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The definition of pij(x) is given in the text of The ExtendedFixedAttributeDynamicsMethodsection.Let s (x), s (x), s (x), and s (x) denote the conditional 11 10 01 00 survivalprobabilityfromagextox + nforthosewhoare bothcarrierandexposed,carrierandnot-exposed,non- carrierandexposed,andneithercarriernorexposed,re- spectively (the terms “carrier, noncarrier, exposed and not-exposed”aredefinedinTheExtendedFixedAttribute Dynamics Method section in the text); N(x), the total numberofpersonsagedx; S(x),theoverallaveragecon- ditionalsurvivalprobabilityfromagextox + nforall personsagedx.

ij ij birthamongtheindividualsbornx + ago,respectively;N1 (0)andN2(0),theinitialsize(num- ij ij berofbirths)ofthecohortsoftheindividualsbornx + yearsagoandxyearsago,respectively;S1(0→x)andS2 0toagexamongalloftheindividualsbornx + andxyearsago,respectively;C1 ij ij thei – age0toxforallofthecohortmembersbornx + ago and x years ago, respectively; p1 (x) and p2 (0), ij ij geneticvariantandthefixedexposureamongallindividu- alsagedxinthecohortsbornx +

BecauseN(x) pij(x) s ij(x) = N(x) S(x) pij(x + that

Wechoosethegroupofindividualswhoareneitheracar- riernorexposedasareferencegroup,anditsimilarlyfol- lowsthat

survivalfromagextoagex + nofthosewiththei–jcombi- carriernorexposed:

Notethattheabovemathematicalanddemographicderi- x + agextox + p ij

appliedtothecross-sectionaldatafromthelong-livedindi- vidualsagedx + fact assumes that: p1 (x) = p2 (x) or p1 (0)C1 = p2 (0)C2 ij ij ij ij ij ij namelyassuming(a)theinitialdistributionsofthecombi- twocohortsbornx + same,thatis, p1 (0) = p2 ij ij

Notethatassumingadditiveversusmultiplicativeinter- actionsisnotpurelyastatisticalquestionbutratheraone basedonwhatweknowaboutthesocialenvironment,be- havior,andbiology(25).Inotherwords,itiscontentbased, andweshoulddeterminewhetherweadoptanadditiveor multiplicativeassumptioncasebycasebasedonempirical dataorthroughlearningfromliterature.

same,C1 =C2 .Iftheseassumptionsaremet,the ij ij survivalRS / ij 00 cross-sectional data adequately provides the estimate oftheORofsurvivalarenotdirectlyaffectedbythediffer- bornx + violatedbecauseofpopulationstratificationand/ortheun- knowninteractionsamongthegeneticvariant,fixedexpo- havebeennopreviouslypublishedstudiesabouttheinflu- ences of FOXO genes on mortality from birth to age)inthedifferentcohortsbornx + this study, we control for population stratification in andmiddle-agecontrolsbyrestrictingouranalysistoasin- cohortdatasourcesbecomeavailable.

standardstatisticalsignificancetests(23)toassesswhetherthe interactioneffectsestimatedbyFormulas(A6)and(A7)are statisticallysignificantbecausethefunctionaldistributionsof Formulas(A6)and(A7)areunknown,atleasttoourknowl- edge.Intheory,wemayusemultivariateDeltamethodtodo thestatisticalsignificancetests.However,inthiscasetheDelta methodisnotfeasiblebecausewehaveeightvariables(ie, P (x)andP (x + n));thesamplesizemayusuallynotbelarge ij ij enoughtoestimatethe72parameters(8variancesplus64(=8 acarefulliteraturesearch,webelievethatabootstrapsimula- intervalsandPvaluesofISmandISa(32,33).

Appendix 3. Potentially Confounding Factors Included in the Cox Hazards Regression Models

Wechoosethefollowingvariablesasrelevantcovariates basedonreviewsofpreviousstudiesinthisfieldandour understandingoftheChinesesocialcontext.

Demographic and socioeconomic variables.—Demographic andsocioeconomicvariablesincludeage,gender,residence (ruralvsurban),education(noschoolingvs≥1yearof schooling;notethattwo thirdsoftheChineseoldest old aged80+reported“noschooling”)andeconomicindepen- dence(havinghisorherownretirementwageand/orearnings vsnoincome).

Appendix 2. Estimating the Effects of Interactions Between the Genetic Variant and the Fixed Exposure on Longevity Following the FAD Approach

Social and family support and villageoronthesamestreet,vsthosewhohadneitherco- residentchildrennorchildrenlivingnearby).Wealsoexam- socialsupportwasmeasuredbyaquestionthataskedpartici- basedonfrequencyofparticipationinsevenactivities:per- and/orpets,playingcardsormah-jong,participatinginorga- andbooks,andwatchingTVandlisteningtotheradio.Re- sevenscores;theresultingscalehasarangefrom0to7.

gene-fixedexposureinteractionsonsurvivalfromagextox + n,usingtheORsofsurvival(RS , i = 0,1; j = 0,1)de- ij/00 rivedbyFormula(2).

Onamultiplicativescale,theInteractionIndexofSur- vival(ISm)isestimatedas:

Onanadditivescale,theInteractionIndexofSurvival (ISa)isestimatedas:

IfISmorISaissignificantlygreaterorlessthan1,the interactionbetweenthegeneticvariantandthefixedexpo- surepositivelyornegativelyaffectssurvivalfromagexto x + n.

The resilience score.—The resilience score was con- structedbasedonthemorestraightforwardandsimplified definitionofresiliencespecifiedinLamondandcolleagues (34),namely,resilienceconnotestheabilitytoadaptposi- tivelytoadversity(i.e.abilityofcopingandadjustment). Weselectedsevenavailableandrelevantitemsinthedemo- graphic study CLHLS to construct the resilience score, whichisactuallyasimplifiedversion(focusingoncoping andadjustment)oftheresilienceindexusedinthein-depth psychologicalstudy(35).

as a proxy for biological age; as a predictor of healthchange,andutilizationofhealthservicesandhasim- andispositedtocapturethecumulativedeficitsofthesam- basedon39variablesthatincludecognitivefunction,instru- mental activities of daily living, activities of daily physical performance functional limitations, interviewer-reported health, hearing loss and vision the past 2 years, and reports of specific chronic inthepastyearforwhichascoreof2wasgivenifthenum- usedtoconstructtheDIisavailableelsewhere(37).