General Linear Model.docx

General Linear Model.docx

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GeneralLinearModel

GeneralLinearModel

4/16/02

Announcements

Background

TheGeneralLinearModel

Therearethreereasonsforcoveringthismaterial.

∙Thismaterialprovidesanintroductiontotheuseof"dummy"variables.

oThesevariablesareveryusefulwheneveryouhaveacategoricalvariable,andareactuallymoreusefulinstandardmultipleregression.

∙Thismaterialemphasizestheimportanceofmodels

oItcauses（conducts,leads）ustothinkabouthowwewanttogoabout（embark）testingmodels,andthealternativeswaysthatwecanlookatproblems.

∙Itmakesitmucheasiertotalkabouttheanalysisofcovariance,andrelatedtechniques,andtotalkaboutunequalsamplesizesandhowwewanttotestthem.

Thistopicstartsoutasamoredifficultwayofdoingwhattheyalreadyknowhowtodo.Butitthengoesontopresentotherstuff（thing）inamuchsimplerwaythanitcouldbepresentedinanyotherway.

Ihavetriedtoremovemuchofthestuffthatdoesn'tfocusonthethreereasonsthatIgaveabove.Iwantstudentstounderstandthegeneralconcepts,andbeabletoseethattheycouldbeapplicableinothersettings.Iamnottryingtoshowpeopleaharderwaytorunananalysisofvariance.

Theapproachtakenhereisbasicallytheapproachthatanystatisticalpackagetakes,whichmayhelpexplainsomeofthesubtletiesofthosepackages.

Amajorsubthemeistoshowthattheanalysisofvariance,theanalysisofcovariance,theanalysisofmultipleregression,andawholebunch（host）ofotherthingsarejustvariationsonacommontheme.

Iwantstudentstounderstandthebasicideaofcoding（dummy）variables,butthespecifics（details）arenotimportant.

Example

IamgoingtouseoneofthesmokingexamplesfromSpilichthatwehaveseeninothercontexts.Thedatafile（Spilich.sav）containsdataonallgroups,butweareonlygoingtolookatthegroupthatwasgivenastandardrecalltask--acognitivetask.

Threebasicgroups.

∙Nonsmokers（peoplewhoneversmoked）

∙Delayedsmokers（Smokerswhohadnothadacigaretteforseveralhours）

∙Activesmokers（Smokerswhosmokedduringthetask.）

Thedependentvariablewasthenumberoferrorsmadeduringtherecalltask.

StandardAnalysisofVariance:

DescriptiveStatistics

Grandmean=38.778

Plot

Plotwitherrorbars（barsrepresent95%CI）

Anova

Itisclearthattherearesignificantdifferencesbetweengroups.IwillevengoaheadandcomparetheNon-smokerswiththecombinedsmokinggroups,andthenthetwosmokinggroupswitheachother.Thisisforcomparisonpurposeslater.

Ididthiswiththeone-wayprocedureandstandardcontrasts.

HerewecanseethatNon-smokersdifferfromsmokers,butthatthetwosmokinggroupsdonotdifferbetweenthemselves.

TheGLMapproach

FirstweneedtocodethedatatoindicateGroups.

∙WealreadyhaveGroupsas1,2,and3,butwearegoingtodoitdifferently.

oThereasonthatwehavetodoitdifferentlyisduetothefactthatourcodingiscompletelyarbitrary.Wecouldhavecodedthemas2,1,and3.Anyregressionagainstgroupmembershipwouldbeentirelydependentontheorderinwhichwecoded--that'sabadthing..

∙WewillsetupdummyvariablesthattelluswhetherasubjectisinGroup1ornot,andwhetherhe/sheisinGroup2ornot. 

oIhavecalledthesenewvariablesNonSmokeandDelayed,becausetheyidentifythosewhoareinthosetwogroups.

∙Wedon'tneedtocodeforGroup3,becauseifyou'renotin1or2,youmustbein3.

∙The"filter"variablebelowjustselectedtheCognitivetask,andignoredtheothertwotasks.

Task   Group   Errors   distract   filter NonSmoke  Delayed

2.00   1.00       27.00   126.00       1   1.00       .00

2.00   1.00       34.00   154.00       1   1.00       .00

2.00   1.00       19.00   113.00       1   1.00       .00

                               omitted

2.00   2.00       48.00   113.00       1   .00       1.00

2.00   2.00       29.00   100.00       1   .00       1.00

2.00   2.00       34.00   114.00       1   .00       1.00

                               omitted

2.00   3.00       34.00   108.00       1   -1.00   -1.00

2.00   3.00       65.00   191.00       1   -1.00   -1.00

2.00   3.00       55.00   112.00       1   -1.00   -1.00

                               omitted

（ExplainwhyIused-1foreachdummyvariableforpeopleinthelastgroup.

Thismakestheinterceptcomeouttobethegrandmean,andexpressestheresultsindistancefromthegrandmean,ratherthandistancefromthemeanofsomearbitrarygroup.

Thisideaisimportant,becauseifwearen'tcarefulitiseasytogetanswerstotellusaboutdeviationsfromsomesinglegroup,andthatusuallyisn'twhatweareafter.

