Table Of ContentCanada
ssc Statistical Society of
Societe Statistique du Canada
2000 Proceedings ofthe
Survey Methods Section
Recueil 2000 dela
Section des methodes d'enquete
Paperspresentedatthe28thAnnualMeetingofthe
StatisticalSocietyofCanada, Ottawa, Ontario, Canada,
June4-7, 2000
Communicationspresenteesau28ifemeCongresannueldela
SocietestatistiqueduCanada, Ottawa,
Ontario, Canada
du4au7juin2000
SSCAnnualMeeting,June2000
ProceedingsoftheSurveyMethodsSection,pp.185-190
ANONLINEARHIERARCHICALMODELLINGAPPROACHFORCENSUS
UNDERCOVERAGEESTIMATION
YongYou1
ABSTRACT
Area-levelnonlinearmixedeffectsmodelsareconsideredinthispaperforCanadacensusundercoyerageestimation.
Wefitanarea-levelnonlinearmixedeffectsmodeltotheprovince-levelundercoveragesurveyestimates.Inparticular,
thesamplingmodelisbasedonthesurveyestimateoftheundercoveragecount,andthelinkingmodelisalog-linear
modelfortheundercoveragerate. AfullhierarchicalBayes(HB)approachisdevelopedtoobtaintheposterior
estimatesofthecensusundercoverageusingMarkovChainMonteCarlo(MCMC)samplingmethods. Ourresult
showsthattheproposedmethodcanprovideefficientmodel-basedestimates. Analysisofmodelfittingisalso
presentedusingposteriorpredictivedistributions,andthecorrespondingresultindicatesthattheproposedmodelfits
thedataquitewell.
KEYWORDS:Censusundercoverage,Gibbssampling,Modelchecking,Nonlinearmixedmodel,Posterior.
RESUME
LesmodelesnonlinSairesaeffetsmixtesauniveaudesregionssontconsidereddanscetarticledanslecadrede
l'estimationdelasous-couverturepr6senteaurecensementcanadien.Unmodelenonlineaireaeffetsmixtesauniveau
desr6gionsestutilise'afindemoddliserlesestimesdesous-couvertureauniveauprovincial.Enparticulier,lemodele
d'echantillonnageestbas6suruneestimationdelasous-couverture,etlemodeledeliaison,servantamodeliserletaux
desous-couverture,estunmodelelog-lin6aire.UneapprochehierarchiquedeBayesestdeveloppeeafind'obtenirdes
estimateursaposterioridelasous-couverture,enutilisantlesm&hodesdeMonteCarloavecchainedeMarkov
(MCCM).Lesr^sultatsobtenusindiquentquelamSthodeproposeeengendredesestimateurs,bas6ssurlemodele,qui
sontefficaces.Deplus,uneevaluationdumodele,utilisantlesdistributionspredictivesaposteriori,estpresenteeetles
rgsultatsindiquentquelemodeleproposesembleadequat.
MOTSCLES:sous-couvenireauniveaudurecensement,echantillonneurdeGibbs,evaluationdumodele,modelenon
lineaireaeffetsmixtes,aposteriori.
1.INTRODUCTION thesecoveragestudies,thecollectionmethodologyis
adjusted in order to improve coverage in the
InCanada,acensusisconductedeveryfiveyears. succeeding census. More details on the coverage
However, the census does not enumerate all the studiescanbefound,forexample,inGermainand
inhabitantsthatshouldfillacensusformonCensus Mien(1993).
Day. Inthe1991Canadiancensus,itisestimatedthat
about 3% ofthe population were not enumerated. In1991,thepopulationestimateswerebasedonthe
Thus the census needs to be adjusted for census counts adjusted for the estimated net
undercoverage in order to properly represent the undercoverageinthecensus. Thebasepopulationwas
demographicpictureofthecountryonCensusDay. formedbyaddingthenetprovincial undercoverage
Since 1966,theReverseRecord Check (RRC) has estimatetotheprovincialcensuscount. Thiscreated
beenusedbyStatisticsCanadatomeasurethegross anadjustedbaseuponwhichalltheotherpopulation
numberofpersonsmissedbythecensus. Starting figures were derived using modelling and
1991, an Overeoverage Study was conducted to demographic metvhods. Rivest (1995) proposed a
measure,the gross number ofpersons erroneously composite estimator to estimate the provincial
includedinthecensus. In1991,forthefirsttime,the undercoverageusingthenationalundercoveragerate
RRC results were combined with those of the asasyntheticestimate. Theeffectofthecomposite
Overeoverage Study to produce the direct survey estimatoristoshrinkallprovincialratestothenational
estimatesofthenetundercoverageforthenationand rate. Rivest'scompositeestimatorperformspoorlyat
allprovinces. Throughtheanalysisoftheresultsof extreme provinces, namely, P.E.I. and Ontario, the
1YongYou,HouseholdSurveyMethodsDivision,StatisticsCanada,Ottawa,Canada,KlA0T6,[email protected].
