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AIDS

    AIDS

    TransactionsofTianjinUniversity

    ISSN1006-4982pp001-007

    V0I.13No.1Feb.2007

    AIDSTreatmentsEfficiencyAnalysisBasedonCostEfficiency

    DataEnvelopmentAnalysisModel

    BIANFuping(边馥萍),XUYin(许茵)

    (SchoolofSciences,TianjinUniversity,Tianjin300072,China)

    Abstract:Threedataenvelopmentanalysis(DEA)modelswereusedtoanalysetherelativeeffi

    cienciesoffourAIDStreatmentsinAIDSClinicaITriaIGroup(ACTG1Studyl93A(1309patientsin

    totaI.classifiedinto4agegroups).ResultsfromlheoutputorientedBCCmodeIshowthatTreat

    ment4(600mgofzidovudineplus400mgofdidanosineplus400mgofnevirapine)isparticularly

    efficientforagegroupl425,butnotefficientfortheolderagegroups;Treatment1(600mgOf zidovudinealternatingmonthlywith40()mgofdidanosine)andTreatment2(60()mgofzidovudine

    plus2.25mgofzalcitabine)areefficientfortheagegroups3545and45':agegroup25

    35

    doesnothaveaparticularlyefficienllreatment.bulTreatmentsland2arerelativelygood.The costefficiencyBCCmodeI.whichtakesthetreatmentcostintoaccount,givessimilarresultsasthe

    outputorientedmodeI.ResultsfrOmtheindirectoutput

    orientedBCCmodeI.whichallows1ere.

    placementamongmedicines.showthattheefficiencyofTreatment2hasgreatlydecreasedco

m.

    paredwiththatoftheoutput-orientedmodeI.andasetofoptimalmedicineamountsfordiffere

    nt

    agegroupsisobtained.

    Keywords:dataenvelopmentanalysis(DEA);decisionmakingunit(DMU):coslefficient:A

    IDS

    treatment

    Acquiredimmunedeficiencysyndrome(AIDS), causedbyinfectionofhumanimmunodeficiencyvirus (HIV).isoneofthefataldiseasesthathavespread acrosstheworldsinceitfirstoccurred.In2003.fol

    lowingaiointsurveywithWorldHeahhOrganization (WHO)andJointUnitedNationsProgrammeon HIV/AIDS(UNAIDS).theChinesegovernmentesti. matedthatChinahadthenabout840000HIV.infec. tedpeopleincludingabout80000HIV/AIDSDa. tients,andtheestimationdidnotincludethosewho hadalreadydied,andtherewasahugeunder.repo~. ingofAIDScases.especiallyintherura1areasJJ.A lotofresearcheshavebeendonetoestimatethetotal numberofHIV.infectedpeople3j.

    Theaimofthis

    paperistopresentanassessmentofdifferentAIDS treatmentsthroughDEAmodels.

    DEAisawidelyusedoptimization

    niquethatallowsmeasuringtherelativeperformance ofdecisionmakingunits(DMUs),whicharecharac. terizedbyamultipleobjectives(outputs)and/ora multipleinputsstructure.WearegoingtouseDEA modelstomeasuretherelativeperformanceoffour

    AIDStreatmentshere.Thepreviousworkinthisarea (usingDEAtoassessAIDStreatments)considered eithertheundesirableoutputsl6j,orthestochastic factorsoftheinputsandoutputslofasoletreatment overasinglepatient,whilethispaperintroducesthe costefficiencyoffourAIDStreatmentsoverpatientsof differentagegroups.

    Thestructureofthispaperisasfollows.Firstly, weintroducethebackgroundoftheAIDStreatments studyandthedataresource.Secondly,wegetthe DMUsforassessment.ThenweintroducethreeDEA basedtech-models,andapplythemtotheassessment. Accepteddate:2006-10-30.

    BIANFuping,bornin1947,female,Prof.

