing chronic illness and maintainingvigour.

Addressfor correspondence:A. R. Thatcher,129 ThetfordRoad, New Malden, Surrey,KT3 5DS, UK.

? 1999 Royal StatisticalSociety 0964-1998/99/162005 6 A. R. Thatcher These viewswere controversial from the beginning and therehas been a livelydebate which is stillin progress.We have space to mentiononly a fewreferences, to give the flavour. Mantonet al. (1991) said that,although traditionalists might not expectthe average length of lifeto riseabove 85 years,there were nevertheless reasons for thinking that, with healthier lifestyles and continuingmedical progress and intervention,it mightbe possibleto raise the expectationof lifein the USA to 95 or even 100 years. Olshanskyet al. (1990, 1991) weremore sceptical, arguing that it would requirevery large fallsin death-ratesat highages to have muchfurther effect on theexpectation of lifeat birth. Longer lifemight often be attainableonly at the cost of a prolongedperiod of frailtyand dependence.More researchwas needed to improvethe quality of life of the elderlyby amelioratingnon-fatal diseases. Vaupel, quoted in Baringa (1991), challengedFries. Swedishdata showed that mortality ratesat ages 85 yearsand over had been fallingfor 50 years.Data on the lengthof lifeof a millionfruit-flies showed that their probability of dyingstarts by increasingwith age but then levelsoff. There was no evidenceof a biologicallimit to life.Gavrilov and Gavrilova (1991) reachedthe same conclusionindependently.

1.2. The maximumlength of life If thereis a virtuallyfixed upper limit to thelength of humanlife, then further falls in death- rates will make the survivalcurve even more rectangularand deaths will eventuallybe compressedinto a narrowerband of ages. In contrast,if thereis no fixedupper limit, or if a fixedlimit exists but is far higherthan the ages whichhave so farbeen attained,then there will be less rectangularizationand less compressionof theages at death,but people may live even longer. These differentpossible outcomeswill be reflectedin differentpatterns for the probabilitiesof dyingabove age 80 years.In principlethere are threemain possibilities. The firstis that the probabilityof dyingwithin 12 monthsmay reach 1 at a finiteage. There will thenbe a definite,fixed upper limit to the lengthof human life.Many people believethere to be such a fixed,definite limit. The French demographerVincent (1951) thoughtthat therewere enormousodds againstanyone exceeding the age of 110 years.Fries (1980) and thebiologist Hayflick(1994) placed the upper limitat about 115 years,and indeed Hayflickused this supposed limit as a reason for rejectingvarious medical theories. Most recentlythe mathematicalstatisticians Aarsen and de Haan (1994) have inferred(though fromvery limiteddata) thatthere is an upperlimit to lifewhich lies, with 95% confidence,between 113 and 124 years. The secondpossibility is thatthe probability of dyingwithin 12 monthsdoes not reach 1 at a finiteage, but neverthelesstends asymptotically to 1. In thiscase thereis no definite,fixed upperlimit, but as age increasesthe probability of furthersurvival eventually becomes so tiny that for practicalpurposes it can be ignored.This is the scenariowhich is impliedby the originallaw of Gompertz(1825) and also by the more recentmodels of Weibull (1951) and Heligmanand Pollard (1980). The thirdpossibility is that the probabilityof dyingwithin 12 monthsmay continueto increasemonotonically as age increasesbut may tendto a limitwhich is less than 1. This is what is impliedby the 'logisticmodel', whichwas firstapplied to mortalityby Perks (1932) and later,under different names and notations,by severalothers. In thismodel thereis no definite,fixed upper limit to lifeand thereis not even an age whenthe probabilityof further survivalis negligible.However high the age, therewill always be a substantialprobability of Mortalityand HighestAttained Age 7 survivingfor at least 1 more year. This model tends towards a Poisson process, so the survivorswill graduallyapproach (but not quite reach) a statelike thatof radioactiveatoms awaitingdecay, with a half-life.Nevertheless, all the survivorswill die in the end and there willbe a probabilitydistribution for the highest age whichwill be attainedin a givengroup of survivors.This will depend on the absolute size of the group. Each of these theorieshas its supportersamong biologists,so there is at presentno consensusin thatquarter. These variousmodels, if considered solely from a theoreticalpoint of view withoutany referenceto actual data, providealmost unlimitedscope forargument about whetherthere is an upper limitto life.

1.3. New data Previousstudies of theforce of mortalityat highages have been based on data forindividual countries.More detailedanalyses can be made when thereare largernumbers, obtained by pooling the data for similarcountries. In 1990 Peter Laslett set up the CambridgeGroup project on the maximal lengthof life,and its members(Kannisto, Thatcherand Vaupel) assembledand computerizeda new database. This is part of theArchive on PopulationData on Aging,funded by the US National Instituteon Agingand the Danish researchcouncils. The database now containsall the available officialstatistics on deathsat ages 80 yearsand overin 30 countriessince 1960,and in manycases earlier.The fulldata are currentlyheld at the Universityof Odense and at the Max Planck Institutefor Demographic Research at Rostock. Copies will shortlybe placed in the Danish State Archiveand publishedin other ways so that theywill be available to both institutionsand individualresearch workers on request. Resultsfrom the archiveare beingpublished in a seriesof monographs(Kannisto, 1994, 1996;Jeune and Vaupel, 1995). The latestin the seriesis The Force ofMortality at Ages 80 to 120 (Thatcheret al., 1998). In thisbook, six possiblemodels for mortality above age 80 years are fittedto each of eight data sets, each comprisingpooled data for 13 industrialized countries.The data sets are for the overlappingperiods 1960-1970, 1970-1980 and 1980- 1990, and for the cohorts born in 1871-1880, for males and femalesseparately. As an indicationof size, the 13 countriesinclude almost 40 millionpeople who reached80 yearsof age and over 120000 who reached 100, duringthe period 1960-1990. They cover over 32 milliondeaths at ages 80 yearsand overin thisperiod. The data used in theseanalyses, which provide one of the foundationsfor the presentpaper, are given in full in Thatcheret al. (1998). At thesehigh ages, thenew data are foundto be clearlycloser to thelogistic model thanto the Gompertz,Weibull and Heligman and Pollard models. This findingis taken as the startingpoint forthe presentpaper.

1.4. Aimsand outlineof thepresent paper The firstsubstantive question to be addressed is whymortality should follow the logistic model. We shall describethree relevant theories, namely the so-called 'fixedfrailty' model (Beard, 1971;Vaupel et al., 1979),the stochastic process model of Le Bras (1976) and a recent controversialtheory based on genetics(Mueller and Rose, 1996). Next,we shall investigatehow well a three-parameterform of the logisticmodel fitsdata forthe past as wellas thepresent, and at all adultages, notjust highages. For this,the model is fittedto some widelyspaced historicallife-tables, from Halley's life-tableonwards, and also to some evenearlier estimates for the mediaeval period. The model fitsthese data ratherwell. 8 A. R. Thatcher Moreover, the parametersreveal a long-termpattern of adult mortalitywhich on one interpretationimplies that the so-called 'rate of aging' has been remarkablystable over many centuries. The fittedparameters can also be used to tackle two difficultproblems. The firstis to estimatethe expectations of lifeat highages in thehistorical periods on a uniformbasis. The second is to estimatethe probabilitydistributions of the highestages which one may reasonablysuppose to have been attainedin Englandfrom the Middle Ages onwards.We are here dealing withthe highestages whichwere actuallyattained in the circumstancesof the time,not withthe potentialages whichmight have been attainedif the circumstanceshad been different. As a finalillustration, we show the impliedtrends over the centuriesof the mode of the distributionof the highestattained ages in stationarypopulations of various sizes.

2. The logistic model 2. 1. Notation We shall use the terminologyand notationwhich is traditionalin thisfield. The probability that a person who has just reachedage x will die within12 monthsis denoted qg, in the actuarial notation.The probabilitythat he or she will die in the instantbetween age x and age x + dx is ,uxdx, where,ax is known as the forceof mortality.In the literature,this is sometimesdescribed as the instantaneousdeath-rate, and it is also the same as the hazard rate in survivaltheory. It is sometimeswritten as p(x) and sometimesdescribed as the mortalityfunction. For adult mortality,the numerical values of ,u, can be estimatedor derivedin severalways, fromage-specific death-rates, from the survivalcurves in life-tablesor cohortdata, or from the probabilityof dyingwithin 12 months(q,). Conversely,qx can be derivedfrom pu,. The relevantformulae are givenin AppendixA.

2.2. Statementof themodel The logisticmodel assumesthat the forceof mortality,ux is a logisticfunction of the age x. The mostgeneral model of thistype has fourindependent parameters and can be expressedin the form

P. =I -Yz+ (1) 1 + Z where

z = ae exp(3x) (2a) = exp{l(x - q)}, = -ln(oa)/O. (2b) However,for studying long-term historical trends and predictinghighest attained ages, this fullygeneral model is found to be less usefulin practicethan a simplerbut more robust version,with only three parameters. When thegeneral model (1) was fittedby Thatcheret al. (1998) to the new data for 13 industrializedcountries, the fittedparameters showed thatti is near 1. This leads us to considerthe three-parametermodel z (3) Mortalityand HighestAttained Age 9 wherez is definedas before.If we use equation (2a) theparameters are oa, and -y.If we use equation (2b) theyare ,B,-y and q. These two ways of definingthe parameters are completely equivalentand each will be foundto have its advantages. Model (3) has two importantproperties. When z is small(as at ages up to 70 yearsor so) it impliesthat p., - ae exp(3x) + -y, (4) whichis the'law' proposedby Makeham (1860), and thiswas foundto be stilluseful on much more recentdata frommany countries by Gavrilovand Gavrilova(1991). This also includes as a special case the originallaw of Gompertz,namely

c.t, 0 exp(3x). (5) The second importantproperty is thatwhen -y is smallcompared with z/(l + z), whichwas true at high ages even in the historicalperiods (and at almost all ages today, when -yis negligible),model (3) impliesthat

logit(,u_); ln(oa)+ Ox. (6) This is theapproximate relationship for old age mortalitywhich was firstnoted by Kannisto (1992), used independentlyby Himes et al. (1994) and since overwhelminglyconfirmed as a good approximationby Thatcheret al. (1998). What is just as importantfor the present applicationis that model (6) is a robustmodel whichhas been found to give good results whenfitted to data below age 100 yearsand thenextrapolated to higherages, whichis exactly the operationthat we shall require.The same advantagewill thereforeapply to model (3). Accordingly,we shall adopt equation (3) as thepreferred model forinvestigating the long- termhistorical trends of adult mortality.

2.3. Comparisonswith other models It is of interestto see how the logisticmodel (3) and the Gompertzmodel (5) comparewith the two othercontending models whichwere mentionedin Section 1. One of these is the model of Weibull(1951), namely

,U = ax3. (7) The otheris the law of mortalityused forgraduation by Heligmanand Pollard (1980). This has threeterms and eightparameters, but at highages the firsttwo termscan be neglected and the thirdcan be writtenin the form

logit(q,) = ae+ Ox, (8) where ae and f are constants. We may note that, as x - oo, qx -- 1 and hence logit(qx) -ln( - q). Also we have - ln(l - qx) i,u(x+ 2), from expression (23) in Appendix A. On substitutingthese approximations in equation(8) we see thatin theHeligman and Pollard model theforce of mortalitytends asymptotically to thestraight line lIx = a - / + Ox. This can also be provedmore formally. When thesefour models are fittedto actual data, theyare all relativelyclose to thedata at theages wheremost of the deathsare concentrated,and hencerelatively close to each other. At higherages, thefour models necessarily diverge because theytend to differentasymptotes (an exponential,a power function,a sloping straightline and a horizontalstraight line). 10 A. R. Thatcher

0 80 90 100 110 120 Age

Fig. 1. Fourmodels for the force of mortality: Gompertz;*- - Weibull;...... Heligman and Pollard; ------logistic

Fig. 1 illustratesthe size of thisdivergence in a practicalcase, whenthe four models are fitted to the same data at ages 80-98 yearsand thenextrapolated to 120 years.In thisparticular examplethe data werefor the deathsof over 8 millionfemales in 13 industrializedcountries in 1980-1990,but thisis immaterial.Whatever the data set,the fitted models always diverge. As already stated,the new pooled data for 13 industrializedcountries are close to the logisticmodel. If we use one of theother models, we shall overestimatethe force of mortality at the highestages and hence underestimatethe highestage which will be attained in a populationof givensize.

