RepeatedMeasuresANOVAMultivariateANOVA

andTheirRelationshiptoLinearMixedModels

EPSY905:MultivariateAnalysisSpring2016

Lecture#12– April20,2016

EPSY905:RMANOVA,MANOVA,andMixedModels

Today’sClass

• RepeatedmeasuresversionsoflinearmodelsØ HowRMANOVAmodelsaremixedmodelswithvaryingassumptions

• MultivariateANOVAØ HowMANOVAmodelsaremixedmodelswithvaryingassumptions

• Anintroductiontomultilevelmodels

• Foralltoday:ModelsassumedataiscompleteØ Everyobservation isrecorded ::nomissingdata

EPSY905:RMANOVA,MANOVA,andMixedModels 2

ExampleData

• Ahealthresearcherisinterestedinexaminingtheimpactofdietaryhabitsandexerciseonpulserate

• Asampleof18participantsiscollectedØ Dietfactor(BETWEEN SUBJECTS):

w Ninearevegetariansw Nineareomnivores

Ø Exercise factor(BETWEEN SUBJECTS) withrandomassignment:w Aerobicstairclimbingw Racquetballw Weighttraining

Ø Threepulse rates (WITHINSUBJECTS):w Afterwarm-upw Afterjoggingw Afterrunning

EPSY905:RMANOVA,MANOVA,andMixedModels 3

REPEATEDMEASURESANOVA

EPSY905:RMANOVA,MANOVA,andMixedModels 4

RepeatedMeasures

• Insteadoffocusingonasingledependentvariable,wecanfocusonallthree

Ø Repeatedmeasures analysis

• InrepeatedmeasuresGLM,theeffectsofinterestare:Ø Between subjectseffects

w Differences betweenDietandExercise TypeØ Withinsubjects effects

w Differences betweenwhenpulseratewastakenw Notusuallyaconsideration inMultivariateGLM(MANOVA)

Ø Interactions betweenwithinandbetween subjectseffects

• Aswewillsee,repeatedmeasuresGLMusesmultivariatedataandmultivariatedistributions

EPSY905:RMANOVA,MANOVA,andMixedModels 5

RepeatedMeasuresDistributionalSetup

• Becausewehaverepeatedobservations,wenowhaveasetofvariablestopredictinsteadofjustone

Ø Ourexampledatasethadthree response variablesperpersonØ Moregenerally,wewillhave𝑝 response variables perperson

𝐲# = 𝑦#& 𝑦#' ⋯ 𝑦#)

• TheGLMisextendedtomodelmultivariate outcomes𝑓 𝐲# 𝐗# ∼ 𝑁) 𝐗#𝛃, 𝐕

• Now,𝛃 issize(1 + 𝑘)𝑥𝑝

• LSEstimates:𝛃8 = 𝐗9𝐗 :&𝐗9𝐘Ø Where𝐘 issize𝑁𝑥𝑝

EPSY905:RMANOVA,MANOVA,andMixedModels 6

UnpackingtheRepeatedMeasuresDesign

• Thedistributionalassumptionsareverysimilar𝑓 𝐲# 𝐗# ∼ 𝑁) 𝐗#𝛃, 𝐕

• MODELFORTHEMEANS:Ø Themodelforthemeans isthesame(onlyhasmore terms)Ø IVs𝐗# andlinearmodelweights𝛃 providepredicted values foreachobservation

• MODELFORTHEVARIANCES:Ø Becausewehaverepeated observations, themodelforthevariances isnowacovariancematrix ofsizepxp

w Diagonalelements: varianceoferrorforeachoutcomew Off-diagonalelements: covarianceoferrorsforpairsofoutcomes

EPSY905:RMANOVA,MANOVA,andMixedModels 7

MoreonVariances

• ThekeytorepeatedmeasuresandmultivariateANOVAisthecovariancematrix𝐕 structure

Ø Typeofmodeldictates typesofoptionsforcovariancematrixØ Theclassicalstatisticsdiscussed todayareoneofthethree

w Moremodernapproachesmakethismoreflexible

EPSY905:RMANOVA,MANOVA,andMixedModels 8

ThreeStructuresofClassicalGLM• Threestructures:1. Independence:𝐕 = 𝜎='𝐈)