    E-mail:fpbian@tju.edu.on(BIAN),xuyin1982@yahoo.com.en(XU), SuppoaedbyNationalNaturalScienceFoundationofChina(No.10571134)

    TransactionsofTianjinUniversityVo1.13No._,2007 DatafromACTG

    TheAIDSClinicalTria1Group(ACTG)isthe

    largestHIVclinicaltrialsorganizationintheUS. whichplaysamajorroleindefiningthestandardsof carefortreatmentsofHIVinfectionandopportunistic diseasesrelatedtoHIV/AIDSaroundtheworld.Pa. tientsinACTGStudy193Awererandomizedtodual ortriplecombinationsofHIV.1reversetranscriptase inhibitors.Specially.patientswererandomizedto oneoffourdailyregimenscontaining600mgof zidovudine:zidovudinealternatingmonthlywith400 mgofdidanosine;zidovudineplus2.25mgofzalcit-

    abine;zidovudineplus400mgofdidanosine:or zidovudineplus400mgofdidanosineplus400mgof flevlraplne(tripletherapy),

    TheCD4cellsarethesubstancesinhumanim. munesystemsthatfightagainsttheHIV.andthe CD4counts(cells/ram)inthebloodisthemost importantindexofthehealthlevelofanAIDSDa tient.InStudy193A.measurementsofCD4counts werescheduledtobecollectedatbaselineandat8. weekintervalsduringfollow.up.However,theCD4 countsdataareunbalancedduetomistimedmeas. urementsandmissingdatathatresultedfromskipped visitsanddropout(seeTab.1),

    Tab.1DataforfourrandomlyselectedsubjectsfromACTG Study193A

    ——1——

    ThenumberofmeasurementsofCD4counts

    duringthefirst40weeksoffollow.uDvariedfrom1 to9,withamedianof4.Theresponsevariableis thelogtransformedCD4counts,log(CD4counts

    1).availableon1309patientsJ.

    Thecategoricalvariabletreatmentiscodedas: Treatment1,zidovudinealternatingmonthlywith 400mgofdidanosine;Treatment2,zidovudineplus 2.25mgofzalcitabine:Treatment3.zidovudine plus400mgofdidanosine;Treatment4,zidovudine plus400mgofdidanosineplus400mgofnevirap- iBe.Thevariableweekrepresentstimesincebase. 1ine(inweeks),andthevariablegenderiscoded as:1formale:0forfemale.

InRef.9],astochasticmethodwasusedto

    analysethedataofStudy193At9J.Herewearego. ingtousethenon.parameterDEAtechniquetoas

    sesstherelativeefficienciesofthefourtreatments overpatientsofdifferentagegroupsinStudy193A. 2DMUs(age-classifiedassessment)

    Thereare1309patients.whichmeans1309 groupsofdatacharacterizedbytreatment,age,and gender.SinceDEAcannotdealwithlargesamples, asimpleclusteringofthedataisnecessaryforusto gettheDMUs.Sincethetreatmenteffecthasa10tto dowiththeageofthepatient.thedatawillbeclas. siftedbytreatmentandage.Afterthat.themean valueofthevariableswillbeusedintheassessment (ageclassifiedassessment),whichwillalsoreduce therandomfactorsofthedata.

    First.thepatientsinStudy193Aareclassified intofouragegroups:14-25,2535,3545,

    and45.withagevariableaccordingly1iesinin. tervals(14,25],(25,35],(35,45]and(45, +..1.16DMUsarethengainedbyclassifyingthe patientsbythefouragegroupsandthefourtreat

    mentsused.InTab.2.Treatment1.14_25stands

    fortheDMUwhichwererandomizedtoTreatment1. andagedbetween14and25.andtheDMUwilltake themeanvalueofthegroupasitsinputsandout. puts.ThemeaningoftheotherDMUscanbegained likewise.Sinceitistheassessmentoftreatment effect,boththeinputsand

    shouldbemeasuredduring

outputsofalltheDMUs

    thesameintervals.We

    willtakethefourintervals(week),(0,10),(0, 20),(0,30),and(0,40)forassessment.In fact,theactualmeasuringintervalsofeachDMU willnotbeexactlythesame,andweshouldmake

    BIANFupe?.l:AIDSTre.tmenciYAnalysisBased.CostciYDaEnvelopmentAnalysisModel

    themclosewhenprocessingthedata.Weusethe meanamounts(mg)ofthefourmedicinestaken duringtheintervalsasinputs,andtheproportionbe

    tweentheCD4countsattheendandthatatthebe- ginningoftheintervalastheoutput.

    1amountsofzidovudine;

    ,——am0untsofdidanosine;

    1am0untsofzalcitabine;

    x4j--amountsofnevirapine;

    .g(m

    1

    Y1j

    e

    l.g(cI)41

    wherelog(CIM+1)Iandlog(CD4+1)?repre

    sentthel0gtransformedCD4countsatthebeginning andthatattheendoftheinterva1,respectively. ThismatchestheDEApropertythattheoutput (treatmenteffect)iSmeanttobemaximizedandthe inputs(medicines)minimized.Themeanvalueof thevariablesandtheinputs.outputsdataofthe

    DMUsduringtheinterval(0,10)arepresentedin Tab.2andTab.3.