2.4. Explanatorytheories 2.4.1. Thefixed frailty model The firsttheory which was put forwardto explainwhy ,u, should be a logisticfunction of age was the fixedfrailty model, originallyproposed by Beard in 1959 (see Beard (1971)). It is based on the assumptionthat the populationis heterogeneous.The ith memberof a birth cohortis subjectto a personalhazard functionwhich follows Makeham's law

Hi(X)= aitexp(/3x) + -y, (9) whereali (the frailty)varies fromindividual to individualbut is fixedfrom (say) 30 years of age onwards.The constants 3 and -yare assumed to be the same forall individuals.The relativevariation in the ai betweenindividuals is assumed to followa gamma distribution withvariance a. As the cohortbecomes older, those withhigh frailty are more likelyto die firstand thereis an effectwhich may be describedas survivalof the fittest.The resultis that the mix of values of ,pigradually changes. It can be shown that the average value of tti(x) among the survivorsfollows the generallogistic model (1), withti givenby Mortafityand HighestAttained Age 11

52 = (10)

This model was laterdiscovered independently by Vaupel et al. (1979). It is oftencalled the 'gamma-Makeham' model.

2.4.2. The Le Bras stochasticprocess Le Bras (1976) assumed that aging can be modelled by a stochasticprocess, in which individualsprogress by jumps at randomtimes through a successionof steadilydeteriorating states. He considereda cohortwhich startsas homogeneous,with all the membersin the same state.As theygrow older, the membersgradually become spread over manystates and the cohortbecomes increasingly heterogeneous. No assumptionis made about how theforce of mortalitydepends on age. Nevertheless,when the cohortreaches age x therewill be an averageforce of mortality,i(x) which,on certainassumptions, follows the logistic model (1). A slightlygeneralized version of the Le Bras model was developed by Gavrilov and Gavrilova (1991), pages 247-250. In any intervalof timedt, the individualsin the ith state will (in theirnotation) have a probability(po + ip) dt of dyingand a probability(A0 + iA)dt of jumpingto the (i+ 1)th state,where po, ,u,AO and A are constantparameters. It can be shown that,if this model is to fitthe data, a modernperson would have to pass through about 100000 statesbefore becoming a centenarian.Also puomust be small compared with A0,p mustbe small comparedwith A and the parametersin equation (1) will satisfy

S , ~~~~~~~~~~(11) K A0.

Thus ti will depend on the initialrate of transitionbetween states fromthe homogeneous opening start,whereas 3 will depend on how fast this rate of transitionchanges between successivestates.

2.4.3. Biologicaltheories of aging Thereis an enormousliterature on biologicaltheories of agingand herewe can mentiononly those points whichare known to the author and which seem particularlyrelevant to the models of mortalitydiscussed in thispaper. For life to continue,the internalworkings of the body have to be kept in balance (homeostasis).This is achieved by compensatingmechanisms in the heart,lungs, kidneys, liveretc. If the balance is disturbed,the body draws on its reservesof capacity to restore homeostasis.In youngadults, the functional capacity of theorgans is 4-10 timesthat needed to sustainlife. After about 30 years of age, however,the capacity of the organs startsto decline steadily.Measurements of vital capacity,maximum heart rate, maximumoxygen consumption,the rate of cell renewal,etc., all declinewith age, and withthem the abilityto restorehomeostasis. Why does the reservecapacity decline with age? Accordingto evolutionarytheory, the body mustspend some of its resourceson the immediateneeds of survival(e.g. the abilityto findfood and to avoid predators),some on reproductionand some on the maintenanceand repair of its bodily functionsas they sufferdamage. This damage can arise eitherfrom externalcauses, or fromwear and tear or fromthe accumulationof DNA mutationsand damage at themolecular and cellularlevel. (Whenever a cell dividesits DNA moleculesmust be copied,and on each occasion thereis a verysmall chance of damage.) However,the more the resourceswhich must be spenton immediatesurvival and reproduction,the less are the 12 A. R. Thatcher resourceswhich are available forrepair and maintenance.A balance mustbe struckbetween the two. This balance is struckby natural selection.If a species is to survive,it is imperative that enough of its membersshould live sufficientlylong to reach the age of reproduction. What happens afterthat does not mattervery much, from the point of view of evolution. Accordingly,when we pass the age of reproduction,we are leftwith a body witha repair mechanismwhich was primarilydeveloped to reach the age of reproduction,not to live beyondit. In fact,the deteriorationin reservecapacity starts after the beginningof the reproductive period but well beforeits end. Two processesthen come into play. In the first,genes which have positiveeffects on fitnessearly in lifecan sometimeshave negativeeffects on survivallate in life(antagonistic pleiotropy). In the second,detrimental mutations which act only late in lifeare not efficientlyeliminated by selectionand can accumulate(mutation accumulation). Medawar (1952) describedthe post-reproductiveperiod as a 'geneticdustbin'. Both these processesare made possible by the declinein the force of selectionwith age. The physi- ological and biochemicalmanifestations of aging have evolved as the resultof a reduced investmentin maintenanceand repairbrought about by the two effectsmentioned (Williams, 1966;Kirkwood, 1981). The highlycondensed summary given in thisparagraph is takenfrom Stearns(1992), p. 182. These processesare believed to account for the Gompertz-typeincrease in the forceof mortalityafter the age of reproduction.However, there are now some biologicalreasons for believingthat this deterioration may not continueindefinitely. Extensive work on fruit-flies has shownthat their force of mortalityincreases sharply with age but thenreaches a plateau, or even shows a fall (Carey et al., 1992; Curtsingeret al., 1992). Geneticistshave now pro- duced some mathematicalmodels to explain how such a plateau can arise, fromparticular combinationsof the effectsof selection,accumulated mutation and antagonisticpleiotropy (Mueller and Rose, 1996). One of the findingsof theirstudy is that it is possible for the plateau to be reachedwhen the probabilityof dyingq, is less than 1, whichimplies that the forceof mortalitycan tend to a finitelimit. This theoryis controversialand will no doubt receivecritical examination. If theirarguments (or an amended version)command support and can be applied to humans,they would supplya biological reason forthe upperlimit for p, in the logisticmodel.

2.4.4. Comments No doubt all the theoriesoutlined in Sections 2.4.1-2.4.3 contain elementsof truth.The population is heterogeneous.Genes are important.Also, although responses to life- threateningevents and bodily developmentswill depend on geneticmake-up, the actual incidenceand timingof theevents may resemblea stochasticprocess. However, none of these theoriesseems fullysatisfactory as it stands and thereis scope for furtherresearch on the reasonswhy the logisticmodel fitsthe mortalitydata. An evenmore difficult question is whythe data shouldbe fittedby theparticular model (3), i.e., to put the question anotherway, why is the parameteri, in the more generallogistic model (1) apparentlyquite close to 1, whenage is measuredin years?Equations (10) and (11) suggestthat possible relevantfactors include the degreeof heterogeneityin the population and/orthe patternof transitionsbetween states, whereas the biologicaltheories suggest that it depends on the balance betweendifferent types of genes. In the presentpaper, we shall simplyregard i, = 1 as a convenientworking hypothesis. Mortalityand HighestAttained Age 13 3. Long-termtrends

3. 1. Historicaldata In thissection, we shallbe concernedwith trends in mortality(at ages 30 yearsand over) over verylong periodsindeed. The simplestway to obtain a pictureof the main changeswhich have takenplace overthe centuries is to examinea selectionof mortalitydata at verywidely spaced dates. For this,the most convenientsources of data are publishedlife-tables. We shall beginwith English Life Tables No. 14, whichrelate to England and Wales in the period 1980-1982.These tableswere prepared by the GovernmentActuary and publishedby theOffice of PopulationCensuses and Surveys(1987). They are based on deathsin the three years 1980-1982combined with population estimates derived from the 1981 censusof population. In the threeyears 1980-1982 therewere about 834000 deaths of males and about 828000 deaths of femalesat ages 30 yearsand over. We now jump back over 140 years to EnglishLife Table No. 1. This was compiled by WilliamFarr fromthe 1841 censusand thethen newly established civil registrations of births and deaths in England and Wales. It was publishedby the RegistrarGeneral (1843). The table was based on the deaths in England and Wales in the singleyear 1841, duringwhich therewere about 71 000 deathsof males and about 74000 deathsof femalesat ages 30 years and over. Another 150 years back and we have Halley's life-table,based on the ages at death in Breslauin 1687-1691.This has been extensivelydiscussed by, among others,Pearson (1978) and Hald (1990). Pearsonconcluded that Halley's life-tableprovided the best of theavailable estimatesof mortalityfor this historical period. It is estimated,from the informationquoted by Hald, thatthere were about 2675 deaths at ages 30 yearsand over in Breslau in the five years 1687-1691. We now come to a veryrecent development. Wrigley et al. (1997) have publisheda new life-tablefor the period 1640-1689,compiled by familyreconstitution from the entriesin parishregisters for married couples who were born and marriedin 26 parishesin England. Their database included3133 deaths at ages 30 yearsand over,which is slightlymore than Halley's. On comparingtheir results with his, we findthat they are remarkablyclose. This is a veryencouraging agreement on the forceof mortalityin the second halfof the 17thcentury. Beforethe 16thand 17thcenturies no life-tablesare generallyaccepted as reliable,except forsmall groups like monks. For thelate mediaevalperiod (the 13th-I5th centuries) there are some life-tablesconstructed by Russell (1948) fromthe informationwhich was recorded whenfeudal tenancies changed hands. For the early mediaeval period, a life-tablewas constructedby Acsadi and Nemeskeri (1970), table 130, fromages at death estimatedfrom forensic examinations of 2300 adult skeletonsin Hungariancemeteries of the 10th-12thcenturies. These cannot be regardedas veryreliable, because a recentstudy has shownthat the forensic methods used by Acsadi and Nemeskeritended to overestimateadult ages below 60 yearsand to underestimateages above 60 years(Molleson and Cox, 1993). However,this does not mean thatthe life-table cannot be used. If the methodunderestimated high ages, we can at least expectthat the highestages attainedin themediaeval period will have been higherthan those predicted from Acsadi and Nemeskeri'slife-table. Acsadi and Nemeskeri(1970) also constructedlife-tables for the Roman era, based on inscriptionson tombstones,as distinctfrom the examinationof skeletons,but we shall not make use of these.Doubts have been expressedabout theaccuracy of theages and about the representativenessof thosewho were giventombstones. Also, the resultingestimates of the 14 A. R. Thatcher Table 1. Historicalestimates of the force of mortalitycompared with values fittedby the logisticmodel (3)t

Age Observedforce Fitted force of Relativeerr or Observedforce Fitted force of Relativeerror group(years) ofmortality mortality (%) ofmortality mortality (%) A, Hungary,10th-12th centuries 30-35 0.0235 0.0232 - 1.2 35-40 0.0291 0.0281 -3.6 40-45 0.0337 0.0351 4.2 45-50 0.0402 0.0453 12.6 50-55 0.0696 0.0598 - 14.1 55-60 0.0814 0.0804 - 1.3 60-65 0.1033 0.1090 5.5 65-70 0.1485 0.1481 -0.3 70-75 0.1877 0.1996 6.4 75-80 0.3008 0.2651 - 11.9 80-85 0.3442 85-90 0.4341 90-95 0.5295 95-100 0.6237 -0.4 B, England,1640-1689 C, Halley,Breslau, 1687-1691 30-35 0.0171 0.0171 0.2 0.0164 0.0183 11.8 35-40 0.0205 0.0188 - 8.2 0.0195 0.0198 1.4 40-45 0.0195 0.0214 9.6 0.0233 0.0220 - 5.6 45-50 0.0244 0.0253 3.7 0.0282 0.0255 -9.7 50-55 0.0307 0.0313 1.9 0.0342 0.0308 -9.9 55-60 0.0459 0.0403 - 12.1 0.0383 0.0390 1.9 60-65 0.0513 0.0540 5.3 0.0474 0.0516 8.8 65-70 0.0701 0.0743 6.0 0.0630 0.0705 11.9 70-75 0.1129 0.1039 -7.9 0.0995 0.0986 -0.9 75-80 0.1445 0.1461 1.1 0.1589 0.1392 -12.4 80-85 0.1974 0.2040 3.3 0.1958 85-90 0.2794 0.2708 90-95 0.3716 0.3638 95-100 0.4757 0.4700 -0.3 -0.3 (continued)

forceof mortalitydo not look verycredible: at ages 30-40 yearsthey are in linewith the early mediaevalestimates but at ages 70-90 yearsthey are below the modernestimates. For examiningthe very-long-term trends in theforce of mortalityat ages 30 yearsand over, we shall use the 'snapshots'provided by the seven sets of data in the life-tablesdescribed above. We shall treatmales and femalesseparately where this is possible,i.e. in 1980-1982 and 1841. For each of theseseven data sets,the averageforce of mortalityover 5-yearage groupshas been derivedby themethod described in AppendixA. The resultsare givenin the columnsheaded 'Observed'in Table 1, and theyare plottedin Fig. 2. Observedvalues are not givenfor the veryhighest age groups,where the life-tablesare constructedby approximate methods. Many importantfeatures of the long-termchanges in theforce of mortalitycan be seen fromFig. 2. Startingfrom the top, line A gives the estimatesfor the 10th-12thcenturies, whichof course are the least reliableof our data. However,no-one will be surprisedto find that the estimatedforce of mortalityin thesecenturies was higherthan in the 17thcentury (lines B and C). Mortalityand HighestAttained Age 15 Table 1 (continued)