Ø ModelsifallobservationsasiftheycamefromseparatepeopleØ NomorestatisticalparametersthanoriginalGLMapproachØ Don’tuse:shownforbaselinepurposes

2. RepeatedMeasures:AssumessphericityofobservationsØ Sphericityisaconditionthatismorestrictlyenforcedbycompoundsymmetry

of𝐕 havingtwoparameters:Ø Sphericityiscompoundsymmetryofpairwisedifferences

w Diagonalelements:samevariancew Off-diagonal elements: samecovariance

Ø Nosphericity?AdjustmentstoFtests

3. MultivariateANOVA/GLM:Assumesnothing– estimateseverythingØ Everyuniqueelementin𝐕 ismodeledØ Needmorepower(i.e.,samplesize)tomakeworkwellØ Mostgeneralprocedure

EPSY905:RMANOVA,MANOVA,andMixedModels 9

WorkingExample

• Wereturntoourexample,onlythistimetobuildtherepeatedmeasuresandmultivariateversions

• Wewillbeginwiththeunconditionalmodel(“emptymodel”)

Ø Providedasabaseline toshowhowrepeatedmeasuresworks, conceptually–youlikelywouldnever runthismodel

• Torunthismodel,Iconvertedourdatafromwideformat(alldataforoneobservationperrow)tolongformat(onlyoneobservationpercolumn)

EPSY905:RMANOVA,MANOVA,andMixedModels 10

ExampleSyntax:DataTransformation

EPSY905:RMANOVA,MANOVA,andMixedModels 11

UnconditionalModel:IndependenceAssumptionSyntax

EPSY905:RMANOVA,MANOVA,andMixedModels 12

UnconditionalModel:IndependenceAssumption

• Parsingthroughtheoutputofthepreviouspage,wecandeterminethefollowingforourdata:

𝑓 𝐲# 𝐗# ∼ 𝑁) 𝐗#𝛃, 𝐕

• Where𝛃89 = 87.5 46.6 102.1

• And

𝐕 = 𝜎='𝐈G =507.7 0 00 507.7 00 0 507.7

EPSY905:RMANOVA,MANOVA,andMixedModels 13

RepeatedMeasuresModel• Therepeatedmeasuresversionoftheunconditionalmodelchangesthestructureofthe𝐕matrix:

𝐕 =𝜎=' + 𝜏 𝜏 𝜏𝜏 𝜎=' + 𝜏 𝜏𝜏 𝜏 𝜎=' + 𝜏

• ThedesignaboveiscalledcompoundsymmetryØ Diagonalelements:variance ofoutcomeshaserror varianceandcompoundsymmetryparameter

Ø Off-diagonal elements: compoundsymmetryparameter givescovariance ofobservations

• Becauseobservationscomefromthesamepersonobservationsarelikelytobecorrelated

Ø Repeatedmeasures incorporates thiscorrelation through sphericity (aweakerformofcompound symmetry)

EPSY905:RMANOVA,MANOVA,andMixedModels 14

UnconditionalModel:RepeatedMeasuresCSAssumption

EPSY905:RMANOVA,MANOVA,andMixedModels 15

ImpossibletoseecovariancestructurefromRoutput

ComparingIndependencevsCSOutput

EPSY905:RMANOVA,MANOVA,andMixedModels 16

TheMixedModelVersionofRMANOVA(Part1)

• Wecangetthesameresultsfromthegls()function:

EPSY905:RMANOVA,MANOVA,andMixedModels 17

ComparingRMANOVAwiththeLMEModelVersion

EPSY905:RMANOVA,MANOVA,andMixedModels 18

InvestigatingtheVMatrixfromtheLMEModel

• UsingtheGetVarCov()functionfortheLMEmodel,wecanseetheformofthe(residual)covariancematrix:

EPSY905:RMANOVA,MANOVA,andMixedModels 19

UnconditionalModel:IndependenceAssumption

• Parsingthroughtheoutputofthepreviouspage,wecandeterminethefollowingforourdata:

