ltm

raschDescription

FittheRaschmodelundertheItemResponseTheoryapproach.Usage

rasch(data,constraint=NULL,IRT.param=TRUE,start.val=NULL,

na.action=NULL,control=list(),Hessian=TRUE)Arguments

dataconstraint

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adata.frame(thatwillbeconvertedtoanumericmatrixusingdata.matrix())oranumericmatrixofmanifestvariables.

atwo-columnnumericmatrixwithatmostprows(wherepisthenumberofitems),specifying?xed-valueconstraints.The?rstcolumnrepresentstheitem(i.e.,1denotesthe?rstitem,2thesecond,etc.,andp+1thediscriminationpa-rameter)andthesecondcolumnthevalueatwhichthecorrespondingparametershouldbe?xed.SeeExamplesformoreinfo.

logical;ifTRUEthenthecoef?cients’estimatesarereportedundertheusualIRTparameterization.SeeDetailsformoreinfo.

thecharacterstring\oranumericvectorofp+1startingvalues,wherethe?rstpvaluescorrespondtotheeasinessparameterswhilethelastvaluecor-respondstothediscriminationparameter.If\randomstartingvaluesareused.IfNULLstartingvaluesareautomaticallycomputed.

thena.actiontobeusedondata.Incaseofmissingdata,ifna.action=NULLthemodelusestheavailablecases,i.e.,ittakesintoaccounttheobservedpartofsampleunitswithmissingvalues(validunderMARmechanismsifthemodeliscorrectlyspeci?ed).Ifyouwanttoapplyacompletecaseanalysisthenusena.action=na.exclude.alistofcontrolvalues,

iter.qNthenumberofquasi-Newtoniterations.Default150.

GHkthenumberofGauss-Hermitequadraturepoints.Default21.

methodtheoptimizationmethodtobeusedinoptim().Default\verboselogical;ifTRUEinfoabouttheoptimizationprocedureareprinted.

IRT.paramstart.val

na.action

control

Hessian

logical;ifTRUE,thentheHessianmatrixiscomputed.Warning:settingthisargumenttoFALSEwillcausemanymethods(e.g.,summary())tofail;settingtoFALSEisintendedforsimulationpurposesinorderrasch()torunfaster.

Details

TheRaschmodelisaspecialcaseoftheunidimensionallatenttraitmodelwhenallthediscrimina-tionparametersareequal.Thismodelwas?rstdiscussedbyRasch(1960)anditismainlyusedineducationaltestingwheretheaimistostudytheabilitiesofaparticularsetofindividuals.Themodelisde?nedasfollows

log??πi1?πi

??

=βi+βz,

50rasch

whereπidenotestheconditionalprobabilityofrespondingcorrectlytotheithitemgivenz,βiistheeasinessparameterfortheithitem,βisthediscriminationparameter(thesameforalltheitems)andzdenotesthelatentability.

IfIRT.param=TRUE,thentheparametersestimatesarereportedundertheusualIRTparameteri-zation,i.e.,

????

πi?

log=β(z?βi).

1?πiThe?tofthemodelisbasedonapproximatemarginalMaximumLikelihood,usingtheGauss-Hermitequadraturerulefortheapproximationoftherequiredintegrals.

Value

Anobjectofclassraschwithcomponents,coefficientslog.Likconvergencehessiancountspatterns

amatrixwiththeparametervaluesatconvergence.Thesearealwaystheesti-matesofβi,βparameters,evenifIRT.param=TRUE.thelog-likelihoodvalueatconvergence.theconvergenceidenti?erreturnedbyoptim().

theapproximateHessianmatrixatconvergencereturnedbyoptim().

thenumberoffunctionandgradientevaluationsusedbythequasi-Newtonal-gorithm.

alistwithtwocomponents:(i)X:anumericmatrixthatcontainstheobservedresponsepatterns,and(ii)obs:anumericvectorthatcontainstheobservedfrequenciesforeachobservedresponsepattern.

alistwithtwocomponentsusedintheGauss-Hermiterule:(i)Z:anumericmatrixthatcontainstheabscissas,and(ii)GHw:anumericvectorthatcontainsthecorrespondingweights.

themaximumabsolutevalueofthescorevectoratconvergence.thevalueoftheconstraintargument.thevalueoftheIRT.paramargument.acopyoftheresponsedatamatrix.thevaluesusedinthecontrolargument.thevalueofthena.actionargument.thematchedcall.

GH

max.scconstraintIRT.paramXcontrolna.actioncallWarning

IncasetheHessianmatrixatconvergenceisnotpositivede?nite,trytore-?tthemodelusingrasch(...,start.val=\.