Age ObservedforceFittedforce of Relativeerror ObservedforceFittedforce of Relativeerror group(years) ofmortality mortality (%) ofmortality mortality (%) D, Englandand Wales,1841, males E, Englandand Wales,184], females 30-35 0.0108 0.0110 2.2 0.0107 0.0107 0.1 35-40 0.0123 0.0119 -3.2 0.0118 0.0114 -3.5 40-45 0.0140 0.0133 -4.7 0.0131 0.0125 -4.4 45-50 0.0159 0.0157 -1.2 0.0145 0.0145 -0.3 50-55 0.0181 0.0196 8.4 0.0162 0.0177 9.2 55-60 0.0254 0.0260 2.4 0.0220 0.0231 5.0 60-65 0.0375 0.0364 -2.9 0.0331 0.0321 -2.9 65-70 0.0553 0.0532 -3.8 0.0493 0.0470 -4.6 70-75 0.0815 0.0797 -2.2 0.0736 0.0712 -3.3 75-80 0.1201 0.1205 0.4 0.1097 0.1094 -0.3 80-85 0.1771 0.1807 2.1 0.1638 0.1674 2.2 85-90 0.2617 0.2643 1.0 0.2448 0.2502 2.2 90-95 0.3884 0.3710 -4.5 0.3674 0.3583 -2.5 95-100 0.4933 0.4847 -0.5 -0.2 F, Englandand Wales,1980-1982, males G, Englandand Wales,1980-1982, females 30-35 0.0010 0.0007 -23.0 0.0006 0.0007 21.2 35-40 0.0014 0.0015 8.4 0.0009 0.0010 10.4 40-45 0.0024 0.0027 15.4 0.0016 0.0016 -1.2 45-50 0.0043 0.0047 8.9 0.0028 0.0025 - 10.4 50-55 0.0079 0.0080 0.8 0.0047 0.0040 - 13.8 55-60 0.0138 0.0133 -3.5 0.0076 0.0067 -11.2 60-65 0.0227 0.0220 -3.3 0.0119 0.0114 -4.5 65-70 0.0365 0.0358 -1.9 0.0187 0.0193 3.0 70-75 0.0587 0.0577 - 1.7 0.0308 0.0326 5.8 75-80 0.0930 0.0916 -1.5 0.0527 0.0548 3.9 80-85 0.1432 0.1422 -0.7 0.0919 0.0908 - 1.2 85-90 0.2110 0.2141 1.5 0.1567 0.1469 -6.3 90-95 0.2900 0.3092 6.6 0.2374 0.2290 - 3.5 95-100 0.3894 0.4236 8.8 0.3215 0.3390 5.4 1.1 -0.2 tAveragevalues of btt in 5-yearage groups.

The fallin theforce of mortalitybetween the 17thcentury (lines B and C) and 1841(lines D and E) is quite clear,at least up to 80 yearsof age. However,this fall is small comparedwith the large relativechanges between 1841 (lines D and E) and 1980-1982 (lines F and G). By far the most importantsingle reason for thischange was the virtualconquest of infectious diseases, followingimprovements in livingconditions, water-supplies, sanitation, hygiene, vaccinationand later medical improvements.This period also shows the emergenceof the sex ratioof mortality,which appears as thegap betweenlines F and G. Thereis, of course,a vast amount of detailed informationavailable about the changes in mortalitysince 1841, includingthe changes in causes of death,which has been admirablysummarized by theOffice forNational Statistics(1997).

3.2. Fittingthe model Figs 2-9 givea veryclear picture of how theforce of mortalityhas changedover the centuries and how well the logisticmodel (3) fits. 16 A. R. Thatcher

0.00

-0.50

-1.00

0 E 0 0 0 0 0

-2.50-3.50 - A

-3.00-

h.oD

-3.50 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Age

Fig. 2. Observed values of the forceof 1Oth-1 mortality: A, Hungary, 2th centuries;B, England, 1640-1689; C, Halley,Breslau, 1687-1691; D, Englandand Wales, 1841, males; E, Englandand Wales, 1841,females; F, Englandand Wales,1980-1982, males; G, Englandand Wales, 1980-1982,females

0.0

-0.2

-0.4

-0.6

0 E -0.8 0 S 2-1.0 0

.0 -1.2

-1.4-

-1.6-

-1.8 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Age

Fig. 3. Forceof mortality inHungary, 10Oth-i 2th centuries:-...... observedvalues; ~, fittedlogistic model Mortalityand Highest AttainedAge 17

-0.2

-0.4

-0.6

-0.8 0 E o -1.0 0

0 -1.2

-1.4

-1.8 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Age Fig. 4. Forceof mortality in England, 1640-1689: .*.--.-, observedvalues; ,fittedlogistic model

-0.2

-0.4

-0.6

> -0.8

0 E 1.0 0 o -1.2 0

0 * -1.4

-1.6-

-1.8

-2.0 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Age Fig. 5. Forceof mortality in Breslau, 1687-1691: .---.-., observedvalues; , fittedlogistic model

The model was fittedby the methodof maximumlikelihood, as describedin AppendixB. The data whichwere used are givenin fullin thecolumns headed 'Observed' in Table 1; these werederived by theauthor directly from the life-tablesas publishedin the referencescited in Section3. 1, usingequation (21) in AppendixA. The fittedparameters of the modelsare given in Table 2. The publishedlife-tables give estimates of q,, and functionswhich can be derivedfrom q,. These are sufficientto estimatethe parameters, but not to estimatetheir standard errors. For 18 A. R. Thatcher

-0.2

-0.4

-0.6

-0.8

E -1.0 / 0 0 -1.2 cm 0 -1.4 -

-1.6

-1.8

-2.0 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Age

Fig. 6. Force of mortalityin England and Wales, 1841, males: .---.--, observed values; ,fittedlogistic model

-0.2

-0.4

-0.6

-0.8-

o -10. E

0 0 -1.2-

0 -1.4-

-1.6-

-1.8-

-2.0 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Age Fig. 7. Force of mortalityin England and Wales, 1841, females:...... , observed values; , fittedlogistic model Mortafityand Highest AttainedAge 19

0.0

-0.5

-1.0

E 0

0 -2.0

0cm

-2.5

-3.

-3.5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Age Fig. 8. Forceof mortalityin England and Wales, 1980-1982,males: ...... , observedvalues; , fitted logisticmodel

0.0

-0.5 -

-1.0

-1.5 E 0 S -2.03 - cm 0

-2.5

ft-~ ~ ~ ~ ~ ~ ~ ~ ~~~g

-3.5

30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Age Fig. 9. Forceof mortality in England and Wales,1980-1982, females: ...... observedvalues; , fitted logisticmodel these,we need moredetailed information than is available about the absolute numberswho wereat riskin thedata fromwhich the q, wereestimated. Accordingly, some simulateddata on thenumbers at riskwere constructed by themethod described in AppendixB and used to calculate the estimatedstandard errors in Table 2. No singlemeasure of goodness of fitis always appropriate.However, Table 1 shows in 20 A. R. Thatcher Table 2. Parametersof the fitted logistic modelst

Data set Deathsat ages30 Valuesof the followving parameters inmodel (3): yearsand above io%a

A, Hungary,mediaeval 2300 8.7378 0.0769 0.0127 91.63 (3.6103) (0.0062) (0.0031) (2.11) B, England, 1640-1689 3133 1.8749 0.0864 0.0140 99.28 (0.5973) (0.0043) (0.0012) (1.38) C, Halley, Breslau, 1687-1691 2675 1.4397 0.0888 0.0158 99.56 (0.5218) (0.0049) (0.0012) (1.50) D, England and Wales, 1841,males 71000 0.5035 0.1008 0.0097 98.15 (0.0299) (0.0008) (0.0001) (0.19) E, England and Wales, 1841, females 74000 0.3231 0.1050 0.0097 98.45 (0.0190) (0.0008) (0.0001) (0.18) F, England and Wales, 1980-1982,males 834000 0.4657 0.0992 -0.0004 100.59 (0.0048) (0.0001) (0.0000) (0.04) G, England and Wales, 1980-1982,females 828000 0.1208 0.1093 0.0003 103.62 (0.0013) (0.0001) (0.0000) (0.03) tThe parameterswere fittedto the data in the second column of Table 1. The standarderrors, in parentheses,use estimateddata on numbersat risk(see AppendixB). detailthe relative differences between the observed and fittedvalues, described conventionally as 'errors'.When theseare large,they provide warnings that care mustbe taken. In fact,the verylargest of theserelative errors (at ages 30-35 yearsin 1980-1982,but with opposite signs for males and females)are only quite so large because in these cases the absolute values of p, are verysmall. The other relativeerrors are not sufficientlylarge to disturbthe verybroad conclusionswhich we shall draw later in this section.For the later calculationsin Section4, about the highestattained age, it is not the relativeerrors but the cumulativesum of the absoluteerrors which is relevant.

3.3. The long-termpattern Long-termchanges in the patternof mortalityshow up as trendsin the parametersof the model. We can see fromTable 2 thatthe parameter ae has fallendramatically. The parameter /3,although rising,has only changed from0.08 to 0.11 over this very long period. The parametery has become negligible.The parameterX has increased.We need to considerthe meaningof theseparameters and thereasons for the changes. It is convenientto startwith .

3.3.1. The parameter-y When z is small we have pl t , so y will be close to the forceof mortalityat youngadult ages. In fact,Tables 1 and 2 show that y is almostas largeas the averagevalue of A, at ages 30-35 years.The observedfall in ytherefore reflects the virtual elimination of thecauses from which people used to die at ages 30-35 years in the past, which overwhelminglywere infectiousdiseases and causes associatedwith childbirth.

3.3.2. Theparameter 3 The relativerate at whichthe force of mortalityincreases with age correspondsto theslope of thelines in Fig. 2. This relativerate of increasewith age is sometimesused as a measureof a hypothetical'rate of aging'. In the Gompertzmodel (5) we have Mortalityand HighestAttained Age 21

1 d,At B (12) , dx so the relative rate of increase is the same at all ages. However, in the logistic model (3) things are more complicated. Even in the simple case where -y= 0, we find that I d- - 3(1 - t). (13) ALdx

Thus the relative rate of increase is close to 3 at young ages, where M, is small, but then it gradually slackens as age increases. By the very high age where ,u = 2' the relative rate of increase has fallen to 12p. It is the parameter /3which ultimately governs the relative rate at which the force of mortalityincreases with age. Table 2 shows that its values were about 0.10 for males and 0.11 for females in 1980-1982, still between 0.10 and 0.11 in 1841, about 0.09 for each of the independent estimates for the 17th centuryand (if the data can be believed) about 0.08 in the 10th-12th centuries. This is a remarkable stability over such a very long period, which calls for explanation. It would obviously be desirable for this finding to be checked, if further historical data can be analysed or collected, but at firstsight these results appear to support the views of those who believe that there is an underlying pattern of aging, genetically determined.

3.3.3. The parameter q When x = q in equations (2) and (3) we have z = l and hence I,k =j2 + Y, which is close to 0.5. Thus q is simply the age at which the force of mortality reaches 0.5 (and hence the probability of dying q, reaches about 0.4). It is also the age at which the logistic curve (3) has its point of inflection. The value of 0 on row A of Table 2 will be too low, because the forensic method underestimatedhigh ages. However, this problem does not affectrows B, C, D and E, which all show estimates between 98 and 100. This means that these four logistic curves all pass quite close to the point where A, = - at 99 years. It is also related to a feature of the cohort mortalityrates for England and Wales from 1841 to 1981, which also show a close approach to convergence at very high ages (Charlton (1997), p. 26). Our interpretationof these findings is that there was a verylong period when the force of mortalityabove 90 years of age changed very little,despite the changes at lower ages. However, this long static period came to an end in the 1950s, when mortalityrates at very high ages started to fall, even (slightly) at ages over 100. This has occurred not only in England and Wales (Thatcher, 1992) but also in most other industrialized countries (Kannisto, 1994; Kannisto et al., 1994).