𝑓 𝐲# 𝐗# ∼ 𝑁) 𝐗#𝛃, 𝐕

• Where𝛃89 = 87.5 46.6 102.1

• And

𝐕 =𝜎=' + 𝜏 𝜏 𝜏𝜏 𝜎=' + 𝜏 𝜏𝜏 𝜏 𝜎=' + 𝜏

=74.2 + 433.5 433.5 433.5

433.5 507.7 433.5433.5 433.5 507.7

EPSY905:RMANOVA,MANOVA,andMixedModels 20

(RE)MLAllowsforTestingCovarianceStructures

• WiththeLMEmodelwecanseeiftheCScovariancestructureimprovedmodelfitfromtheindependencemodel:

• Thenullhypothesisisthat𝜏 = 0

• Werejectthenullhypothesis

EPSY905:RMANOVA,MANOVA,andMixedModels 21

RepeatedMeasures:FullAnalysis

• Thefullrepeatedmeasuresanalysisofourdatagives:Ø Between subjectseffects

w Differences betweenDietandExercise TypeØ Withinsubjects effects

w Differences betweenwhenpulseratewastakenw Notaconsideration inMultivariateGLM(MANOVA)

Ø Interactions betweenwithinandbetween subjectseffects

• Withinsubjectseffectscanhaveadjustmentstop-valuesØ JustincasedatadonotmeetassumptionofsphericityØ SASandSPSShaveadjustments built-in…Rdoesnot

• WithLMEmodels,wedon’tneedadjustmentsØ Wehave theabilitytofit(andtestthefitof)therightmodel!

EPSY905:RMANOVA,MANOVA,andMixedModels 22

FullRepeatedMeasuresANOVAModel

EPSY905:RMANOVA,MANOVA,andMixedModels 23

SameAnalysiswithLMEModel

EPSY905:RMANOVA,MANOVA,andMixedModels 24

TheANOVATable:SameResults(forF-values)

• Thep-valuesdifferduetoLMEhavingmultiplemethodsfordeterminingthedenominatorDF

EPSY905:RMANOVA,MANOVA,andMixedModels 25

TheLMEResidualCovarianceMatrix

• FromtheEmptymodel:

• FromtheFullmodel:

• Note:withCSassumptionR2isthesameforallDVs:Ø =507.66-386.15/507.66=.239

• Multivariateversion:

EPSY905:RMANOVA,MANOVA,andMixedModels 26

CLASSICALMANOVAVSLMEMODELS

EPSY905:RMANOVA,MANOVA,andMixedModels 27

MultivariateApproaches

• Likerepeatedmeasures,multivariateapproachesfocusonalldependentvariables,simultaneously

Ø Repeatedmeasures analysis

• InmultivariateGLM,theeffectsofinterestare:Ø Between subjectseffects – Impliesdifferences inmeanvectors

w Differences betweenDietandExercise Type

• WithinsubjectseffectsarelesscommonlyinspectedØ Butcertainlycanbe

• Keydifferencebetweenmultivariateapproachandrepeatedmeasuresapproachcomesfromassumptionsabout𝐕matrix

EPSY905:RMANOVA,MANOVA,andMixedModels 28

MultivariateApproachAssumptions

• MANOVAestimatesallelementsoftheVmatrix:

𝐕 =𝜎=J' 𝜎=J=K 𝜎=J=L

𝜎=J=K 𝜎=K' 𝜎=K=L

𝜎=J=L 𝜎=K=L 𝜎=L'

• Contrastthatwiththerepeatedmeasuresversionofcompoundsymmetrystructureforthe𝐕matrix:

𝐕 =𝜎=' + 𝜏 𝜏 𝜏𝜏 𝜎=' + 𝜏 𝜏𝜏 𝜏 𝜎=' + 𝜏

EPSY905:RMANOVA,MANOVA,andMixedModels 29

MultivariateVersusRepeatedMeasures

• Multivariate:Ø Moremodelparameters; lessparsimonyØ FewerassumptionsØ Better ifassumptions ofRMareviolated (LMEmodelsdon’tneedthis)

• Repeatedmeasures:Ø Fewermodelparameters;moreparsimonyØ StrictassumptionsØ Morepowerthatmultivariate ifassumptions aremet

EPSY905:RMANOVA,MANOVA,andMixedModels 30

MultivariateTestStatistic(s)