185
.
smallestandthelargestprovincesofCanada (You, design-unbiased assumption may be restrictive, in
1997). particular the estimates of missed persons y,are
probably subjecttosomeunknownbias(Zaslavsky,
In recent years, modeling techniques have been 1993). However,sincewedon'thaveestimateforthe
appliedtoimprovethedirectundercoverageestimates possiblebiasasintheUScase(Zaslavsky,1993),we
from sample surveys. Datta et al. (1992) and assumethatthesurveyestimate y,isdesign-unbiased
ZbmpaarossodlepeadolvssosenkdfyoUraS(1trc9heee9gn3rs)aeudssjspiurcosoontpumonemstnoetaddneodlhfiseiurcttaeuirnalcstihizuioiscnnasgl.unaBdDnaeirycecekmospv(ie1r(r9iHa9cBg5a)e)l fdsiomsratlrlmi,bu.atrieToanhiasesssitsuimmaaplttsiiooonnaoc(fRoamtohm,eosn1a9m9ap9s)ls.iunmgpteTirhorenorsunsoaetrdmtahilen
Bayes(EB)methodtoestimatetheundercoveragein province level seems quitereasonable'"dueto large
small domains. DickandYou(1997) employed a samplesize. Thesamplingvariances£2areestimated
linearFay-Herriotmodel(FayandHerriot,1979)for throughgeneralizedvariancefunctionmodelsofthe
ctthheiensspuprsaopvueinrndcewer-elceoavvreeelrapcgaerentisrcuaustleasurnliydneiranctoHevrBeesrtafegrdeaimcneowumonortdkse..linIIgnn fkonromwnVri(ny,t)hoecmCo|?'del(D.icTkh,us19£925)isaanctduatlhlyenatsrmeaotoetdheads
particular, weproposeanonlinearmixedmodel in estimateofthesamplingvarianceofy,
which the sampling model is based on the
undercoveragecounts whereas thelinking model is Instead of using (1) directly to model the
btoDeavirsecmerksdcoaoomnnfedrusanYtsdeoesou.rmceTo(vh1le9iemr9iua7tn)gadeteaicronocndusonvctoeasfr.nathgpeTerhomrevaotidepdereilospmepooxrsdpoeerpdleos-smsbeoeadddsebeilydn uturnnaddneesrrfccooorvvmeearrtaaiggoeencraotuen.tanNuda,m,eDliyc,ckolnaestnidddeiYr=oeduw,(/1(u9,9,m7+o)dc,eu)lsleaidnndga
estimates for undercoverage counts and rates r.=y./(y.+c,), where 0,-is defined as the true
mbIsoSniueaomtcsduSteelaeidltscosatnonfineoo3flonotlurhosw4ewlesycp.w.erienoIlslnpTuudossheSstseereccadurttrneiiendbomoeetnanhrilect2inhondeGvweeeiaerirbrmabofpghsfoliereetsmrmhaaaeeerslmnctlptahiylarimittcapiniatcrgloileonesmnme.oioensdtftehllttaoshIhid.ende uttfrrnoealdanletsorefwcdoiornvamgresar,tmiat=oohgd0neee,,ld+:riDearit„eeccktfOoea^rsnxtdtiJhmYe/aot3oiu-r+th(vo,1f,p9r9o7fvt)=i.nclBc,oea.n,.ss.e,imadd,nedroen(d2f)ttthhiiess
GSfiineitcbtStibeniscgo.tnisoa5Fnmiwpn6ale.leldrye,sacnwrdeibpeprredesiseaengntntotsshtoeimcesesutcsioemndacttliouodnaisrnsegessursletmsm.oadrekIlsn wnwohhreemrraeel<fr7ia2insisdaoumvnekecnftfooewrcnto;fwierttehg~rNEe((s0vs,,ii)o/n/=?p)a.raanmdTethVee(rvss,;a)mv=p;liaisn1ga„
2.UNDERCOVERAGEMODELS vaarianceys?Tawyalsortreatedaskanpoprwonxiamnadticaolnculatedfroams
