specific

t.994)

43,No.2,pp.371-384

Subject-specificandPopulation-averaged

ContinuationRatioLogitModels

for

MultipleDiscreteTimeSurvivalProfiles

19931

SUMMARY

Subject-specificandpopulation-averagedcontinuationratio

0are

tion,generalized

logit(CRL)models

tofordinal

multinomialresponprobabilitiessummingto1,thecontinuationratioisdefined

tobetheratioofamultinomialprobabilityoverthepartialsumoftheremaining

multinomialprobabilities(e,forexample,Agresti

tAddress

Hershey,PA17033,USA.

htsRerved.

372TENHAVEANDUTTAL

the10trials,eachsubjectwasallowedthreeattemptstofindthetoyaftereing

bjectwasrandomlyassignedto

oneoftwogroups:oneinwhichthemapwasrotatedwhenprentedbyan

mary

questionofinterestis:didtherotatedmapgrouprecoverandeventuallyfindthe

toyassuccessfullyasthenon

-rotatedmapgroup?Hencethefocusoftheanalysis

isonmodellingandcomparingthetrendsincorrectnessacrossthe10locationsin

thetwogroups,wherelocationreprentsadiscretetimevariable.

TheCRLmodel,whichwasintroducedbyCox

(1988)udmiparametric

pieet

(1990),pages

319

-321and337).

Thesubject

-specificandpopulation-averagedCRLmodelsconsideredhereare

extensionsofthelogisticregressionmodelsdiscusdbyZegeretal.(1988).The

subject

-specificCRLmodelisamixedeffectsmodelwheretheexpectationofa

responisconditionalonasubject

-

thepopulation

-averagedCRLmodel,theexpectationofaresponisobtainedby

integratingoutthesubject

-specificrandomeffectsandhenceisinterpretedasan

averageforthepopulationofinterestasoppodtoanindividualsubject.

ThemixedeffectsCRLmodelprentedhereisdistinguishablefromthe

(GEES),which

arescore

-likefunctionsthatarederivedbyincorporatingaworkingcorrelation

matrix

htsRerved.

CONTINUATIONRATIO

LogitModel

Let

i=1,2,3)atthejthhidden

location(forj=1,...,0otherwiforalli,j,k,t

Y4jk/=

allj,k,I,c;=IYUkl=

jthlocation,assumethatthe

randomvector

rk),i=rijkl=1),where

k'srandomeffectsparametersinthesubject-specificCRLmodeldefined

below.

Nextdefine

rijkl

j'4

i=j,kand=

lthgroup,andgiventhatheorshehasfailedtofindthe

toyatthejthhiddenlocationontheprevious

hUkl

i.e.

(1990),

p.337,forahazardrateinterpretationofaratioofprobabilitiessimilartoAuk/.)

Conditionalontheeffectofthekthsubjectinthe

log-

likelihoodsinvolvingthe

ijk+log(l-*ijkl)Yi,jkl

i=l=;

for1,2,3andall1,andwhere$ijkl=1-

isacontinuationratioandcanbeinterpretedastheconditionaloddsthat

subjectkingroup

1,2,3,which

acco

variableindicatingthelocationatwhichthetoyishiddenisordinal,inthatthe

childrenimprovebylearningastheyproceedacrossthe10hiddenlocationsbutthen

Copyright

,b,,PL1X1,

T!*'XIj*2X2j,

0

IthT

rk1

pO,

0;

i.e.

P

flC-'(4)

1

0.

rilrk2

0s52isa3x3variance-covariancematrix.

R~ghtsRerved

CONTINUATIONRATIO

h2-',suchasa

Wishartdistribution,andRandpareanapriori

specified3x3non-singularsymmetricmatrixandapositivescalarrespectively.

tionofSubject-specificModel

ParameterestimationforthemixedeffectsCRLmodelcanbeviewedinterms

ofestimatingunivariateposteriordensities(e,forexample,Stiratellietal.(1984)

andZegerandKarim(1991))ofthe

rkand

52,becauthesimulatedvariancecom-

r,convergenceappearedtobeachieved

(althoughafter1500iterations)withthe

F()denotesthemultivariatenormaldistributionof7,(eAppendixC).

tion-averagedContinuationRatio

(jectsingroup

hesubject-specificmodel

definedinequations(1)and

hFwithamodelthatisanalogoustothemixedeffectsmodelin

equation(3)

376TENHAVEANDUTTAL

logrC/;=pL1*XU+pL2*X2j+

=A;/(1,2,3andalljand

1vector

=Oisthetestofthehy-

pothesisthat,foragivenlocation,theexpectedproportionofallsubjectsinthe

lthgroupwhofindthetoyonthefirstattemptdoesnotdifferfromtheexpected

proportionthatfindthetoyontheithattemptamongthosubjectsinthe

1attempts.