• Themultivariateapproachhasanewclassofstatisticsusedfortestingmultivariatehypotheses

Ø Relyonsummaries ofkeymatrixproducts

• MorethanoneteststatisticisavailableØ Noneareuniformlymostpowerful (across allsamplesizesandtypesofindependent variables)

• Teststatistics:Ø Wilks’lambdaØ Pillai’s traceØ Hotelling-Lawley traceØ Roy’slargest root

EPSY905:RMANOVA,MANOVA,andMixedModels 31

FormingMultivariateTestStatistics

• Aswithunivariatelinearmodels,multivariatelinearmodelsdecomposevariabilityintovarioussources

Ø Butnowdecomposition ismultivariate

• Univariatedecomposition:Ø Sumsofsquares treatment:∑ 𝑛O 𝑦PQO − 𝑦PQQ

'SOT&

Ø Sumsofsquares error:∑ ∑ 𝑦#O − 𝑦PQO'S

OT&UV#T&

• F-ratiowasformedbycomparingtheseterms(dividedbytheirdegreesoffreedom)

EPSY905:RMANOVA,MANOVA,andMixedModels 32

MultivariateDecomposition

• MultivariatedecompositionisbasedonmeanvectorsØ Eachvariable isrepresented

• Sumsofsquaresandcross-productsfortreatment(theHmatrix…standsforhypothesis):

𝐇 = X 𝑛O 𝐲PQO − 𝐲PQQ 𝐲PQO − 𝐲PQQ9

S

OT&

• Sumsofsquaresandcross-productsforerror(theEmatrix…standsforerror):

𝐄 =XX 𝐲#O − 𝐲PQZ 𝐲#O − 𝐲PQO9

S

OT&

UV

#T&

EPSY905:RMANOVA,MANOVA,andMixedModels 33

Wilks’Lambda

• Oncebothmatricesareformed,Wilks’lambdacanbeobtained:

𝜆∗ =𝐄

𝐇 + 𝐄

• Equivalently(andshowntotrytorelatetounivariateteststatistics),Wilks’lambdacanbeformedfromtheeigenvalues(𝜆) of𝐄:&𝐇:

𝜆∗ =^1

1 + 𝜆#

_

#T&

• Where𝑠 = min(𝑝, 𝑔 − 1)EPSY905:RMANOVA,MANOVA,andMixedModels 34

MultivariateEmptyModel

• Returningthefocusawayfromtheteststatistics,let’sexaminethemultivariateapproachusingourempty(unconditional)model

• Theresultsareprettysparse(noeffectstested):

EPSY905:RMANOVA,MANOVA,andMixedModels 35

LMEModelVersionofMANOVA:EmptyModel

EPSY905:RMANOVA,MANOVA,andMixedModels 36

Note:ThediagonalelementsareequaltotheMSEsfromthepreviousslide

EmptyModel:UnstructuredCovarianceMatrix

• Parsingthroughtheoutputofthepreviouspage,wecandeterminethefollowingforourdata:

𝑓 𝐲# 𝐗# ∼ 𝑁) 𝐗#𝛃,𝐑

• Where𝛃8 = 87.5 46.6 102.1

• And

𝐑 =1

𝑁 − 1 𝐄 =𝜎=J' 𝜎=J=K 𝜎=J=L

𝜎=J=K 𝜎=K' 𝜎=K=L

𝜎=J=L 𝜎=K=L 𝜎=L'

=279.9 333.5 395.7333.5 473.0 571.2395.7 571.2 770.0

EPSY905:RMANOVA,MANOVA,andMixedModels 37

BuildingtheFullModel:ClassicalMANOVA

EPSY905:RMANOVA,MANOVA,andMixedModels 38

BuildingtheFullModel:LMEModel

EPSY905:RMANOVA,MANOVA,andMixedModels 39

THEBENEFITSOFMIXEDMODELSOVERCLASSICALRMANOVA ANDMANOVA

EPSY905:RMANOVA,MANOVA,andMixedModels 40

So..Which V MatrixisRight?