    Tab.2Meanvalueofvariablesduringinterval(0,10) 3DEAmodels

    ThefirsttwoDEAmodelswillaccordinglyhigh- 1ightthetreatmenteffectandthetreatmentcost.The thirdmodeltakesintoaccountthetreatmentcost whenmeasuringthetreatmenteffect.

    3.1Output.orientedBCCmodel

    InAIDStreatmentsassessment.wepaymore attentiontotreatmenteffect,whichaccordinglyisthe output,andweassumethattheempiricalproduction setis

    T={(1,,4,Y1)

    ?Ayyv?Y.,?=1,,=1J=1

    A?0d=1,,}

    Tab.3Inputs-outputsda!adu~nginterval(0,10) Theoutput-orientedBCCmodelwith

    nonArchimedeaninfinitesimaliS

    (M1)

    max[+(l+2+3+4+l)]

    s.t.?A+s=,i=1,2,3,4J=l

    ?Ays=otyJ1

    ?Aj.=1

    A?0,

    s一?0,

    l?0

    Theorem1Let

    infinitesima1.Thesolution

    1

"

    ,

    then?1,and

    J=1,,

    i=1,2,3,4

    beanonA:chimedean 0f(M1)is,A,s?,

    If=1,DMUisweakDEAefficient(BCC);0

    

    3

    4

    =

    ?

    V

    A?

TransactionsofTianfinUniversityVo1.13No.2007

    If'=1,?=0,i=1,2,3,4and DMUj0

    isDEAefficient(BCC). Definition1Let=

    I,2,3,4,=

    (i,,,)

    ?J=I

    I

    "

    =0,

    一?

    S,

    A=y+sl.WecalYl1/weal.

sl-

    projectionofDMU~oontothe

    efficiencysurfaceoftheproductionfunction. Theorem2TheprojectionofDMUontothe

    efficiencysurfaceoftheproductionfunction(i,,,

    )isDEAefficientrelativetothepreviousDMUs. ProofSeeRef.[11].

    FromTheorem2wecanseethattheexplanation nfthesolutionof(MI)isthat,ass=0,i=1,2,3,

    4,andl"=0,theoutputofDMU:

    0

    canbeamplified

    '

    timeswithoutincreasinganyoftheinputs,andif theoutputcannotbeamplifiedwecallDMUDEA emcient.

    3.2CostefficiencyBCCmodel

    ThecostinAIDStreatmentisamainfactorfor patientswhenchoosingatreatment,whichaccordingly isthecostofthefourmedicinesusedintreatments. Thepricesofthefourmedicinesusedherearethebest offermadebythemanufacturerstothe1eastdeveloped countries,regardlessofmarketandtaxfactorsL12_. P=(P12,P3,P4)=

    (,,,)USdollar/nag,6Oo'40o'225'40o

    wherePl,P2,P3,P4arethepricesofzidovudine, didanosine,zaleitabine,nevirapineseparately. ThecostefficiencyBCCmodelallowsallthe inputstoreducethemselvesfreelyinordertogetthe bestcostefficiency,withoutworseningtheoutput(the

treatmenteffect).

    ThecostefficiencyBCCmodelis

    44

    (M2)

    minEoipf/Ep=ci=1i=l

    s.t.?=1

    A=Oixi=1,2,3,4

    ?Ajy=Y

    0 ?A=1,?

    4——

    Since0?0i?l,wehavec=Eoipixljo/

    4

    ??1.Andifc=l,obviouslywehave01=1,

    i=l,2,3,4

    Definition3Let',A,1beasolutionof

    (M2).Ifthereisonesolutionof(M2)satisfyingc<1, thenDMUiscostinefficient;ifallthesolutionsof (M2)satisfyc=1,DMUjisweakcostefficient;ifall thesolutionsof(M2)satisfyc=1andl=0,DMUj iscostefficient[13_.

    Thecostefficiencyindicatorcmeansthatthe 4

    treatmentcosIofDMUj

    .canbelessenedto?.If

    DMUiscosteffieient,itmeansthatnoneofitsinputs canbedecreasedwithoutworseningtheoutput,i.e. DMUisoptimalincostefficiency.

    3.3IndirectoutputorientedBCCmodel

    Theprevioustwomodelsconsidertreatmenteffect andtreatmentcostrespectively,whiletheindirect

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