SAS-PROC-MIXED

SASPROCMIXED1SASPROCMIXED(asinthestandardlinearmodel)buttheirvariancesandcovariancesaswell.TheprimaryassumptionsunderlyingtheanalysesperformedbyPROCMIXEDareasfollows:•Thedataarenormallydistributed(Gaussian).•Themeans(expectedvalues)ofthedataarelinearintermsofacertainsetofparameters.•Thevariancesandcovariancesofthedataareintermsofadifferentsetofparameters,andtheyexhibitastructurematchingoneofthoseavailableinPROCMIXED.SinceGaussiandatacanbemodeledentirelyintermsoftheirmeansandvariances/covariances,thetwosetsofparametersinamixedlinearmodelactuallyspecifythecompleteprobabilitydistributionofthedata.Theparametersofthemeanmodelarereferredtoasfixed-effectsparameters,andtheparametersofthevariance-covariancemodelarereferredtoascovarianceparameters.Thefixed-effectsparametersareassociatedwithknownexplanatoryvariables,asinthestandardlinearmodel.Thesevariablescanbeeitherqualitative(asinthetraditionalanalysisofvariance)orquantitative(asinstandardlinearregression).However,thecovarianceparametersarewhatdistinguishesthemixedlinearmodelfromthestandardlinearmodel.Theneedforcovarianceparametersarisesquitefrequentlyinapplications,thefollowingbeingthetwomosttypicalscenarios:•Theexperimentalunitsonwhichthedataaremeasuredcanbegroupedintoclusters,andthedatafromacommonclusterarecorrelated.•Repeatedmeasurementsaretakenonthesameexperimentalunit,andtheserepeatedmeasurementsarecorrelatedorexhibitvariabilitythatchanges.Thefirstscenariocanbegeneralizedtoincludeonesetofclustersnestedwithinanother.Forexample,ifstudentsaretheexperimentalunit,theycanbeclusteredintoclasses,whichinturncanbeclusteredintoschools.Eachlevelofthishierarchycanintroduceanadditionalsourceofvariabilityandcorrelation.Thesecondscenariooccursinlongitudinalstudies,whererepeatedmeasurementsaretakenovertime.Alternatively,therepeatedmeasurescouldbespatialormultivariateinnature.PROCMIXEDprovidesavarietyofcovariancestructurestohandletheprevioustwoscenarios.Themostcommonofthesestructuresarisesfromtheuseofrandom-effectsparameters,whichareadditionalunknownrandomvariablesassumedtoimpactthevariabilityofthedata.Thevariancesoftherandom-effectsparameters,commonlyknownasvariancecomponents,becomethecovarianceparametersforthisparticularstructure.Traditionalmixedlinearmodelscontainbothfixed-andrandom-effectsparameters,and,infact,itisthecombinationofthesetwotypesofeffectsthatledtothenamemixedmodel.PROCMIXEDfitsnotonlythesetraditionalvariancecomponentmodelsbutnumerousothercovariancestructuresaswell.PROCMIXEDfitsthestructureyouselecttothedatausingthemethodofrestrictedmaximumlikelihood(REML),alsoknownasresidualmaximumlikelihood.ItisherethattheGaussianassumptionforthedataisexploited.OtherSASPROCMIXED2estimationmethodsarealsoavailable,includingmaximumlikelihoodandMIVQUE0.Thedetailsbehindtheseestimationmethodsarediscussedinsubsequentsections.Onceamodelhasbeenfittoyourdata,youcanuseittodrawstatisticalinferencesviaboththefixed-effectsandcovarianceparameters.PROCMIXEDcomputesseveraldifferentstatisticssuitableforgeneratinghypothesistestsandconfidenceintervals.Thevalidityofthesestatisticsdependsuponthemeanandvariance-covariancemodelyouselect,soitisimportanttochoosethemodelcarefully.SomeoftheoutputfromPROCMIXEDhelpsyouassessyourmodelandcompareitwithothers.BasicFeaturesPROCMIXEDprovideseasyaccessibilitytonumerousmixedlinearmodelsthatareusefulinmanycommonstatisticalanalyses.InthestyleoftheGLMprocedure,PROCMIXEDfitsthespecifiedmixedlinearmodelandproducesappropriatestatistics.SomebasicfeaturesofPROCMIXEDare•covariancestructures,includingvariancecomponents,compoundsymmetry,unstructured,AR(1),Toeplitz,spatial,generallinear,andfactoranalytic•GLM-typegrammar,usingMODEL,RANDOM,andREPEATEDstatementsformodelspecificationandCONTRAST,ESTIMATE,andLSMEANSstatementsforinferences•appropriatestandarderrorsforallspecifiedestimablelinearcombinationsoffixedandrandomeffects,andcorrespondingt-andF-tests•subjectandgroupeffectsthatenableblockingandheterogeneity,respectively•REMLandMLestimationmethodsimplementedwithaNewton-Raphsonalgorithm•capacitytohandleunbalanceddata•abilitytocreateaSASdatasetcorrespondingtoanytablePROCMIXEDusestheOutputDeliverySystem(ODS),aSASsubsystemthatprovidescapabilitiesfordisplayingandcontrollingtheoutputfromSASprocedures.ODSenablesyoutoconvertanyoftheoutputfromPROCMIXEDintoaSASdataset.SeetheChangesinOutputsection.NotationfortheMixedModelThissectionintroducesthemathematicalnotationusedthroughoutthischaptertodescribethemixedlinearmodel.Youshouldbefamiliarwithbasicmatrixalgebra(refertoSearle1982).Amoredetaileddescriptionofthemixedmode