General Linear ModelWord格式.docx

General Linear ModelWord格式.docx

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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 

.00

34.00 

154.00 

19.00 

113.00 

omitted

2.00 

48.00 

.00 

1.00

29.00 

100.00 

114.00 

3.00 

108.00 

-1.00 

-1.00

65.00 

191.00 

55.00 

112.00 

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