3.3.4. The parameter ae The force of mortalityis dominated at low ages (30-35 years) by the parameter 'y and at very high ages (90 years and over) by the parameter 0. In the middle range of ages, the parameter ae is dominant. As an illustration, equation (2b) shows that aE = z2 when x = q/2. In a modern example, where q/2 ;: 50 and -y is small, it then follows from model (3) that ; A50 Vc' . The force of mortalityat 50 years of age thereforedepends primarilyon ae (or vice versa). There is an important relationship between ae, 3 and b. The definitionof X in equation (2b) can be rewrittenas 22 A. R. Thatcher Table 3. Expectationof lifecalculated from the logistic model (3)

Age Expectationof lifeat thefollowving mortality levels: (years) Hungary10th- England Englandand Wales 12thcentuiries 1640-1689 1841, males 1841,females 1980-1982, 1980-1982, males females 30 21.1 27.6 33.1 34.3 42.8 48.4 40 15.7 22.1 26.6 27.7 33.3 38.8 50 11.0 16.6 19.9 21.0 24.3 29.4 60 7.2 11.5 13.7 14.5 16.5 20.8 70 4.6 7.4 8.5 9.0 10.1 13.3 80 2.9 4.5 4.8 5.1 5.7 7.6 90 2.0 2.7 2.7 2.8 3.2 4.0 100 1.5 1.8 1.7 1.7 1.9 2.2 110 1.3 1.4 1.3 1.3 1.4 1.5

3= -ln(ae)/q. (14)

In theperiod when 0 hardlychanged but aewas falling,we can see fromequation (14) that 3 would necessarilyrise. Thus therewould be a negativecorrelation between ae and 3. The size of thefalls in mortalityin middleage could also explainthe apparent slight increase in / over the centuries.

3.3.5. The expectationof life The fittedparameters can be used to producesome new estimatesof thepast expectationsof life at ages 30 years and over, given in Table 3. Their distinctivefeature is that all the expectationsare calculatedon a comparablebasis. It is also possible to extendthe estimates to thevery highest ages, and thishas been done fortheoretical interest. The resultsshow that, whereasthe improvementsin the expectationof lifeat ages 30-50 yearshave been measured in decades, by age 80 theyare measuredonly in yearsand by age 100 only in months. Because Halley's life-tableand the new life-tablefor England in 1640-1689 are almost indistinguishable,only the latteris shown separately.We have alreadynoted thatthe estimates forthe mediaevalperiod are less reliablethan the others.As the forensicmethods used by Acsadi and Nemeskeri(1970) are now believedto have underestimatedages over60 years, the estimatesof mediaevalexpectations of lifeat ages 60 years and over are probablytoo low.

4. The highest attained age 4.1. Concept and methods 4.1.1. Distributionof thehighest attained age Considera cohortof individuals,all born in thesame year. Suppose thatN membersof this cohortsurvive to reach age xo. When all theseN membershave died, theirN ages at death will have a highestvalue WN. This is the highestage whichwas attainedby the membersof thiscohort. It is importantto notethat WN is thehighest age whichwas attainedby themembers of the cohortin the circumstancesin whichthey actually lived. It is not the potentialhighest age Mortalityand HighestAttained Age 23 whichmight have been attainedif the circumstanceshad been different.WN is an observable quantityand a propersubject for statistical study. The observedvalue WN can be regardedas the extremevalue of a sample of size N, drawn fromthe distributionof theages at death.Thus WN willitself have a probabilitydistribution, whichcan be derivedfrom N and the mortalityor hazard function,u(x). The theoryof such extremevalues goes back to Fisherand Tippettin the 1920sand it was specificallyapplied to human mortalityby Gumbel (1937). Let s(x) be theexpected number of membersof thecohort who willsurvive to reachage x, startingfrom the initialvalue s(xo) = N. It is a standardresult that

s(x) = Nexp {- i(t) dt}. (15)

Now the probabilitythat a randomlychosen memberof the cohortwill have a lengthof life whichexceeds x is s(x)/N. Also WN will be less than x if all theN membersof the cohortdie beforereaching age x and the probabilityof thisis

Pr(wN < x) = {1 - s(x)/NIN t exp{-s(x)} whenN is large. (16)

The two equations(15) and (16) give the (almost) exact distributionfunction for WN. By integratingequation (3) and substitutingin equation (15) we obtain

s(x) = Nexp[-y(xo- x) + f-' ln{1 ?+a exp(fxo)}_- 1T ln{1 ? oaexp(L3x)}]. (17) From equations (16) and (17) we can calculate the 'exact' probabilitiesfor each completed year of age. This is the methodwhich was used to constructTables 4 and 5. The probabilitydistribution of WN has a mode, usuallydenoted by WN. At least formodels in whichs(x) and itsfirst derivative tend to 0 as x oo, and whenN is large,it can be shown that WN iS close to the value of x whichsatisfies

s(x) = 1. (18) This is not surprising,since at the highestage thereis only one survivor. We are heredealing with the classical problemconsidered by Gumbel,in whichthere is a knownmodel forthe distribution of ages at death. In more recentdevelopments of extreme value theory,no assumptionis made about a model. Instead, observedmeasurements of extremevalues (such as floods) are fitteddirectly to theoreticaldistributions for extreme values. However, this modern approach cannot be used when, as for mortalityin the historicalperiods, we do not have any properreliable data forthe extremeages. In periods beforethe presentcentury we have only isolated observationsof veryhigh ages, validated withgreat difficulty. There is thereforeno alternativeto usingthe classical approach. Later, we shall compareits predictionswith some of the highages whichare known.

4.1.2. Sensitivityto errors The value of WN depends on N, but it is not verysensitive to its exact value. In a typical situation,even doubling N willnot increaseWN by morethan 1 or at most2 yearsof age. (The reason forthis is thatat thehighest attained age theprobability of survivalfor 1 moreyear is generallyabout -. If two people reachage X, we expectthat one of themwill reach age X+ 1. If fourpeople reachage X, we expectthat two will reachX+ I and thatone willreach X+ 2. 24 A. R. Thatcher Thus doublingthe size increasesthe expectedhighest age by 1 year.) In consequence,we do not need to have a highlyaccurate value forN to make predictionsabout WN. Nor does the estimateof WN dependvery critically on thechoice of the age x0 at whichthe calculationis started.In a testcomparison it was foundthat it made hardlyany difference whetherthe calculationwas startedat age x0 = 50 or at age x0 = 90 years.Nevertheless, in principleit is bestto startfrom the highest age forwhich the size of thecohort is known,since thiswill minimizeany errorswhich might result from errors in the mortalityfunction ,l(x).

4.1.3. Methodsfor estimatingsizes of cohorts In all exceptone of thehistorical cohorts to be consideredlater, proper estimates of the sizes of the cohortsat appropriateages can be derivedeither from the censusesof populationor fromthe population reconstitutionsmade by the Cambridge Group for the History of Population and Social Structure. In the earliestcohort, before the censusesand parishregisters, we can stillmake a rough estimateof the size of the cohortwhen it reachedage 50 years.This happens to be possible because it appears to be an empiricalfact that the number of people who were50 yearsold at theirlast birthdayis neververy far from 1 % of the total population,over an astonishingly wide range of circumstances.It was observedby Wilmoth(1995), analysinghigh mortality modellife-tables with expectations of life at birthas low as 20-30 years,that the number of 50- year-oldpeople was between0.6% and 1.2% of thetotal population. At theother extreme, in a modernlow mortalitypopulation, all theBritish censuses from 1931 to 1991 inclusiveshow thatthe numberaged 50 yearswas between1.0% and 1.5% of the totalpopulation, for both males and females.At theintermediate levels of mortalitystudied by the CambridgeGroup, approximateestimates of thenumbers aged 50 yearsare about 1.1% of thetotal population in 1651 and 1.0% in 1751.Thus as a roughrule of thumb,in theabsence of betterinformation, we can assume thatthe numberaged 50 yearsis about 1% of the total population.

4.2. Applicationto historicaltrends As an illustrationof the methods describedabove, we shall now apply them to some historicalcohorts in England and Wales. These are the cohorts born in five particular calendaryears, chosen largely for interest and the availabilityof data. The resultsare shown in Table 4, whichis based on data drawnfrom many sources. We firstdescribe how Table 4 was constructed.The top row shows the earliesthistorical cohortfor which it was possibleto make a roughestimate of size. (Its memberswere aged 50 years in 1086, the year of the DomesdayBook, when the population of England was 1.1 million.Following Section 4.1.3, thesize of thecohort at 50 yearsof age was takenas 1% of thisfigure.) For the cohortsborn in 1600 and 1700, the sizes at ages 70 and 80 yearswere derivedfrom the Cambridge Group estimatesof thepopulation in 5-yearage groups.The size of the 1811 cohortat age 80 yearswas similarlyderived from the 1891 census.The size of the 1881 cohortat age 90 yearsis givendirectly in the 1971 census. The predicteddistribution function for the highest attained age was thencalculated by the exactmethod, equations (1 5)-(17), usingthe nearest relevant parameters in Table 2. It is here assumedthat the logisticmodel, having been fittedto the age groupsshown in Table 1, will continueto hold forjust a fewage groupsfurther. We now examinethe results in Table 4. On thefirst row, the predicteddistribution implies thatthe highestattained ages in the cohortswhich became extinctin the 12thcentury were generallyin the 90s. This predictionis strengthened,rather than weakened,by the factthat Mortalityand Highest AttainedAge 25 Table 4. Predicteddistribution ofthe highest attained age in particularhistorical cohorts in England and Wales, comparedwith some knownhigh agest

Yearcohort Estimatedsize at Predicteddistribution Knownhigh ages bornin age shown Mode Percentiles 1% 99% 1036 11000at 50 years 92 88 99 1600 23000at 70 years 103 101 110 At least97 in 1701 1700 27000at 70 years 104 101 110 100around 1800 1811, males 14500at 80 years 105 102 111 103-107in 1911-1920 1811,females 19000at 80 years 105 103 111 103-108in 1911-1920 1881,males 7800at 90 years 109 106 115 106-112in 1985-1994 1881,females 24500at 90 years 112 110 118 109-115in 1985-1994

tHighestattained ages in completedyears.

Acsadi and Nemeskeri's(1970) mediaevallife-table was foundby forensicmethods which are now believedto have underestimatedhigh ages. There werecertainly mediaeval people who were believedby theircontemporaries to be over 90 years old but it is difficultto validate individualcases by modernmethods. We must also rememberthat only one person out of each annual birthcohort reaches the highestattained age forthat cohort, so such cases will have been exceedinglyrare and almost impossibleto identifyand verifybefore the days of compulsoryregistrations of birthsand deaths. The predicteddistributions for the cohortsborn in 1600 and 1700 implythat the highest attainedages in 1700 and 1800 wereprobably above 100 yearsof age. This is an interesting calculation.Hynes (1995) foundseveral possible centenarians who died in England between 1700 and 1800, but the evidencewas not conclusive.The observed value in Table 4 is conservative;it showsonly the death at age 97 yearsof Sir JohnHolland (1603-1701), whose documentationis describedas indisputable.There is a presuniptionthat the attainedhighest age around 1800 was higherthan this,perhaps at least 100 yearsby 1800 or soon after. For the cohortsborn in 1811 and 1881 we are on muchfirmer ground. The observedhigh ages are fromthe official death registrations from 1911 onwards,and theiraccuracy has been discussedby Thatcher(1992) and by Wilmothand Lundstrom(1996). Ages of 110 yearsand overare includedonly when they have been verifiedby tracingthe originalbirth registration. In each of the 10 calendaryears 1911-1920 (and 1985-1994)there was a highestage at death, and the last columnin Table 4 shows the range of these 10 highestages. The predicted1 % and 99% percentilesare forthe highestage in a singleyear, not forthe rangeover 10 years.

4.3. Applicationto stationarypopulations The method can be applied to a much wider range of population sizes than has been experiencedin Englandand Wales. Purelyfor illustration, some resultsare shownin Table 5 forstationary populations. The calculationsare easy in thiscase, because thecohort sizes and the mortalityfunction are constantfrom year to year,and the age distributionis stableand consistentwith them. For simplicity,Table 5 showsjust thetotal population size. The cohort sizes at age 50 yearswere foundby the 1% approximationin Section 4.1.3, assumingalso thatat age 50 yearsthe numbersof men and womenare equal. The illustrativeresults in Table 5 show themode of thedistribution of thehighest attained age in a randomlychosen year. It would, of course,be possible to calculate also the distri- 26 A. R. Thatcher Table 5. Mode of the distributionof the highestattained age in a stationarypopulation in a randomyeart

Totalpopulation Ages(years) at thelevel of mortality inthe following: (males+ females) Hungary, England, Englandand Wales 10th-12th1640-1689 centuries 1841, 1841, 1980-1982,1980-1982, males females males females 10000 81 90 90 91 93 98 100000 87 96 96 97 99 103 1000000 92 100 100 101 103 107 10000000 96 104 104 105 107 111 100000000 99 108 107 108 110 114 1000000000 103 111 110 111 113 117

tAgesin completedyears. bution of the highestage whichwould be attainedin a longerperiod, such as a decade or a century,but forthe present purpose it is sufficientto considera randomyear. It would also be possible to calculatethe percentiles as well as the mode, but theseall move togetherand the mode is sufficientto show the trends. Table 5 showsthat the transition from mediaeval to modernmortality increases the highest attainedage by about 14-17 years.In addition,the differencebetween a population of size 10000 and a populationof 1000million can add a further20-22 yearsto thehighest attained age. The boldest step in Table 5 is the extensionof the mediaeval estimatesto very large populations,implying ages whichare far higherthan those observedin the originaldata. However,some unexpectedreassurance can be gleanedfrom Zhao (1995). He foundreliable recordsof a fewdeaths at ages 96-102 yearsin the 13th-17thcenturies in China, when the population was over 100 million.At least, the predictedages in Table 5 are not totally withoutsupport.