RestrictiveAssumptions

FewParameters

RelaxedAssumptions

MoreParameters

RegularANOVA RepeatedMeasuresANOVA

CompoundSymmetry UnstructuredIndependence

MultivariateANOVA

279.9 333.5 395.7333.5 473.0 571.2395.7 571.2 770.0

507.7 0 00 507.7 00 0 507.7

507.7 433.5 433.5433.5 507.7 433.5433.5 433.5 507.7

Classicalmethodstaughttodayhavenogoodwayofdeterminingwhichismostappropriate𝐕matrix

EPSY905:RMANOVA, MANOVA, andMixed Models 41

LMEModelsCanTestfortheBestStructure• WecanuseLikelihoodRatioTeststotestforthebestcovariancematrixstructureinLMEmodels:

• TheLRTaboveisnotrescaled(notavailableinnlmepackage:BADR!)

• WecouldalsocomeupwithanewstructurealtogetherØ Notatallpossible inclassicalapproach

• Onnextslide:HeterogeneousCompoundSymmetryØ Different covariances perDVØ Samecorrelation

EPSY905:RMANOVA,MANOVA,andMixedModels 42

NewModel:HeterogeneousCompoundSymmetry

EPSY905:RMANOVA,MANOVA,andMixedModels 43

LMEMODELEXTENSIONS:RANDOMCOEFFICIENTS

EPSY905:RMANOVA,MANOVA,andMixedModels 44

AnÀ lacarteApproachtoModelBuilding

Choices for Model for the Variances

Empty Means

Saturated Means

eti only1a 1b

BP ANOVA

U0i + etiCompound Symmetry (CS)

2a 2bUniv. RM ANOVA

All variances and covariances (Unstructured; UN)

3a 3b Multiv. RM ANOVA

Choices for Model for the Means

The labels for the models (1a – 3b) correspond to example 2c.

MLM generally begins with 2a, which is used as a baseline model.

EPSY905:RMANOVA,MANOVA,andMixedModels 45

EmptyMultilevelModel(U0i +eti):NewTerminology

VarianceofYà 2sources:

Level2Random InterceptVariance (ofU0i):

à Between-Person Variance(τU02)à DifferencefromGRANDmeanà INTER-Individual Differences

Level1ResidualVariance (ofeti):à Within-Person Variance(σe2)à DifferencefromOWNmeanà INTRA-Individual Differences

140

120

100

80

60

40

20

EPSY905:RMANOVA,MANOVA,andMixedModels 46

EmptyMultilevelModelModelforMeans;ModelforVariances

GeneralLinearModel

yi =β0 + eiMultilevel Model

L1: yti =β0i +eti

L2: β0i =γ00 +U0i

Sample Grand Mean Intercept

Individual Intercept

Deviation

3 Model Parameters1 Fixed Effect:

γ00 à fixed intercept

1 Random Effect (intercept): U0i àperson-specific deviation

à mean=0, variance = τU02

1 Residual Error:eti à time-specific deviation

à mean=0, variance = σe2

Combined equation: yti = γ00 + U0i + eti

EPSY 905: RM ANOVA, MANOVA, and Mixed Models 47

EmptyMultilevelModel:UsefulDescriptiveStatisticà ICC

0

0

2U

2 2U e

τ=

τ + σ

IntraClass Correlation(ICC):

• ICC=Proportion ofvariance thatisbetween-persons• ICC=Average correlation acrossoccasions

=Intercept VarianceICC

Intercept Variance + Residual Variance

Between VarianceICCBetween Variance + Within Variance

=

EPSY905:RMANOVA,MANOVA,andMixedModels 48

MatricesUsedinLongitudinalModelsfortheVariances

• Twomatricesofvariancecomponents(piles):Ø GMatrix:Between-person U0i varianceØ RMatrix:Within-person eti varianceØ G andR combinetomakeV,thetotalvarianceofY

• Themostbasic+WPmodel(justU0i +eti)canbeestimatedintwoequivalentways:

Ø “Random InterceptOnlyModel”:GandRtogether =Vw Randomintercept onlyinGw “Identity”(SPSS)or“VC”(SAS)R(uncorrelated, homogeneous var)

Ø “Compound SymmetryModel”: JustuseR(soR=V)w Nothing inG,Rhas“compoundsymmetry”form

