Multilevel modelling of ordered and unordered …

1 Slide1'&$%Multilevelmodellingof orderedandunorderedcategoricalresponsesS ophiaRabe-HeskethInstituteof PsychiatryKing'sCollegeLondonJointwork withAndersSkrondal,NorwegianInstituteof PublicHealth,OsloandAndrewPickles,TheUni versity of ManchesterInstituteof ChildHealthLondon,February 2003 Slide2'&$%Outline Orderedcategoricalresponses{Single-level models{Multilevelmodels Two-levelrandominterceptmodels Cluster-speci candpopulationaveragee ects Multilevelrandomcoe cientmodels{Estimationmethods{Analysisof clusterrandomizedtrialof sexeducation Unorderedcategoricalresponses{Single-lev elmodels{Multilevelmodels{Analysisof abuseof antibioticsin youngchildrenin China1 Slide3'&$%I.}}}}}}}

2 Orderedcategoricalresponses Onlya smallnumber of responsesor `categories'are possible,ys,s= 1; ; S Thecategoriesare ordered ,y1< y2< < yS;we0ll setys=s Examples:{Severity of symptoms( ):`none',`moderate',`severe'{Frequencyof symptoms:`never',`occasionally',`nearlye veryday',`everyday'{Responseto treatment:`progressivedisease',`nochange ',`partialremission',`completeremission' {Diagnosis:`non-autistic',`PDD-NOS'(perv asivedevelopmentdisorder),`autistic'{Tes tresults( ):`normal',`borderline',`abnormal'{Satis factionwithtreatment:`verydissatis ed',`dissatis ed',`satis ed',`verysatis ed'Slide4'&$%Latentresponsemodels Underlyingtheobservedordinalresponseyifo r subjectiis alatent(unobserved)continousresponsey i.}}}}}}

3 A thresholdmodeldeterminestheobservedrespo nse:yi=8>>>>>>>>>>> <>>>>>>>>>>>:1ify i 12if 1<y i S 1<y i; For a probitmodel,y i N(0;1), withS= 3categories:latent k1 k2Pr(y=y1)Pr(y=y2)Pr(y=y3) Thelatentresponseis modelledas a linear regressionwithoutan intercept(for identi cation)y i= 1x1i+ 2x2i+ + i= 0xi+ i2 Slide5'&$%Generalizedlinear models Thecumulativeprobabilitiesare modeledasPr(yi> s) =F( 0xi s); s= 1; ; S 1; theinverselinkfunction Proportionaloddsmodel(logitlink):Pr(yi> s) =exp( 0xi s)1 + exp( 0xi s)log24Pr(yi> s)1 Pr(yi> s)35= 0xi s{Theexponentiatedregressioncoe cientscanbe interpretedas oddsratiosfor highversuslow scoresregardlessofcut-point(proportional odds):Pr(yi> s)=[1 Pr(yi> s)]Pr(yj> s)=[1 Pr(yj> s)]= exp( 0fxi xjg) Cumulativemodelsare equivalentto latentresponsemodels:ModelLinkF 1 Distributionof iVarianceof iProportionaloddslogitlogistic 2=3 Ordinalprobitprobitstandard 2=6 Slide6'&$%Two-levelrandominterceptmodels Subjectsinestedin clustersj( ).}

4 Includearandominterceptujfor clustersin thelatentresponsemodely ij= 0xij+uj+ ij; uj N(0; 2); ujindep:of ij: Thetotalresidual ij=uj+ ijhasvariancevar( ij) =8>> <>>: 2+ 1forprobitmodels 2+ 2=3 forlogitmodels Thecovariancebetweenthetotalresiduals ijand i0jof twosubjectsin thesameclusteris 2andtheintraclasscorrelationis Cor( ij; i0j) =8>> <>>: 2=( 2+ 1)forprobitmodels 2=( 2+ 2=3)forlogitmodels Thelatentresponsesfor two unitsin thesameclusterareconditionallyindependen tgiventherandomintercept:Cor(y ij; y i0jjxij; uj) = 0If we donotconditionontherandomintercept,theco rrelationis theintraclasscorrelationCor(y ij; y i0jjxij) = 3 Slide7'&$%Cluster-speci cversuspopulationaveragee ects For a probitmodel,the`marginal'or populationaverageresponseprobabilitiesar ePr(yij> s) =Pr(y i> s) = Pr( 0xij+ ij> s)=Pr( ij 0xij s)=Pr0@ ijp 2+ 1 0xij sp 2+ 11A= 0@ 0xij sp 2+ 11A;where ij=uj+ ij.