5. Discussion It is perhapsnot surprisingthat a logisticmodel of the form(3) shouldbe foundto fitrecent data on adultmortality, since it is veryclose to two othermodels which are alreadyknown to fit(namely Makeham's law at ages 30-70 yearsand Kannisto's logitmodel at ages 80 years and over). What is more surprisingis that this model should fit,rather well, not only the recentdata but also estimatesof the force of mortalityderived from historical life-tables for the 19thand 17thcenturies, and some even earlierestimates for the mediaeval period. Moreover,the parameterwhich governsthe relativerate at which the force of mortality increaseswith age (the so-calledrate of aging) appears to have changedvery little over many centuries,though it has showna slighttendency to increase. These findingsclearly call forexplanations. Section 2.4 describessome publishedtheories whichmay perhapshelp to explainwhy adult mortalityshould followa logisticmodel, but these requirecritical examination. Perhaps furthertheories will be forthcoming,both on processeswhich can generatelogistic models and also on the rate of aging. The long runningsearch for a law of mortalityhas recentlybeen reviewedby Olshansky and Carnes (1997). In severalapproaches, deaths are analysedby cause and an attemptis Mortalityand HighestAttained Age 27 made to distinguishbetween intrinsicand extrinsiccauses (or, in another terminology, betweenendogenous and exogenouscauses). However,such methodscannot be applied to historicaldata and in any case thefindings in thepresent paper do not seemto requirethem. It is the 'all-cause' mortalitywhich is foundto followthe logisticmodel. The model,once fitted,can be used to make estimatesof how theexpectation of lifeat high ages, and the highestattained age, may reasonablybe supposed to have evolvedin England over the centuries.The resultsinclude the quite specificpredictions that ages over 90 years wereattained in themediaeval period and thatthe age of 100 yearswas probablyattained by the end of the 17thcentury. We now turnfrom the past to the future.Although the methodin Section4 enables us to predictthe probabilitydistribution of the highestage which will be attained,given the mortalityfunction, it does not producea predictionfor the mortality function itself. Thus the methoddoes not answerone of themajor controversialquestions of themoment, about how much furthermortality rates at highages are likelyto fall withinthe nextfew decades. The investigation,however, provides evidence which is relevantto anothercontroversial question, about whetherthere is a fixedupper limitto the lengthof human life. The viewthat there is a definite,fixed upper limit to thelength of humanlife is verywidely held, particularlyby manybiologists and gerontologists.However, it has not gone unchal- lenged.Gavrilov and Gavrilova(1991) commentedthat the idea of a fixedlimit, unchanged overtime, has becomeestablished by a processof 'mutualcitation' rather than by convincing proof.It is thereforeworthwhile to reviewthe arguments. If a definitefixed limit exists, call it X. Values of X are of threekinds. The firstkind consists of values like X = 110 and X = 115 whichhave been seriouslyproposed or assumed,e.g. by Vincent(1951), Fries (1980) and Hayflick(1994), but which we now know, in 1998, have alreadybeen exceeded. We have thedirect evidence of thecase of Mme JeanneCalment (born 1875,died 1997) who reachedthe age of 122 years.The documentsauthenticating her birth and lifehave been carefullyverified by Frenchscientists and the detailshave been published (Allard et al., 1994). The second kind of X is somewhereabove 122 years of age, but stillsufficiently close to have a visibleeffect on the shape of the distributionof ages at death. If X exists,then this distributionwill come to an abrupt end at age X and ,u,,at this age will be infinite.Even beforeage X, we would expectthe observedvalues of,u, to startto risewith age fasterthan theexpected values givenby the Gompertzmodel. In fact,the new data forthe industrialized countriesshow thatthey are fallingbelow the Gompertzmodel. Moreover,the age reached by Mme Calmentwas even higherthan would be expectedfrom the logisticmodel, let alone the Gompertz model (Thatcher et al., 1998). There is no visible evidence here that the observedages at death have been restrictedby the effectof a fixedupper limit X. The thirdkind of X is one thatexists, but at so highan age thatit has no visibleeffect on our observations.This is a hypothesiswhich can neitherbe proved nor disproved.It is also indistinguishable,statistically, from the hypothesisthat thereis no fixedlimit. Of course,the idea thatthere may be no definitelimit to lifeis not new.The alternativeto a fixedupper limit is not thatimmortality is possiblebut simplythe moremundane hypothesis that therewill continueto be a probabilitydistribution for the highestage which will be attained in given circumstances,i.e. with a given mortalityfunction and a given size of population, and which can be calculated by the method used in this paper. Since the populationcannot riseindefinitely and the mortalityfunction cannot fall indefinitely,there mustbe some highage whichis unlikelyever to be exceeded,but it is not predeterminedand it is not fixedand definite. 28 A. R. Thatcher It is perhapsworth observing that even if there is no strictlimit the length of humanlife will stillbe governedby an interactionbetween biology and circumstances.It is biologywhich determinesthe probability that people ofa givenage willdie in givencircumstances. Seen from thispoint of view,Fig. 2 illustrateshow thesecircumstances have changedover the centuries.

Acknowledgements This work would not have been possible withoutthe meetingsand workshopson old age mortalityarranged by PeterLaslett and JamesVaupel, and thenew data assembledby Vaind Kannisto. I am verygrateful to all three,for many reasons, and to the refereesfor their criticaland constructivecomments. I am much indebtedto TrevorSweeting for fitting the logisticmodel (3) to the data fromthe historical life-tables and forproducing Figs 2-9. I am indebtedalso to KirillAndreev for calculating the standarderrors in Table 2. The paper was writtenas part of the project on maximallength of life of the CambridgeGroup for the Historyof Population and Social Structure.

AppendixA: Formulaeand approximationsfor the force of mortality

If ,u(x)is theforce of mortalityat age x, thenthe probability that a randomlychosen individual who has just reachedage x willdie beforereaching age x + dx is ,u(x)dx. The forceof mortalityis also knownas the hazard rate and as the instantaneousdeath-rate. Let s(xo) be thenumber of membersof a givencohort who reachage xo, and let s(x) be theexpected numberwho will surviveto reach age x, wherex > xo. Then

= exp { -1(x) dx (19) s(xo) Xi

ds (x) I- sdx (20) Now let fi(xo,x) be the averagevalue of,(x) betweenages xo and x. Then

P(xO, x) = ln{s(xo)/s(x)} (21) x - X This is the formulawhich was used to derivethe 'observed' values in Table 1 fromthe publishedlife- tables,identifying s(x) withl. in life-tablenotation. The forceof mortalityat age x + 2 is veryclose to the averageforce of mortalitybetween ages x and x + 1. Thus fromequation (21) we have

8t(X+ 2)ln{s(x)/s(x + 1)1 (22) -ln( - qx.) (23) whereq., in life-tablenotation is theprobability of dyingwithin 12 monthsof reachingage x. Whenq., is knownexactly, model (23) givesa veryclose approximationfor the force of mortality.Even in extreme cases, whenthe force of mortalityincreases exponentially by 10% witheach year of age, thenumerical errorin expression(23) is only0.04%. In a large population,,u(x) is also close to the age-specificdeath-rate found by takingthe annual numberof deaths betweenages x - 2 and x + ', and dividingthis by the average populationbetween theseages. Hence [u(x+ 2) is close to theage-specific death-rate based on deathsbetween exact age x and exact age x+ 1, whichis m, in actuarialnotation. Hence Mortalityand Highest AttainedAge 29 This approximationis givenby Pollard (1973). In thesame extremecase as before,the numericalerror in expression(24) does not exceed 0.8%. The approximationsin thisappendix apply to adult mortalitybut not to infants,because thepattern of deaths is verynon-uniform in the firstyear of life. Throughoutthe paper, age is measured in years. Thus [u(x) and j(xo, x) measure the force of mortalityin unitsof deathsper person-year.

AppendixB: Methodsof fittingthe model

The moderndata describedin Section 1.3 show theestimated numbers at riskas well as thedeaths. If N,, people reachage x duringa givenperiod, of whomD, die within12 months,then we can regardDv as theoutcome of N, binomialtrials, each withprobability q, as givenby themodel. The likelihoodof the observeddata setis theproduct of all therelevant binomials. The parametersof the model can thenbe estimatedby maximizingthe likelihood.This methodcan be applied wheneverwe know N,. and D, whetherthe populationis stationaryor non-stationary. The historicallife-tables present a differentproblem, because they do not showthe absolute numbers at risk.The life-tables,on theirown, simply show the behaviour of a hypotheticalstationary population which has probabilitiesof dyingqv, as estimatedin variousways by thecompilers of the life-tables.From the life-tablewe can derivethe implied relative frequencyf of observeddeaths in theith age group.We also have theprobability pi thata death will fallin thisage group,according to the model. Now considera hypotheticalcohort of arbitrarysize N drawnfrom this hypothetical population. We assume that the ages at deathin thecohort will follow the life-table, so thatthe observed number of deathsin theith age groupwill be Nf . The probabilitythat the model values pi willproduce this observed result is thengiven by the appropriateterm in a multinomialdistribution. The resultinglog-likelihood for the observed resultis ENf ln(pi).The parameterswhich maximize this log-likelihood are the same forall N. These 'bestestimates' of theparameters, calculated in thisway byTrevor Sweeting and givenin Table 2, are all thatwe need to constructTables 3-5. Because the best estimatesfrom the life-tablesdo not depend on N, and are not based on known numbersat risk,they have no associated standarderrors. However, the earlier historical life-tables are based on ratherlimited numbers of observations(of deathsin Breslauetc.). If we had been fittingthe parametersto the originaldata, instead of to the life-tables,the estimatesof the parametersmight have had substantialstandard errors. Unfortunately, we cannotfind this out directly,because in many cases the originaldata are not available in the formrequired. It has, however,been possible to make estimatesof N*, thetotal number of deathsat ages 30 yearsand overwhich were included in thedata on whichthe life-tableswere based. These are shownin the second columnof Table 2. Now considera cohortwhich starts with size N* at age 30 yearsand has subsequentdeaths which followthe values of q, giveneither by thelife-table or by thefitted model. We can calculatethe standard errorsfor parameters fitted to such a cohort.Of course,this is not how the life-tableswere actually constructed,but at least the method gives an order of magnitudefor those errorswhich are un- avoidable, owing to stochasticvariations in the ages at death in a limitednumber of observations. Standarderrors estimated in thisway are shownin Table 2.