GiventheBernoullilog

-likelihood(2)withAuk/and+uk/replacedbyA$and$$

respectively,wedefinethecorresponding

(i.e.

ivj))forthekthofI,A;,

Dkl

Vk/istheworkingvariance-covariancematrixfortheelements

Ykl.

GEEestimatesoftheparametersinmodel(9)thatarebadonindependence

andexchangeableworkingcorrelationmatricesarereportedbelowwithnaiveand

dwichestimatorisaconsistentestimatorof

thevariance

-covariancestructureoftheGEEestimatesregardlessoftheworking

hevectorof

GEE

-badestimatesofthepopulation-averageddiscretehazardrates(eAppen-

dixkingroupvk/andD~~beestimatesof

nalveestimatorofthevariance-covariancematrixoftheGEEestimatorsis

GEEsare

,whenindependenceholdsandtheindependence

workingcorrelationisud,comparingtheGEEestimateswiththeirrespective

htsRerved

CONTINUATIONRATIO77

TABLE1

NaiieRobust

(tersotherthan@$,i=2,3,I=1,2)exceedinmagnitude(upto15%)

thecorrespondingGEEestimatesofthepopulation-averagedmodelunderboth

246810

locationlocation

mixedeffectstimatesofthepopulationaveragesofthesubject-specifichazard

probabilitiesacrosslocationforeachattemptfor(a)therotatedmappopulationand

non-

rotatedmapgroup:

htsRerved

TENHAVEANDUTTAL

TABLE2

Obrvedpercentage

1RotationPercentagesforthefollowingnumberofIIstatus

attempt:

2

Yes24.421.816.5

(42/254)

No53.540.725.2

(32/127)

0.50/0.17=-

0.19/0.16=-1.19)andmixedeffects(Z=-

0.40/0.22=-1.82)incontrastwith

theclearlynon

-significantestimatesoftheotherapproaches.

Thefrequencies(summedacrosslocation)andcorrespondingpercentagesforthe

cross

-classificationoftherotateandattempteffectsaredisplayedinTable2tohelp

toexplainthedifferencesingroup

-specificattempteffectestimatesamongthe

saprecipitousdropinpercentagecorrectbetweenattempts1and

3(53.5%comparedwith25.2%)forthenon

-rotatedmapgroup,whichisshown

ubstantialdropfrom53.5%to

40.7%betweenattempts1and2forthenon

-rotatedmapgroupcorrespondstothe

.

2(a)revealsthatfortherotatedgrouptheindependence

GEEprocedureweightedthespikeatlocation7forattempt1inFig.

(i.e.l?(hiikl);e

AppendixC)withthecorrespondingfittedvaluesfromtheexchangeabilityGEE

estimates

htsRerved

CONTINUATIONRATIO

(a)

-,attempt1;.-.-....,attempt2;-------,attempt3

oftheindependenceGEEestimatesofAtlandthemixedeffectestimatesofAtl,

revealingpooreragreementbetweenthetwotsofestimates.

yofSubstantiveResults

Althoughthefollowingresultsarebadonthesubject-specificestimatesinTable

1,theyaresupportedbyallmodels.

(a)Foragivensubjectintherotatedmapgroup,theconditionalprobability

ofacorrectchoiceimprovedsignificantlyacrosslocations(linear

0.086/0.015

0.095/0.025=-3.8)sothattheconditionalprobabilityof

B(hijkl))versus

thecorrespondingfittedvaluesfromtheexchangeabilityandindependenceGEEestimates,parately

Co~vriaht

TENHAVEAND

Qforthemixed

effectsmodel(3)t

Componentof

nint

0.310.30

0.140.059

0.61

Qint-in0.00850.0083

0.00340.00250.016

%n-quad0.00069

0.000650.00039

0.0000150.0015

Oquad

0.01

97.5Vodesignatethe2.5and97.5percentiles

respectively.

successdidnotreachtheconditionalsuccessrateonattempt1ifthesubject

hadreceivedanon

-rotatedmap.

(b)Foragivensubjectinthenon

-rotatedmapgroup,theconditionalprob-

abilityofacorrectchoicedidnotriasmuchacrosslocationasifthe

subjecthadreceivedarotatedmap(linear

-rotatedZ=-

3.71),whichcontrastswiththenon-significantattempteffectsforagiven

subjectintherotatedmapgroup.

Finally,theposteriorestimatesanddensitiesofthevariancecomponzntspre

-

ntedinTable3andFig.4revealthattheestimateddensityofthevariance

componentforthelineartermismasdnear

2(a)

and

subject-

specifichazardratetheneithermodelistheoreticallyandempiricallyappropriate,

althoughthepopulation-averagedmodeliasiertofit.