Herewecometothefirstimportantidea.Ihavetakenacategoricalvariablewith3（k）levelsandturneditinto2（k-1） newvariables.Thesetwovariablescarryalltheinformationthatthesinglevariabledid,andaremoreuseful.

RegressionApproachusingDummyVariables

IwillnowsimplypredictErrorsusingNonsmokeandDelayedasmypredictorvariables.Thisisastandardmultipleregression.

LookfirstattheAnovatestfortheregression

F=4.744,p=.014

ThisisexactlythesameresultwegotwhenweranthetraditionalAnova.

Explainwhythisshouldbe.

LooknextattheR2value=.184.Thisisnothingbuteta-squared

GotothetableheadedCoefficients

Thefollowing（uptothenextmajorheading）ismaterialthatIfindimportantandhelpful,butifitaddstoinformationoverload,setitasidefornow.

NotethattheIntercept=38.778.Thisisexactlyequaltothegrandmeanofallthegroups.Interceptequalsgrandmean.

NotethattheslopeforNonsmoke=-9.911.ThisisexactlyequaltothedifferencebetweentheNonsmokemeanandthegrandmean.

NotethattheslopeforDelayed=1.157.ThisisthedifferencebetweentheDelayedmeanandthegrandmean.Slopeequalsdifferencebetweencorrespondingpredictormeanandgrandmean.

WhynothaveaslopeforActive?

?

?

Itwouldberedundant（excessive）--ifweknowthegrandmeanandthedeviationoftheothertwogroups,wecancomputethedeviationofthe3rdgroup.

Thesumofthedeviationsfromthemean=0.So,thedeviationofthethirdgroupis0-（-9.911）-1.156=8.755

IfIhadcodedforActiveandDelayed,andleftoutNonSmoke,Iwouldgetaninterceptof38.778,slopeforactive=8.755,slopeforDelayed=1.157,andcouldcomputeslopeforNonSmoke=-9.911.Thisillustratesthatthechoiceisarbitraryandunimportant.

TestingContrasts

Iforgottodooneadditionalthing,soIwentbackanddidit.IaskedSPSStocompute"deviationcontrasts"whenitrantheAnova.

Deviationcontrastsarecomparisonsofeachmeanwiththegrandmean.（Again,itdoesn'tdoallthree--itleavesoutone,whichinthiscasewasthelastone.）

Outputbelow:

Notethatthetestsandtheprobabilitiesareexactlythesameasthetests（andprobabilities）ontheregressionequation.

Whyshouldthisbe?

Whatisallofthisabout?

IwanttoshowthatAnovaandRegressionarebasicallythesameprocedure.Theonlydifferenceherebetweenthisregressionandstandardmultipleregressionistheuseofdummyvariables.

Therearealotofimportantthingshere,buttheirimportancedoesn'tshowupuntilwemovetomorecomplexanalyses.

GLMandFactorialAnova

Nowthingsgetinteresting.

First,wewilltakethesameexample,butwithallthreetasks,andcreatedummyvariablesforthedifferenttasksaswell.（Again,wecreatedummyvariablesforonlytwoofthetasks.）

Thenwecreateinteractiondummyvariablesbymultiplyingourdummiestogethertocreate4newvariables.

Nonsmoke*Patrec,Nonsmoke*Cogit,Delayed*Patrec,andDelayed*Cognit

TheoverallFactorialAnovafollows:

Regressionapproach

Wewillstartwiththecompletemultipleregressionusingalldummyvariablesaspredictors.Herewearetryingtoexplainvarianceinerrorsasafunctionofeverythingweknowaboutgroups,tasks,andtheirinteractions.

Regression

CommentonSSregressionasbeingequivalentto"Model"inregularAnova（Explainwhy8df.）10-2=8

Commentontheerrorterm.

Thiserrortermisallofthevariancein"errors"thancannotbeexplainedonthebasisofgroups,tasks,ortheirinteractions.Thisisthestandarderrorterminthefactorialanalysisofvariance.

NowstudentsshouldunderstandwhySPSSpresentstheAnovasummarytablethewayitdoes,evenifthatisaconfusingwaytohavechosentopresentit.

RemovingtheInteractionTermsgives:

ThedifferenceintheSSregressionis31744.726-29016.074=2728.652.

ThisistheSSfortheinteractiontermintheAnova.

RemovingtheTaskTerms（afterreplacinginteraction）gives:

IfwesubtractthisSSregressionfromtheSSregressioninfullmodel,weget

31744.726-3083.200=28661.526

ThisistheeffectofTask

Lastly,lookatthemodelwithdummyvariablesforTaskandInteraction,butnodummyvariablesforCondition

Herethedifferencebetweenthefullandreducedmodelsis

31744.726-31390.178=354.548

ThisistheeffectofCondition.

Noticethateachofthisisbasicallywhatwecalledahierarchicalmodelearlier.Thedifferencebetweenthefullmodelandareducedmodeliswhattheextravariable（s）explainoveranabove（controllingfor）theothervariables.

FromhereIgetthefollowingmodels:

Model

SSreg

Difference

SSerror

Effect

Full

31744.726

13587.200

Error

Maineffects

29016.174

2728.652

Interaction

Task.+Interaction

31390.178

354.548

Condition

Cond+Interaction

3083.200

28661.526

Task

Butwearen'tdone.

Yesweareforclass.Ihavelefttherest