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M. Murphy(London School of Economicsand Political Science) This paper is at theinterface of demography,statistics and biologythat is producinga wealthof new insights(Wachter and Finch, 1997). As examples,Mr Thatcher'spaper helps to demolishthe idea that thereis a fixedhuman 'lifespan'(or biologicallydetermined maximum lifetime), and he shows that mortalityat the oldest ages has been improvingsteadily in developedcountries, a topic of debate in earlierdecades. This paper is an exampleof statisticsas a core partof thescientific process, rather than a techniquethat is usefulfor otherdisciplines. Statisticians rightly concentrate on the estimationof populationaggregates, such as means,and give littleattention to extremevalues. However,in thecase of humanlongevity, both public interest and scientificinsight come fromthe analysis of one individual out of the 6 billionon thisplanet. This paper showsthat we can be confidentthat the number of olderpeople willincrease substantially in decades to come,a trendthat is sometimesreferred to ratherunfortunately as the'demographic time- bomb'. I would like to concentrateon some of the implicationsfor both the elderlyand the overall populationsarising from this paper's analysis, starting with forecasts of theelderly population. Current Britishofficial projections (Government Actuary's Department, 1998) suggestthat the population aged 60 yearsand overwill increase by 50% between1996 and 2066, and therewill be agingwithin the elderly population, so that of those aged 60 years and over the proportionwho are aged 85 and over will increase from9% to 14%. The projectednumber of centenariansincreases 14-fold,from 6000 to 102000, even thoughthis paper estimatesthat the firstcentenarian in England and Wales may not have been seen until the 18thcentury. Population projections,however, are subject to considerable uncertainty:in the USA, in 1994 therewere 3.7 millionpeople aged 85 yearsand over; by 2050 thisis expectedto increaseto 18.2 millionin thecentral variant projection, but witha rangefrom 9.6 million to 31.1 millionbetween the lowest and highestscenarios (US Bureau of the Census, 1996). Even these changesare modestcompared with what may occur in developingcountries. In the past, officialforecasts in all countrieshave consistentlyunderestimated improvements in mortalityat olderages (Olshansky,1988; Murphy,1995). The parameterizationfor old age mortalityin this paper providesa potentiallyvaluable startingpoint for improvedforecasts and assessmentof forecastingerror. The prolongingof lifeis welcome,but a key issue is how manyof theseadditional years of lifeare likely to be spent in the healthyrather than the unhealthystates (Robine and Ritchie, 1991). Establishingtrends in the healthstatus of older people is difficult(Grundy, 1997). In the 1970s and 1980s, it appeared that the additional years of life were mainlybeing spent in an unhealthystate, althoughoften not of a veryserious nature. More recently,there is evidencethat the healthstatus of older people has been improvingin countriessuch as Canada, France and the USA. If US disability rateshad remainedat 1982levels, there would have been an additional1.2 milliondisabled people in the USA in 1994 (Manton et al., 1997). Elucidatingthe interactionbetween trendsin disabilityand mortalitywould appear to be a priorityfor future work since the numbersand healthstatus of elderly people will have major implicationsfor health care and long-termfinancing. Germany has introduced compulsorylong-term care insurance,and in Britainexpenditure on healthand personalsocial services per personaged 85 yearsand overwas ?3995 comparedwith ?263 perperson aged 18-64 yearsin 1989- 1990 (CentralHealth MonitoringUnit, 1992). Whilepeople live longer,the average age at entryto the labour forcehas been increasingand people leave it at earlierages. In 1971 15% of menaged 60-64 yearswere economically inactive, compared with 57% in 1996 (Officefor National Statistics,1998), a trendreflecting preferences by workersand/or employers,rather than government compulsion. Although the role of agingfor problems in fundingof pensionshas been exaggerated,it continuesto be important(Johnson, 1997). Informalcare of older people is usually undertakenby close familymembers and recenttrends in developed countrieswith declines in marriage and increases in breakdowns of partnershipsand voluntarychildlessness are likelyto reducethe pool of thiskey caring group in thelong term,although not in the shortterm (Wolf, 1995; Wachter,1997). I am verypleased to propose the vote of thanksto this stimulatingand fascinatingpaper by Mr Thatcher.

J. W. Vaupel (Max Planck Institutefor DemographicResearch, Rostock) Roger Thatcherhas presenteda thoughtful,thought-provoking analysis of mortalityat older ages. 32 Discussionon the Paper by Thatcher His contributionis intelligent,judicious and stimulating,in the best traditionof English statistical thinking. As Thatcheremphasizes, 'the populationis heterogeneous'.Frailty models provide a convenientway of roughlyaccounting for unobserveddifferences in susceptibilityto death among individualsof the same age. Thatcher'sresults, as well as researchby Lee and Carter(1992) and Lee and Tuljapurkar (1994), and some earlierfindings about theevolution of Swedishmortality (Vaupel et al., 1979) suggest to me thatthe following frailty model may providesome usefulinsights into thedynamics of mortality over age and time:

,u(x,y, z) = z[po(x)exp{-r(x) 0qy)} + c exp{-0(y)}, (25) where,u(x, y, z) is theforce of mortalityat age x and timey forindividuals with fixed frailty z in some cohort,pi0(x) is the base-lineor standardforce of mortalityat age x, c is a positiveconstant, 0qy) and 0(y) are functionsthat determine how mortalityis affectedby alteredconditions and r(x) is a function that determinesthe relativeeffect of alteredconditions on mortalityat differentages. If z followsa gamma distributionwith mean 1 at the startingage of observationx0, then

[i(x,y)= 10o(x)exp{-r(x)0qy)} + c exp{-0(y)}, (26) 1 + 2(y- x)j po(t) exp{-r(t) 0(y - x + t)} dt

wherej(x, y) is theforce of mortalityat age x and timey and a2(y _ X) iS thevariance at age x0 of the gamma distributionfor the survivingcohort born at timey - x. This formulacan also be writtenas

i(x, y) = ,uo(x)exp{-r(x) 0qy)}s(x0, x, y - x) 0x-v)+ c exp{-Vb(y)}, (27) wheres(x0, x, y - x) is theproportion of thecohort born at timey - x thatsurvived from age x0 to age x. Otherdistributions for frailty z could be used, but the gamma distributionis convenientbecause it leads to the simpleformulae (26) and (27). 'Biology', 'circumstances'and 'heterogeneity',three key factorsin Thatcher'sanalysis, are distin- guishedin formulae(25) and (26). Biology(in specifiedbase-line circumstances) is capturedby po(x) and c, the effectof changingcircumstances by r(x) q(y) and V/{y),and heterogeneityby o2(y_ x). To identifythe model some arbitraryrestrictions must be imposed,such as settingr(0) equal to 1 and X(0) and 4'(0) equal to 0. This model is surelyoversimplistic and hence wrong,but it mightlead to some insightsabout how biology,circumstances and heterogeneityhave interactedto determinemortality over age and time.Thatcher's logistic formula (3) is a special case of equation (26). Models likeequation (26) can be fittedto mortalitysurfaces over age and time.Such surfacescan be constructedfrom reliable data for the past centuryor longer for Denmark, England and Wales, Finland, France, Italy,the Netherlands,Norway and Sweden (Vaupel et al., 1998). Thatcher'sresults demonstratethat research on estimatingmodels of mortalitysurfaces can lead to importantfindings. If data are available for 100 yearsof age and 150 yearsof time,then 15000 death-ratescan be analysed. Values of 0/y),4'y) and a2(y _ X) can be estimatedfor everyyear and r(x) and ,u0(x)for every age. The mortalitysurface can thenbe decomposed,in a flexibleway, into componentspertaining to biology, circumstancesand heterogeneity.Alternatively, functional forms might be specifiedfor 0qy), Vb(y)a2(y - x), r(x) and pN(x).The mortalitysurface can be analysedfrom birth or, followingThatcher, foradult ages above 30 years,where the Gompertzmodel mightbe a reasonablechoice for,u0(x). To capturegeneral trends over age and timeroughly the followingsix-parameter version of equation (27) mightprove useful,at least at ages above 30- or perhaps 50- years and for restrictedstretches of time:

,i(x, y) = a exp{bx - qy} s(x0,x, y - xf + c exp(-Vy). (28)

Thatcher's paper, in sum, is both interestingin itselfand likely to encourage some interesting additionalresearch. Consequently, I would like to second the vote of thanksand to congratulatethe authoron his stimulatinginsights.

The vote of thankswas passed by acclamation. Discussionon thePaper by Thatcher 33 Vaino Kannisto(Lisbon) One of the revelationsof Mr Thatcher'sexcellent paper is thatmortality does not rise as highas has generallybeen thought.Projected to age 120years, the probability of dyingwithin 12 monthswould still be in the neighbourhoodof 0.6. I made a search in the Odense database to see whetherthere are exceptionsto thispattern. In 147 old age life-tablesfrom 31 countriessince 1950 therewere about 2000 q-valuesat ages 100 yearsand over.Among them, qx exceeded0.6 bymore than 1 standarderror in only threeinstances, and in none by more than 1.5 standarderrors. I thinkthat this fits neatly into thepattern described by Mr Thatcherin accordancewith the logistic curve.There is no proofthat the underlyingprobability of dyingever exceeded 0.6 in thismaterial. The same may also applyto highmortality countries where any givendeath-rate may be foundat an age up to 10 yearsor even 15 yearsyounger than in theWest, leaving few survivors to veryhigh ages. In contrast,the subsequentascent of thesehigh rates with advancing age is sloweras is well illustratedin Fig. 2 of Mr Thatcher'spaper.

SidneyRosenbaum (Radlett) Over 40 yearsago I was struckby a reviewby Kendall (1955) of Gumbel(1954) and I acquireda copy of thebook. AlthoughGumbel's main field of applicationwas floods,he had an interestingexample of the oldestages in Switzerlandwhich was closelymodelled by the so-calleddouble-exponential distribution whose cumulativedensity function is exp{- exp(-y)}. His estimatefor the oldestage was based on the mode of the distributionof theseages, plus an add-on factorwhich was a functionof theirstandard deviation(SD). Thus he arguedthat a largeSD could morethan compensate for a low mode,using the analogy of a streamwith a largevariation in its flowsthat could resultin greaterfloods than a bigger, moreplacid river.Hence a countrywith a lowerexpectation of lifemight record a higherage at death. We shall explore this in the followingindicative calculations. I have some difficultywith Gumbel's notationand proposeto use themean ratherthan the mode, which avoids havingto calculatethe latter. If x denotesthe oldestage in each of N years,the standardizedadd-on is max ln(N)- YN ln(N) - forlarge N -\. UN 7r/V/6 wheremax is themaximum age in a period of N yearsand -yis Euler's constant:for N = 100,as in the followingexample, the limitingvalue is quite accurate. To startwith England and Wales, suppose thatthe average of the oldestages over the past century was 108 years,with an SD of 2. Then max is foundto be 114. This is fora populationat mid-century of 40 million. For Uroland, with 10 times the population, i.e. 400 million,we must add a further (SW/UN)ln(l0), withmax= 118. Suppose that the fabulouscountry of Chindostan has a population of 4000 millionwith a mean extremeage, overthe same period,of only98 yearsbut withtwice the SD of 4. Then in comparisonwith Englandand Wales theeffect of theincrease in SD alone yieldsmax = 111,with the addition for the size of thecountry of 15, totalling126. Evidentlya furtherincrease in SD would resultin even greaterages. The effectof a populationreduction can be similarlycalculated; for instance,to get down to the numberof Fellowsof theSociety we mustsubtract (SU/UN) ln(104), say (however,the Society has existed for164 yearsso thesubtraction must be reducedslightly); I calculatethat the maximum age would have been slightlyunder 100 years.To myknowledge, we have had one centenarian,Lt Col WilliamButler, who died in 1969 at 101 yearsof age.

JohnCharlton (Office for National Statistics,London) ProfessorMurphy has eloquentlysaid much of what I was intendingto say, but it is perhaps still interestingto look at the figuresthat I have whichillustrate some of the points. The trendsshown by Mr Thatcherare the resultof both growthin the adult population in the precedingage groups ages and improvementsin mortalityat higherages-there is a multiplicative effect(Fig. 10). If we turnto lookingat mortalityby birthcohort (Fig. 11) we see wideningdifferences betweensuccessive birth cohorts, as mortalitylevels have decreased. If the linesin the graphare projectedforwards (though we have no certaintythat improving trends willcontinue) then life expectancy would be ratherlonger than previously expected. Perhaps it is worth modellingthis graph to considerdifferent scenarios. Even withmore conservativeassumptions about 34 Discussionon the Paper by Thatcher

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tStandard errors are givenin parentheses. improvementsin cohortmortality we can see in Fig. 12 thatat age 75 yearsthe improvements in cohort lifeexpectancy are much greaterthan those calculatedon the basis of period life-tables. The improvementobserved in overallmortality among those aged 75 years and over has not been paralleled in all the underlyingcauses of mortality,however -for example death-ratesfor cancers, musculoskeletaldisorders, mental disorders and diseases of the nervoussystem (including Alzheimer's and Parkinson'sdiseases) and some otherdegenerative diseases are rising(Fig. 13). This rise in mor- talitymay also reflectan increase in morbidityamong those over 75 years old among whom the centenariansstill represent only a small proportion(about 0.1%). Trendsin the burdenof morbidity and disabilityare of considerableimportance to the Health and Social Services,especially if the numbersare growingso fast,but theseare currentlynot verywell understood.

David Bartholomew(Stoke Ash) The logisticmodel is distinguishedby the fact that, at very high ages, it levels offto a horizontal asymptote.This impliesthat, as age increases,age-independent causes of death are becomingrelatively more importantthan age-dependent (i.e. biological) factors.We know thatdeaths among thevery old are oftenprecipitated by fallsor otherkinds of accidentand, giventhe frailtyof all withinthat age group,we mightexpect such 'accidental'death to be largelyindependent of age. If thisobservation is correct,we would predicta levelling-offof the forceof mortalityand only the logisticmodel captures that feature.