5 {Therefore,themarginale ectsofxijare =p 2+ 1. Toachievea givenmarginale ect, mustincreaseif 2increases. The`conditional'probabilitiesfor a givenclusterjarePr(yij> sjuj) = 0@ 0xij+uj s11A:{Therefore,theconditionalor cluster-speci ce ects ofxijare greaterthanthemarginalor populationaveragee ects =p 2+ '&$%Cluster-speci cversuspopulationaveragee ectscluster-speci c, '&$%Multilevelrandomcoe cientmodels Considerclusteredlongitudinaldatawithocc asionsi(level1)nestedin subjectsj(level2) in hospitalsk(level3) Exampleof a three-levelrandomcoe cientmodel:y ijk= [u(2)0jk+u(3)0k] + [ 1+u(2)1jk+u(3)1k]x1ijk+ 2x2ijk+ ijk{u(2)0jkandu(3)0kare randominterceptsat levels2 and3.}}}

6 {u(2)1jkandu(3)1kare randomcoe cientsofx1ijk.{Therandomcoe cientshavezeromeansandthe xede ect 1ofx1ijkrepresentsthemeane ect.{Randome ectsat thesamelevelare correlated,(u(2)0jk; u(2)1jk)is bivariatenormal. Generalthree-levelrandomcoe cientmodely ijk= 0xijk+u(2)0jkz(2)ijk+u(3)0jkz(3)ijk+ ijkSlide10'&$%Estimation Estimationof multilevelmodelswithcategoricalresponses ,alsoknownas generalisedlinear mixedmodels(GLMMs),isnoteasybecausetheli kelihood does notgenerallyhaveaclosedform. MarginalQuasilikelihood (MQL)andPenalizedQuasilikelihood(PQL)are approximatemethods avaiblableinMLwiNandHLM.}}}

7 {Two versionsare available, rstandsecondorder(MQL-1,MQL-2,PQL-1,PQL- 2),thelastbeingthebest.{EvenPQL-2sometim esproducesbiasedestimates,particularlywh entheclustersare small.{Themethods donotprovidea likelihood. Maximumlikelihood estimationrequiresevaluationof integralssincethelikelihood is marginalwithrespectto therandome ects.{NumericalintegrationusingGauss-Her mitequadratureisusedinMIXOR/MIXNO(two-le velonly),aML,SAS PROCNLMIXED(two-levelonly)andgllamm.{Ada ptivequadratureis superior to `ordinary quadrature'whichsometimesdoesn'twork ( ,counts).Thisis availableinSAS PROCNLMIXED(two-levelonly)andgllamm.}}}} }

8 {HLMprovidesa '&$%Clusterrandomizedtrialof sexeducation Schoolswererandomisedto receivesexeducationor not Assessmentspre randomisation,6 monthsand18 monthspostrandomisation Oneoutcomeis a questionrelatingto `Contraceptiveself-e cacy':{\IfmypartnerandI wereaboutto haveintercoursewithouteitherof us havingmentionedcontraception,it wouldbeeasyfor meto producea condom(if I broughtone)"{Thequestionsis answeredin termsof veordinalcategories:`notat all trueof me',`slightlytrueof me',`somewhattrueof me',`mostlytrueof me',`completelytrueof me'. Thedataare multilevelwithresponses(level1) from1184pupils(level2) from46 schools(level3).}}}

9 Only570pupilsalways responded, '&$%Models Occasionst, subjectsj, schoolsk Covariates{x1t[Time](0,1, 3){x2jk[Treat](yes=1,no=0){x3tjk[Treat] [Time] Modeltheprobability of exceedinga categorys,s= 1;2;3;4logit[Pr(ytijk> s)] = 1x1t+ 2x2jk+ 3x3tjk+u(2)jk+u(3)k s Estimationusingadaptivequadratureingllam m:gllammuse treattimetreat_time,i(idschool)/**/ link(ologit)family(binom)adapt Conditionalandmarginalprobabilities:gen u1 = 0gen u2 = 0gllapredp_cond,mu us(u)above(2)gllapredp_marg,mu margabove(2)6 Slide13'&$%EstimatesSingle-levelmodelTwo -levelmodelThree-levelmodelParameterEst( SE)Est(SE)Est(SE) 1[Time] ( ) ( ) ( ) 2[Treat] ( ) ( ) ( ) 3[Time] [Treat] ( ) ( ) ( )var(u(2)jk){ ( ) ( )var(u(2)k){{ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Log-likelihood-2531-2471-2471 Slide14'&$%Conditionalrelationships Probability of respondingat least`mostlytrueof me'(category3) Relationshipbetweentheprobability andoccasionfor thetwotreatmentgroups,whentherandominter ceptis -1,0 and1.}}}}}}

10 Interventiongroup,controlgroupTime in monthsAt least mostly '&$%Observedandpredictedmarginalproporti onsinterventiongroup,controlgroup, predicted, observedPr(y >1)Pr(y >2)Time in monthsAt least slightly in monthsAt least somewhat (y >3)Pr(y >4)Time in monthsAt least mostly in monthsCompletely '&$%II. Unorderedcategoricalresponses Onlya smallnumber of responsesor `categories'are possible,a,a= 1; ; A. Thecategoriescannotbe ordereda priori. Examples:{Treatmentdecision:`chemotherap y',`surgery',`none'{Healthinsurancechoic e:`NHS',`PPP',etc.{Method of birthcontrol:`pill',`condom',etc.}}}