Supposetherearemprovinces. Inthei-thprovince ^=(l-^)VU.)/c,2=(l-i;.)^/c,2. Themodel
nthuembceenrsuus,hpaesrscoonusntaerdecm,-ipsesresdonbsywhtihleecaennsuusn.knoTwhne gm1io9v7de9e)nl.bfyoHr(o2w)semivaslelra,satrwaeenadaenrsodttieampapttlhiaiotcnatmi(ooFdnaeyolfa(Fn2ad)y-HhHeaersrrriiototht,e
coveragestudiesprovideasurveyestimate y,-ofthe following limitations: (i) the zero mean sampling
netundercoveragealongwithanassociatedvariance model may not be true due to the nonlinear
pocw£oerbfnirssts.ettorernvTnusehcdteamis,wntig.rtTuahTteoh=mupeotocd,pc.seu,oal+lvamuetc,pr.eilanoitgnSnreigeonssfcetreoutrndohtireteh,hseiem-uotceshsesetntpsirsoumoafasvmtiptenchlcoeoeeufwnsmcotuiarrsnvkcse,ebyiidesns tbdttirhhafaeesfneoesdkrrfienognoroitnmnwafanttlrihoosesmnua.trtmvrhpeaoTlynshiseenefsogotbrritemvmsaeaauidtrlnetiedssadan(tocfabYertoocauomiyusnatflenhiddedsfbmRraeoaodosmv,eielgt2rnh0ybie0af0isms)cetoa;drndot(eoinlilngy)
whichthroughstandardestimationproceduresproduce assumption, since ysfis indeed a function of the
amsinotdueeastltioinmgyaec,t.naee=nraoub,lfel+ywtrh£ie,utstneehund-maibs,t.eo.r.;dmoe,fs>crmiibses"edthipserseosntsi.m(a1t)Tihoen nuusionnmdrkiemnlraaoclrwovnmmeomradaeyegalensnroa0tat,re;ebw(eihaiiil)casopihnpirumsosopbedredietalwtbee(ye2n)boaetcshaasenurudsmeip1en,ogaplltvte,i.sh,tooutfgbhoheer
where''e,-N(0,^). This'modelassumesthatthe example,Dattaetal.(1992)andZaslavsky(1993). A
direct survey estimator y(is design-unbiased. The morerealisticmodelmayuselog(0,)*=xJ/3+vt. To
186
, ,
oyercometheselimitations, wenowpropose anew • 03\Y,U,a2]~
|M•m»ooddeLSweliahl:nmek1pr:ilenig£ngfm2oimsdoekdlne:ol:wn;y,.=m,+£,, £t-Af(0,£2) IG{<am=[1a+2m\1Y2,,U.bm,=1+P]£~(log^—^-•Mm-vxj0)212).
log(U—,^+C—i)=xj0+Vi, Vi~N(0,a2). Drawing sampli=eis fro"'m• [f3\Y,U,a ] and
IguninvdMeenorcdoevdlierrea1c,gteltyhecobsuyanmt.pl(1i)Tngheamnoddsaemlipsliisnbgaaslevidnaerairaonnmcoedtehiles [[mC,72\1Yy,,p£,/,a/?2]] doeissnotsthraaivgehtafocrlwoarsde.d^Tor|m.HTowoedvrear,w
known. The linking model is a log-linear random samples from [«,\Y,/3,a2], Metropolis-Hastings
effectsmodelfortheundercoveragerate,butherethe updatingscheme(see,e.g.,ChibandGreenberg,1995)
rate0,isexpressedasafunctionofcountm,,thatis, isusedwithintheGibbs'sampler. TheMetropolis-
l0i,n=kimn,g/(m«o,d+elc,)w.itIhnMtohedeslam1p,lwiengcamnondoetlcotmobifnoermthae HSausptpionsgesthuepdMaatrinkgovstcehpainisissautmtmhaerik-ztehditaesratfioonl.lowTso:
Tmshteiatnshdoakdrisdndfloironefalripnrmeoaibrxlemedimxseefdffoercmtossdmemalolsdlelca.arnenaoTthesubtseimsaattpaipnoldniaerdi.ds uspadmaptleem[-u*s+1\)Yf,rpo,ma2N](,y,w,£e2),firtshtendwriatwhpraobacbainldiitdyate