Q. Tan and J. W. Vaupel (Max Planck Institutefor DemographicResearch, Rostock) Thatcher presentsa logisticmodel (his formula(3)) and then fitsit to several sets of data, with parameterestimates shown in his Table 2. We were curiouswhether his model would fitdata thatwe have compiledon historicalpatterns of mortalityin China. Our data consistof theage at deathand year of death of famousChinese men who died between805 BC and 1900AD (Liao et al., 1990). We omitted a small numberof cases forwhich we suspectedthat the age at death mighthave been misreported.We assessed theage at whicheach man had becomesufficiently famous to warranthis inclusionin thebook. We thenused maximumlikelihood methods to fitThatcher's model to theseconditional survival data. The resultingparameter estimates are shown in Table 6. A visual inspectionof graphssuch as those presentedin Thatcher'sFigs 3-9 indicatedthat the fit was reasonable,although the observed values did varyowing to thelimited size of our samples.The parameterestimates are interestingbecause theyare quite similarto thoseof Thatcher.This can be taken as additionalevidence that simple models might provideserviceable approximations to surfacesof mortalityover age and time.

Shiro Horiuchi(Rockefeller University, New York) Thatcherhas shownthat (a) the logisticmodel fitsthe historicalseries of adult mortalityin England and Wales verywell, (b) trendsin the logisticparameters are interpretablebiomedically and (c) the estimatedlogistic models are consistentwith records of the highestage attained. These findingsare important,and I would like to give a fewsuggestions for future work in thisline of research. Discussionon thePaper by Thatcher 37 Firstly,to justifythe use of thelogistic model, it is essentialto comparethe extent to whichthe logistic and othermathematical models fit data. (I presumethat the author is conductingthis type of analysisas well.) However,in theconventional semilogarithmic plot of death-ratesby age, manydifferent models oftenappear to fitdata fairlywell. Thus I recommendthat the life-table aging rate LAR (theage-specific rateof relativemortality increase with age) be used formodel evaluation. Since theimplied LAR pattern differsmarkedly between the mortalitymodels, the observedvariations with age of LAR indicate unambiguouslywhich model reallyfits data well. In our analysisof internationalfemale mortality data, thelogistic equation was theonly model that fitted the observed LAR patternsreasonably well (Horiuchi and Coale, 1990). Secondly,this study focuses on periodmortality only. It seemsinteresting to investigaterelationships betweenperiod and cohortmortality patterns in relationto thelogistic model. Should thelogistic model fitperiod schedulesor cohortschedules better? Probably it fitsboth verywell, but levelsand trendsof thelogistic parameters are likelyto differsignificantly between periods and cohorts.For example,if the mortalitylevel is declining,the 3s forcohorts should be lowerthan the 3s forperiods. Then a question arises:which series of 3s reflectthe underlying genetically determined pattern of aging,and whichseries of Os are statisticalartefacts? Finally,it is wiselyassumed in thispaper that the K-parameterin equation (1) is fixedto be 1. The parameterseems to be determinedmainly by death-ratesat age 85 yearsand over,which are lackingin the threeearly life-tables. I would like to stressthat this assumption should be limitedto data in which oldestold death-ratesare (partlyor totally)lacking or unreliable.The ,c-valueis themaximum senescent mortality,and its stabilityshould always be checkedwith the latestdata.

BernardJeune (Odense University) In hisexcellent and stimulatingpaper RogerThatcher compares Halley's well-knownlife-table (Breslau, 1687-1691)with Wrigley's new life-table (England, 1640-1689)and findsremarkably similar results. As, however,Buffon (1835) emphasized,mortality figures for this period differconsiderably from study to study.Recently Bourdelais (1993) has mentionedseveral new studies that show variationsin adult mortalityin the 1600sand the 1700sbased on differentgeographical areas and social classes.Thatcher's remarkablysimilar results could thereforebe nothingmore than a coincidence. The historicaltrends of the highest attained age are discussed in Jeune and Vaupel (1995). Regrettably,we failedto consideran importantstudy by Charbonneauand Desjardins (1990). Among 210000 burialsregistered in Quebec before1800, 178 listedan age at death of 100 yearsor more.After age validation,only one case could not be refuted.On averagethe alleged centenarianswere in fact88 yearsold. However,among the 20000 people born in Quebec before1700 Desjardins (1998) reportsa dozen people aged 95 yearsand older (the highestconfirmed age was 99 years). I thereforeagree with Thatcherthat people in the 1600scould have reachedthe age of 90 years.But it is stillan open question whetherthe age of 100 yearswas reachedbefore 1800. We stilllack valid documentationfor even one case of a centenarianfrom that time. Accordingto somestudies reported by Bourdelais(1993) theprobability of reachingthe age of 60 years in the 1700sfor young adults was probablynot higher than the probability of livingto 80 yearsold today. In Table 5 Thatchershows thatthe mode of thehighest attained age could have been 100 yearsin a stationarypopulation of 1 millionpeople in the 1600s.This estimationis based on a fittedlogistic model of theforce of mortalityin Englandfrom 1640 to 1689,where the highest age among theobserved cases was below 85 years (Table 1). Surprisingly,the mode of distributionof the highestattained age in a stationarypopulation of 1 millionin England in 1841 is no higherthan this,although the forceof mortalityis clearlylower for all ages-at least up to 85 years-and an observedage of 95 yearswas not reached.This factbegs foran explanation.

RichardL. Smith(University of NorthCarolina, Chapel Hill) My commentsare concernedwith the applicationof extremevalue theoryin Section 4. The author distinguishesbetween 'classical' extremevalue theory,in which the underlyingdistri- bution is assumed known,and more modern statisticalapproaches. He only considersthe classical approach,but theway in whichthis is applied is rathertrivial. Instead, it seems to me that the problem is well suited to modern statisticalmethods based on exceedancesover highthresholds; see forexample Davison and Smith(1990). The advantageof these methods,in comparisonwith those used by the author,is that theyallow attentionto be focused specificallyon thehighest ages, so thatthere is less bias due to a parametricmodel fittingvery well over 38 Discussion on the Paper by Thatcher the bulk of the data but not in themost extreme portion. The authorapparently rejects this approach because of the absence of 'properreliable data' forthe most extreme ages, but thismay be based on a misconceptionof themethodology. What matters is thatwe have a reasonablequantity of data overthe thresholdselected. If a fewof thevery extreme observations are suspect,it mightbe possible to treat themas censoredobservations or in some otherway to downweighttheir influence. The presenceof such observationsdoes not in itselfmake the methodsinvalid. I have two othercomments related to extremevalue theory. (a) Determiningthe distributionof the maximumof a sample of size n, wheren is some large but finitenumber, and whenthe underlying parameters of themodel are unknown,can be regardedas a problemof predictiveinference. Similar issues have arisenin thediscussion of athleticsrecords (Robinson and Tawn, 1995; Smith,1997). Bayesianmethods have a role to play here. (b) Does environmentalvariation play a role?We oftenhear accountsof how people in certainparts of the world live to exceptionalages because of unique life styleor geneticfactors. This may suggestsome kind of random effectsanalysis, instead of treatingthe population membersas independent. In makingthese comments, I do not seek to be criticalof the author's work. The data collection whichhe and his colleagueshave performedis trulyimpressive; I see my own commentsas indicative of the potentialfor furtherresearch. I hope that the data will be made generallyavailable, to allow interestedstatisticians the opportunityto pursuethese fascinating questions.

Nicholas T. Longford(De MontfortUniversity, Leicester) I thinkthat this paper bringsout very nicelyhow crucial is the model specificationin estimating quantitiesabout whichthe data containlittle information. A freakevent (record longevity) will almost alwayshappen, especially if we searchfor it sufficientlyhard. This makes all models forsingle extreme eventsvery vulnerable. Although appreciating the historical importance of the 'highestattained age' as an estimand,I would propose as an alternativea highpercentile of the ages at death, such as 99.9%, because its estimationis likelyto be much more precise,and yet it is a similarcharacteristic of the studiedpopulation.

The followingcontributions were received in writingafter the meeting.

Russell Ecob (Medical ResearchCouncil Medical Sociology Unit,Glasgow) May I congratulatethe author on his formidableundertaking in collatingand modellingmortality, particularlyin olderage groupsover both timeand place? These modelshave potentialuse to aid in the explanationand understandingof differencesin mortalityby social groups,regions and countries,both contemporarilyand historically(incidentally, I wonderhow representativeof nationalpopulations are the life-tablesused in Hungaryand England over the period 1640-1689?). Currentlythere is some concern(Drever and Whitehead,1997) over the differencesin mortalitybetween social groupsin both Britainand otherEuropean countries,differences in lifeexpectancy at birthfor males in theUK during 1977-1981showing differences between social classes I and II and IV and V of 4.5 yearsand showing some signs of an increase since then,and differencesin life expectancyfor males betweenextreme regionsin the UK in 1987 amountingto 3.2 years(Illsley et al., 1990). Differencesbetween regions also varyaccording to age, a decreasein the between-regionalvariance particularlyin age groupsbelow 35 yearsover the period 1931-1981resulting in largerbetween regional varianceover theperiod 1971-1981in the35-64 yearsage group(though with more recent increases at ages 25-34 yearsto similarvalues) (Ecob et al., 1997). Interestinglythere is a relativelack of between- regionalvariation in the age groups75 yearsand over. What would theauthor suggest as themost useful method of extendingsuch modelsto accommodate thesedifferences? In discussionof the 'fixedfrailty' model, he suggeststhat individual frailty ozi is fixed fromthe age of 30 years onwards. However, the above findingssuggest that such models may be usefullyfitted to data disaggregatedby social-economicfactors and possiblyby region,allowing in additionrandom variation at individualand higherlevels in otherparameters, possibly -. Finally,could such extensionsto the model account for the apparent lack of fitof the model in particularage groups (particularlythe underestimationof mortalityin recentperiods for femalesin England and Wales betweenages 45-60 years,shown in set G in Table 1)? Discussionon thePaper by Thatcher 39 T. B. L. Kirkwood(University of Manchester) Thatcherprovides an analysisof mortalitypatterns which will be of undoubtedvalue to thoseinterested in the biology of aging. As he correctlystates, most gerontologistsnow believe that aging results primarilyfrom the gradual accumulationof randomdamage to cells and tissuesin the body, which builds up at a rate thatis determined,on average,by the evolutionaryoptimization of investmentsin somaticmaintenance and repair(Kirkwood, 1996). There is an importantdifference between this view and theearlier idea thataging is programmedby some kindof biologicalclock. If such a clock existed, the concept of a fixeddefinite limit would make more sense. However,there are strongevolutionary argumentsagainst programmed aging. Aging through the accumulation of damage is a processthat one would expectto be susceptibleto modification,if circumstanceschange so thatthe body is subjectto fewerintrinsic and extrinsicstresses. The long-termtrends described by Thatcherare consistentwith this interpretationand it is interestingto ask what biologicalmechanisms may be responsiblefor the long-termstability of theparameter 3. The slowingof themortality rate increase at thehighest ages, as shownby thebetter fit of thelogistic model,has attractedmuch biological interest. There is good reasonto regardthe model by Muellerand Rose (1996), which suggeststhat a mortalityplateau is intrinsicto the predictionsof evolutionary theory,as flawed(Pletcher and Curtsinger,1998; Kirkwood, 1998). Currentexplanations for apparent mortalityplateaus include genetic heterogeneity, non-genetic heterogeneity due to the stochasticnature of the agingprocess and modificationsof behaviour(human centenarians tend, for example, to receive greatersupport than people in their80s and 90s, whereasold fruit-fliescrawl instead of flying).All these factorsmay contribute. Thatcher'sanalysis of thedistribution of the highestattained age is illuminatingand confirmsthat a species'smaximum lifespan is a statisticthat should be used withcaution, a point whichI and others have argued previously(see, for example, Kirkwood (1985)). Comparativebiologists are coming to appreciatethat the parametersof the gamma-Makeham or logisticmodel providea betterbasis for analysingspecies or populationdifferences (but thisrequires larger samples than are oftenavailable). This paper reinforcesthis important message.