EPSY905:RMANOVA,MANOVA,andMixedModels 49

InterceptVariance

Intercept Variance

Random Intercept Model (4 occasions):yti = γ00 + U0i + eti

0

2Uτ⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

2e

2e

2e

2e

σ

0 σ

0 0 σ

0 0 0 σ

0 0 0 00 0 0 00 0 0 00 0 0 0

Unstructured G Matrix(RANDOM statement)

Identity (VC) R Matrix(REPEATED statement)

Each person gets same 4 x 4 R matrix à equal variances and 0 covariancesacross persons & time

211

221 22

231 32 33

241 42 43 44

σ

σ σ

σ σ σ

σ σ σ σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

2e

2e

2e

2e

σ

0 σ

0 0 σ

0 0 0 σ

Level 2, BP Variance

Level 1, WP Variance

Total Observed Data Matrix is called V Matrix

EPSY 905: RM ANOVA, MANOVA, and Mixed Models 50

EquivalentVarianceModels

• “Random InterceptOnlyModel”: GandR=V– Allowsquantifiable estimates ofBPvs.WPvariation(separateestimates) andconstantcorrelationbetweenpersonsovertime

• “Compound SymmetryModel (CS)”:R=V– BPandWPvariancecombined intooneestimate instead,butstillallowsconstantcorrelationbetweenpersonsovertime

“+”

0

2Uτ⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

G matrix2e

2e

2e

2e

σ

0 σ

0 0 σ

0 0 0 σ

R matrix0

0 0

0 0 0

0 0 0 0

2 2e u

2 2 2u e u

2 2 2 2u u e u

2 2 2 2 2u u u e u

σ +τ

τ σ +τ

τ τ σ +τ

τ τ τ σ +τ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Total V matrix

=2T

2T

2T

2T

σ

CS σ

CS CS σ

CS CS CS σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

R matrix

Get “Var(eti)” and “Var(U0i)”

Get “Var(eti)” and “CS”à CS = Var(U0i)

0

0 0

0 0 0

0 0 0 0

2 2e u

2 2 2u e u

2 2 2 2u u e u

2 2 2 2 2u u u e u

σ +τ

τ σ +τ

τ τ σ +τ

τ τ τ σ +τ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Total V matrix

EPSY 905: RM ANOVA, MANOVA, and Mixed Models 51

ANALYSISADVICE

EPSY905:RMANOVA,MANOVA,andMixedModels 52

AdviceAboutPickingAnAnalysisMethod

• UseREPEATEDMEASURESif:Ø Youhavecompletedata(nothing ismissing)Ø YouhaveasmallsamplesizeØ Yourdatameetsphericityassumption

• UseMULTIVARIATEif:Ø Youhavecompletedata(nothing ismissing)Ø YouhavealargesamplesizeØ Yourdatadonotmeetsphericityassumption

• Note:bothmethods,althoughclassical,arestillusefulØ However: wewilllearnamodernapproach totheirestimation

w Allowsformorebroadanalysistypesandmissing dataØ Problem:cannottellwhich𝐑matrix isappropriate

EPSY905:RMANOVA,MANOVA,andMixedModels 53

WhenNotToUseEitherMethod

• DoNOTusethisversionofREPEATEDMEASURESorMULTIVARIATEif:

Ø Youhaveindependent variables youwishtostudythatvarywitheachdependent variable (example: time)

w Usemultilevel approachØ Youhavemissingdata

w Wewillusethistodemonstrate imputationmethodsØ Youhavecategoricaldata

w Use link-functionsØ Youhavedatathatcomefrompsychometricmeasures

w UsestructuralequationmodelingØ Youwouldliketodeterminewhich𝐑 matrixtouse

EPSY905:RMANOVA,MANOVA,andMixedModels 54

CONCLUDINGREMARKS

EPSY905:RMANOVA,MANOVA,andMixedModels 55

WrappingUp

• TodaywecoveredtheworldoflinearmodelsinmatricesØ UnivariateØ RepeatedmeasuresØ Multivariate

• Linearmodelsaccountforthemostfrequentlyusedstatisticalmethodsinthesocialsciences

Ø Wewillcovermoremodernversions ofthesemodelsintheweeks tocomeØ Todaywasanintroduction toclassicallinearmodelsanalyses

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