dHioswceuvsesred,binyRusaiong(a19c9o9m)plaentdeYHoBuaapnpdroaRcahowi(2t0h00t)h.e a-,(/.u(\k)\u\(A+l)%)=_n:u_nf{8g\(uMiw,ft'<H*>,'g"**>)>,n1}' ((C)\ •
Gesitbibmsatessampolfingthemetuhnodde,rcwoevercaagneficnodunttheapnodsterriaoter weacceptthis uf+V);otherwise,set uf+1)=u\k).In
simultaneously. (6),g(-)isafunctionofut,/3,a2givenby
3.GIBBSSAMPLER g(Ui,p,cr2)=^U:+C:s-exp{-(log(M,./(2M,a.z+c,)-^r^)2}•
WenowuseModel1toobtainmodel-basedestimates (6)
ofrfacmeen,sMuosduenlde1ricsoevxeprraegseseudta.s:InahierarchicalBayes Twehleleassti£m(«a,t.io|nnoafndtheV(pmo,st|erKi)ocrandisbteribbutaisoendoofnm,tahse
y,\u,-NiMi4f\ i=l.-,m, (3) samples{u\k)}fromtheGibbssampler.
and
wPurh,ieuo+rirces—\lpoo,fgNa2/dS~ean\onotdgeNs(axa]2Plao,rgeo-rn2os)re,mtail=asd\:e,.n.s.i,;tmr,y(/Tf)u°n=(c4t1)iaonnd. Aacnosaul4niy.tnsIiDsaMi.scPktLWhaeEenMdauuEYxsNioeldTuiaArl(yo1Tg9I-9vat7Or)ri,aNanbwsAlfeeoN,ruDmstahetEaditSo1nTi9sI,9o1xfMuAdta=hTtealIocOgie(nNncs,ouu)sr
a2~IG(a,b) with a and b are known positive then the linking model for m,is
constants.Let Y=(yl,...,ym)T and U=(u1,...,um)T, log(u,/(«,+c,))=PQ+*,,/?!+v,. Toimplementand
we are interested in the posterior distribution of monitortheconvergence oftheGibbs sampler, we
ssuEia,(mmUgupiillvai|etYnneg)amtsnheaedmtphpldooaesdtstae(rGfieoYolr,rfavi«nan,dr.ipaaaTnnrocdteiSciumVlmi(aptruhl,t,e\mpY1eo)9ns9tt.0e)rTtiihhoseerusGGmeiiedbbabbtnsso fet(ith1oaee9lcr9lha2oat)wuio.toftnohscWleoeorebnrfagesteliihaanctcditheao=ppn2espdner,idqoneuawnectitnhhtlcehyegissdwvei=eeqm5nruu0eeli0nan0dct.eGeel,edeltLTwem=dhea.8entsaTfoeinoorqdksutreReen5udvc0beue0ircsn0y,e
sampling,weneedtodrawsamplesfromthefollowing 10thiterationfortheleft5000iterations,leadingto
fullconditionaldistributions: 500 iterations foreach sequence kept for analysis.
|,r./»,a'j-^«P.{-^i- ThuswefinallyhaveL=8sequenceswithsizen=500
. I«| W. foreachsequence.
(lg(«,/(»2,+aC:,)-Xjfi)'-}, i=l,...,m; Fsoimrutlhaetepdarvaamleuteerofofmi,nitneretshteu1-tt,hlesteqku^.endceen.otTehtehnejt-hteh
187
m
' , .
posterior mean of m,is estimated by small provinces such as P.E.I, and Manitoba, the
ui=YL X"_iM«//Ln. ThenwecomputedBlIn,the iesmtpirmoavteemiesnstignoifficpaonstt.eriFoorreOsnttiamraitoeaonvderQutehbeecd,irtehcet
vbaarsieadncoenbent=w5e0e0nstihmeula8tesdeqvuaelnucees moefanust,; twh);a.teaicsh, lalaanrrdggeetshtestapwmoopslteeprrioosvirizneecsetsfiomriantetChaehnsaeavdeat,ewqoutahelprdCoivVrienccdteusee.sttiomatAthnee
tBhet/nav=erXag;Le=](o«,f,.t-heu,)82/w(iLth-i1n)-.seqAulesnoc,elevtariWa,nced,enost\e, piontsetreersitoirngesrteismualtteis«,thhaastlfaorrgerNeCwVBtrhuannswtihcek,dirtehcet
each based on n-1 degrees of freedom; that is, estimate y,,andalso «,isquitedifferentfrom y,.