AnthonyW. Ledfordand Michael E. Robinson(University of Surrey,Guildford) The standardpractice in extremevalue statisticalmodelling is to base inferenceon just the extreme observations,and to modelthese by usinga familyof theoreticallyjustified tail models. This approachis adopted because thenon-extreme and extremeobservations may behavedifferently; thus models which are appropriatefor the main part of thedata maybe inappropriatewhen extrapolated into theextremes and viceversa. As extremeages are of primaryinterest here, extreme value methodshave a usefulrole to play. We focuson one such method:the peaks-over-thresholdapproach of Pickands (1975) and Davison and Smith(1990). Let the randomvariable X denote age at death and u some high age. Then, under standardassumptions, the behaviourof X above u may be describedby the generalizedPareto distribution(GPD) givenby

Pr(X > u + xlX > u) = (1 + (X/T)+/4 forx > 0, wheres+ = max(s,0), a > 0 is a scale parameterand ( E IRis a shape parameter.The parameter( is of particularsignificance: if > 0 thenthe upper tail of X extendsto infinity,but if( < 0 thenX has a finite upper tail. To illustratethe above, we followedthe approach suggestedin AppendixB and fittedthe GPD to themales and femalesdata for 1980-1982(see Table 1) forseveral values of u. For both sexes, our resultssuggest that there is a finiteupper end point for the age at death. However,owing to the limitednature of theavailable data, thisfinding is not conclusive. Modern extremevalue statisticalmethods may be useful here in other ways. For example, the similaritiesbetween the force-of-mortalitycurves in Fig. 2 suggest that a joint analysis may be appropriate,with parameters that vary with timeand depend on sex. Such an analysiswould make more efficientuse of the available data and would quantifyhow the extremesof age at death vary between the sexes and how they have changed throughhistory. However, to address such issues adequately,data recordedat a finerresolution or at an individuallevel may be required.Although the authorremarks that reliable historical extreme age data are not available,a carefuluse of censoringmay allow less reliabledata to be incorporatedappropriately in such methods.We are pleased thatthe full data sets are to be made available to researchers,and we commendthe authorfor his efforts. 40 Discussionon thePaper by Thatcher JohnR. Wilmoth(University of California,Berkeley) Roger Thatcherpresents a conciseand well-reasonedaccount of our presentknowledge about theshape of theage patternof mortalityin adulthumans and about thehighest ages attainedthroughout history. I agree withalmost everythingthat Thatcher says in thispaper, but I would like to make two specific commentsregarding statements made in the discussion. First,Thatcher asserts: 'The resultsinclude the quite specificpredictions that ages over90 yearswere attainedin themediaeval period and thatthe age of 100 yearswas probablyattained by theend of the 17thcentury'. Here, I thinkit is appropriateto underlinethat this conclusion,which emerges from Table 4 of the paper, applies to England and Wales only,not to the entireworld. For example,the modal maximumage at deathfor the single cohort born in Englandand Wales in 1036would have been only 92 years.However, estimates of the size of the world's populationduring the 11thcentury were around 300 million,compared with 1.1 millionfor England and Wales in 1086. Referringto Table 5, we see thatthe modal maximumage at deathbased on mediaevallife-tables in a populationof 300 million would have been around 100 years already in this period. Furthermore,Thatcher notes that these life-tablesare now thoughtto underestimatethe numberof survivorsto older ages. Overall, I find Thatcher'sempirical results to be entirelyconsistent with my previous conclusion that human cen- tenariansshould have occurredregularly once the world's population exceeded about 100 million people, whichis thoughtto have occurredaround 2500 BC. Of course, this conclusionrequires the ratherheroic assumptionthat early human mortalitywas not significantlyhigher than duringthe mediaevalperiod, but theextant evidence from various high mortality populations suggests that such a presumptionis not entirelyunreasonable. Second,Thatcher concludes: 'Since thepopulation cannot rise indefinitely and themortality function cannot fallindefinitely, there must be some highage whichis unlikelyever to be exceeded,but it is not predeterminedand it is not fixedand definite'.After such a carefuldiscussion of theprobabilistic nature of limitsto thehuman lifespan, I foundthis statement to be puzzling.I would say thatthe evidence for finitelimits to theearth's 'carrying capacity' for human life or foran insuperablelower bound on age- specificmortality rates has no firmerscientific foundation than the claim thatthe maximumlength of lifeis fixed.

AnatoliYashin (Max Planck Institutefor DemographicResearch, Rostock) I congratulateMr Thatcher on his very interestingand stimulatingpaper. I have just two brief comments.First, I would like to mentionthat the logisticmortality curve of the LeBras model of changingfrailty is identicalwith the mortalitycurve obtainedin the 'gamma-Makeham' fixedfrailty model (Yashin et al., 1994). This observationshows that survivaldata alone are not enough to distinguishbetween the two fundamentallydifferent mechanisms of agingand survival.It seemsthat data fromlongitudinal studies of aging, which requiremore sophisticatedmodels (Yashin and Manton, 1997), are more appropriatefor thispurpose. Second, thereis some question about the explanatory powerof thegamma-Makeham model in describingthe levelling-off of mortalityat late ages due to the factthat univariate frailty models are non-identifiable.Since thereis no biologicalbackground for the choice of theGompertz-Makeham underlying hazard, other functions can be used formodelling pLo(x) as well. In particular,the logistic pLo(x) can be used. Such a choice,however, may excludeheterogeneity in mortalityfrom the causes of the levelling-offof the mortalityrate. Fortunately,the bivariate correlatedgamma-frailty model (which is identifiable)can be used for the evaluation of both the heterogeneitydistribution and an underlyinghazard fromsurvival data on relatedindividuals. The analysisof twindata shows that the estimatesof ,I0(x) may increasefaster than the Gompertzcurve (Yashin and lachine, 1997).

The authorreplied later, in writing,as follows.

I am gratefulto both the proposerand the seconderof the vote of thanksfor their kind words. Both have adopted a verywide perspective,but theyhave not raised any specificpoints on the paper which call forreply. One may note an interestingdifference of approach betweenmine and Vaupel's. Given a verylarge volume of data, mynatural instinct was to tryto finda simplemodel whichwould fit.Vaupel, seeingthe simple model, has generalizedit. Such actionand reactionare the stuffof progressand I shall be glad if my paper encouragesfurther research. Of the specificissues whichwere raised in the discussion,the most importantis whetherthe force of mortalitytends to a finitelimit as age increases.The paper argues that both the modernand the Discussionon thePaper by Thatcher 41 historicaldata are consistentwith the logistic model, which has thisproperty. This impliesthe corollary discussedin Section5, that,if a fixedupper limit to lifeexists, it mustbe at so higha levelthat it has no visibleeffect on our observations. In any debate about whetherthere is a finitelimit to life,the views of biologistsmust be crucial.I am thereforeextremely glad to see the authoritativecontribution by Kirkwood. He lists threecurrent explanationsfor the levelling-offof mortalityat high ages, namelygenetic heterogeneity, non-genetic heterogeneitydue to the stochasticnature of the aging process and modificationsin behaviour. Bartholomewadds a furtherreason, the pronenessto accidentsat veryhigh ages. In the discussion,two additionalpieces of factualevidence were presented which further support the conclusion.Charlton showed very clearly in Fig. 11 how age-specificdeath-rates (and hencethe force of mortality)have levelledoff at highages in thecohorts born in Englandand Wales since 1841.Kannisto reportedan importantnew finding,that the observedupper limitfor q, has not significantlyexceeded 0.6 in any of the 31 countrieswhich are now includedin the new database on old age mortality. On the other side of the argument,Ledford and Robinson apply one of the modernmethods of extremevalue theory,the 'peaks-over-thresholds'approach. This is based on the fact that under standardassumptions the behaviourof the ages at death above a sufficientlyhigh threshold u can be describedby thegeneralized Pareto distribution (GPD). On fittingthe GPD to thedata for 1980-1982 in Table 1, theyfind that their results support (though not conclusively)a finiteend pointfor the age at death. However,this finding needs to be examinedcritically, because we know thatthe same data can also be fittedby the logisticmodel, which points to exactlythe oppositeconclusion. There is a logical contradictionhere. It arises,I think,because the thresholdsused by Ledfordand Robinson are not sufficientlyhigh for the GPD approximationto be valid. If ,-tudenotes the forceof mortalityat the thresholdage u, and ifmortality truly follows the logistic model (3), thenat all ages x > u we shall have pu < p, < 1 + y.If u is sufficientlylarge for pu to be close to 1 + 'y, thenp,u, will be almostconstant when x > u, whencethe distribution of ages at death will be almost exponentialand so will be close to the GPD with( = 0. Thus theextreme value theorywill work perfectly, but onlywhen u is sufficientlyhigh forpu to be close to 1 + y. Unfortunately,the forceof mortalitydoes not reach thislevel until beyond the rangeof ages whichwe can observe. In Thatcheret al. (1998),my colleagues and I applieda batteryof testswhich would have detectedany seriousdepartures from the logistic model below age 120 years,but we did not findthem. Of course,the modelmay break down somewhere above thisage, but it would stillneed a thresholdabove age 120 years to detectthis. Although I can see the theoreticalattractions of the peaks-over-thresholdsapproach, as advocated by Smith,I thinkthat he will findit difficultto apply thismethod to the data on human mortality,because the required threshold is so high.The methodswhich I used in thepaper may look old fashioned,but at leastthey can be appliedto thedata whichare availablein thisparticular problem. I now turnto thecomments on thehistorical highest ages. I am delightedthat Wilmoth finds that my calculationsare generallyconsistent with his own. He is quite rightto stressthat my Tables 1-4 are confinedto Englandand Wales. He and I are in good agreementfrom the Middle Ages to thepresent, and it is a veryminor difference that we seemto take differentviews about theindefinite future! In my case, I simplythought it unlikelythat mortality rates will fall as faras 0, when our evolutionarypast dictatesthat the humanbody will deteriorateafter the age of reproduction. I also verymuch welcome the informedcomments of Jeune. Of course, I accept that the close agreementbetween Halley's and Wrigley'slife-tables may be partlycoincidental, but it is a happy coincidence.The onlyclear difference between Jeune and me is reallyrather slight, about whetherthere wereany centenariansbefore the year 1800. Both Wilmoth'sand mystatistical calculations suggest that therewere probably a few,though very rare indeed. Also, Desjardinshas now identifieda personborn in Quebec in 1648who definitelyreached 99 yearsof age. The populationof Englandin 1648was much larger than the population of Quebec, so it is reasonable to suppose that theremay have been a centenarianin Englandbefore 1800, let alone somewherein therest of theworld. However, I acceptthat thisremains an open questionuntil a definitecase has been identified.The last pointraised by Jeuneis based on a misunderstanding.The 'observed'data in Table 1 are truncated,for the reason statedin the paragraphbelow Table 1. This does not mean thatthere were no observeddeaths above age 95 yearsin 1841. I was also verypleased to see the commentsof Horiuchi.His excellentsummary of mypaper is far moreconcise than my own and I was glad to receivehis endorsementof mydecision to assumethe fixed value rc= 1 forthe historicalperiods, in theabsence of more reliabledata. Material thatis relevantto some of his questionswill be foundin Thatcheret al. (1998). For example,the coefficients 3 are indeed 42 Discussionon thePaper by Thatcher foundto be lowerfor cohorts than for periods, as theyare bound to be whenmortality is falling.Both, however,can be affectedto some extentby changesin circumstances,so neitheris a completelypure measureof an underlyingpattern. Rosenbaum,in his inimitableand entertainingway, has applied extremevalue theoryto the ages of Fellows in our own Society.His exampleis instructiveand also strikeshome! Tan and Vaupel have producedsome extraordinaryand completelyunexpected data on ages in early China. Perhapsone should not take the resultsfor the Tang dynastytoo literally,but the otherresults show mediaevallevels of mortalityin themiddle range of ages and modernlevels at veryhigh ages. This is similarto data fromRoman tombstones(see Section3.1). Thereis food forthought here. It is good to know thatthese data existand thatthe model can fitthem. I am glad thatCharlton showed us his figuresand I have alreadycommented on his valuable Fig. 11. Fig. 12 mustbe interpretedwith care because thereare timelags. For example,the people who were75 years of age in 1991 (withthe expectationof liferepresented by the last pointson the period curves) weremembers of thecohort born in 1916,not thecohort born in 1941. Ecob asks about applyingthe model to studythe long-standingdifferentials in mortalitybetween regionsand betweensocial classes. This is an importantquestion and we need to proceedin stages,the firstpriority being to discoverwhether the model will be usefulfor this purpose. My suggestionwould be to startvery simply indeed, by takingregional data fora modern(post-1950) period. For thesewe would expect y to be negligible,so thatuse can be made of equation (6). If plots of logit(w,) against x show that the lines are reasonablystraight at ages over 30 years,for individualregions (or social classes), thenthe model will be usefuland we can startto considerthe implicationsof the levels and slopes of the lines.On Ecob's finalquestion, although the relative error for these groups is in one case 13.8%, the correspondingabsolute errorin the forceof mortalityis only 0.0007 and it is not clear whetherthis is a matterfor concern. I am gratefulto Yashin forhis furtherreferences and to Longfordfor his comment,though he will findthat high percentiles are not easy to measurein historicalcohorts. Several contributorswere keen to have more data. The positionis thatthe fullhistorical life-tables can be found in the referencescited in Section 3.1. The pooled moderndata for 13 industrialized countrieshave now also been published,in Thatcheret al. (1998). Thus thereis already plentyof materialfor those who wish to make a start. The full disaggregateddata for up to 31 individual countriesare expectedto be available shortly,on application. I would liketo thankall thecontributors to thediscussion, and particularlythose who travelledfrom overseasto attendthe meeting.

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