Copyright

CONTINUATIONRATIO

--RotatedRotateQuadratic

alposteriordensitiesofreprentativefixedeffectsandvariancecomponent

parametersofthesubject-specificmodel(3):thefullanddottedverticallinesindicatethepositions

oftheindependenceandexchangeabilityGEEestimatesrespectively,correspondingtothedisplayed

fixedeffectsdistributions

TheobrveddifferencesbetweentheindependenceandexchangeabilityGEE

estimatesoftheattempteffectsareattributabletotherelativelysmallnumbersof

subjects(42orfewer)onwhichthecondandthirdattempteffectstimatesare

ferencefortheexchangeabilityGEEestimateconformswithprevious

work(e,forexample,Lipsitzetal.(1991)andLiangetal.(1992)).However,

McDonald(1993)contendsthattheindependenceGEEestimateismore'stable'than

GEEestimatesthataccountforworkingcorrelationswhenfittingbinarylogistic

iningwhichviewholdsforCRLmodels

isasubjectoffuturerearch.

JL0.010.020.03

REvarianceintRE

CA53787whenhewasVisitingAssistantProfessor,Department

ofBiostatistics,awerecollectedunderagrant

horswouldliketothankProfessorMark

,tworefereesandtheAssociateEditorforinvaluablecommentsthat

substantiallyimprovedthepaper.

Copyright

382TENHAVEANDUTTAL

AppendixA

TheextensionofClayton'sbufferedstochasticsubstitutionalgorithmtotheCRLmodel

mthefollowingthree

steps,whichcompriabufferedstochastic

62and

thisstep

(B2th)repetitionofstep(b),

simulatenewP,whichthenenterthepoolofsimulationsof62andPfrom

lly

B1was

increadto100and

62-'.Giventhe

Wishartdistribution

withparametersthatarefunctionsofT~,

p=

Ode11andFeivson(1966).

Asanexampleofsimulatingtherandomandfixedeffectsparametersattheithiteration,

considersimulatingthefirstrandomeffectlementforsubjectk,

)equaltheproductoftheexponentiatedlog-likelihoodfunction

(2)andthepriordefinedindistribution(9,andlet

(pes)ectionsamplingalgorithmud

tosimulateavaluefor

rp,canthenbedescribedasfollows:

(a)

sample);

(b)

generatea

variance-

covariancematrixofthemultivariatesplitt-densityismatchedateachiterationtothe

inverHessianofthelog

-likelihoodpriorsuchthatthenumberofrejectionsforagiven

cisminimizedapproximately(eCarlinandGelfand(1991)fordetails).

AppendixC

FollowingKarimandZeger(1992)andZegeretal.

~teach

E(Aijkl)badonthesimulatedfixedeffectsparametersinthesubject-specificprobability

modelfortheIthgroupontheithattemptatthejthhiddenlocation:

htsRerved.

CONTINUATIONRATIO

(14)

where

Ulthcombination,oj(Q)=I[(16~3)/15~)'~X,~+~~.11-3",

XZi)and(Aijkl)istaken

acrossthe2000post

-convergenceiterationsforeachgroup-location-attemptcombination,

thusyieldingtheplotsoftheposteriorestimatesE(Aw)

ofthepopulation-averaged

discretehazardratesinFigs

2(b).

AppendixD

TheGEE-badestimateofA$[iscomputedas

=$GI()-

$*istheexchangeabilityorindependenceGEEestimateoftheparam-

etersinthepopulation-averagedmodel(9).

AppendixE:ListingofMapData

Eachlineconsistsofasubject'srotationstatus

LIL2L3L4LlORotateLlL2L5L6L7L8L9

htsRerved

384TENHAVEANDUTTAL

RotateLSL9LIORotateL5L10

References

Agresti,A.(1990)k:Wiley.

Carlomethodfor

Soc.B,34,

187-220.

t.

ASS.,83,414-425.

Fienberg,on,W.M.(1978)Identificationandestimationofage

-period-cohorteffects

intheanalysisofdiscretearchivaldata.

Carlointegration.

631-644.

Liang,K.-er,S.L.(1986)Longitudinaldataanalysisusinggeneralizedlinearmodels.

Biometrika,73,13-22.

Liang,K.

-Y.,Zeger,ish,B.(1992)Multivariateregressionanalysforcategoricaldata

(withdiscussion).t.

Soc.B,55,391-397.

.,61,198-203.

Ryan,L.(1992)rics,48,163-174.

Stiratelli,R.,Laird,e,J.H.(1984)Random

-effectsmodelforrialobrvationswith

rics,40,961-971.

Thompson,Jr,W.A.(1977)trics,

33,436-470.

Zeger,im,M.R.(1991)t.

Ass.,86,75-90.

Zeger,S.L.,Liang,K.-ert,

'."--