neWsa,tmie=mlay^t,ed_&fs?bt=yIL(.na-Tw1eh)Wie,g-nh/ttneh+deBpoatsv/tener,raigwoerhevroaefrina=Wn5,c0ea0n.odfNuoBtt,ies TpAwochhstietusreaerliisloydrmiorprerracoettvelioynb0c,veeisisotNuioesmnawlitnyeBdtr2eu.rrna5mts5sew%iocrfftkouriinssdNea3ern.cw2oou5vBte%lrriuaenagrsnewdinircatttkhhe.e,e
thatifonlyonesequenceissimulated, Bi cannotbe model. Moredetailedanalysiswillbegiveninthe
calculated. To estimate the i-th province nextsection.
u6ntd=eructo/v(we,ra+gce,),ratleet 6e'taj,=uw'hyi/c(uh*,j+isci)d,eftihneend thaes 5.TESTOFMODELFITTING
posterior mean of 0,can be estimated by TotesttheoverallfitofModel2,weusethemethod
§i=X/=iSn=i^;^n-Similarly,wecancalculatethe oetfaplo.s,te1r9i9o5r).preLdeitctiTv(ey,p0)valbueea(Mdeinsgcr,ep1a9n9c4y;mGeealsmuarne
posteriorvarianceof0,. Tomonitortheconvergence dependingonthedatayandonparameters6. Let9*v
oftheGibbssampler,wecalculateVJ=&?+BtILn, representadrawfromtheposteriordistributionof8:
andthenfindRt=V^/W, foreachobservation. R, is and let y* representa draw from f{y\9*), then
knownasapotentialscalereductionfactor(Gelman marginally y* is a sample from the posterior
andRubin, 1992). Iffl.'sarenear1forallofthe predictive distribution /(y|yofa), where yobs
parameters m, ofinterest,thenthissuggeststhatthe representstheobserveddata. Theposteriorpredictive
desiredconvergenceisachievedintheGibbssampler. p value is defined as
eInquoaulrtostourdyv,ervyalculeosseoftoR1t,'sthfeorytshteron1g0lyprsouvgignecsetstahraet ppro=baPbirl(itnyyi's,weit)h>rneyspobesc,tdt)o\ythobes)p.osterNioortedistthraitbuttihoen
the Gibbs sampler converged very well. Table 1 giventheobserveddata.
shows the direct undercoverage estimates and the
posterior undercoverage estimates as well as the Thisisanaturalextensionoftheusualpvalueina
associated coefficients ofvariation. The posterior Bayesiancontext. Ifamodelfitstheobserveddata,
esssmutarilvmleaeytreecsootefifmfaittcheieencyt,enofsofursvaalrulinadptreioorvncion(vcCeerVsa)geetxhcacenoputtnhtOentmda,irrhieaocs,t tfsiihtmesinltathrh.eeoItbnwsooetrhvveearldwueomsroddsoe,fl,itfhetthheednigsicvTre(enypaombnoscd,y0e)lmeasadhseouqurulaedtealbryee
QuebecandNewBrunswick..Particularlyforsome
Table1.Censusundercoverageestimation
Province yt cvty,) u, CV(h,.) rt% CV(f,) e;% cvw,)
NFLD, 11566 0.16 10782 0.14 1.99 0.16 1.86 0.13
PEI' ; 1220 0.30 1486 0.19 0.93 0.30 1.13 0.19
NS 17329 0.20 17412 0.14 1.89 0.20 1.90 0.14
NB 24280 0.14 18948 0.17 3.25 0.13 2.55 0.17
QUE ; 184473 0.08 189599 0.08 2.58 0.08 2.65 0.08
MONATN,! ,.3•8•120160941, 00..20-18- 32618540244 00..0184 31..6846 00..0280 31..5923 00..0184
SASK 18106 0.19 18822 0.14 1.80 0.18 1.87 0.13
BACLTA . ,5912822356 00..1150 5859392992 00..0192 22..0713 00..1104 22..1637 00..0192
188
