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Statistics 203: Introduction to Regression and Analysis of ...

- p. 1/19 Statistics203 s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p. 2/19 Today s mixed& s s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p.

In ordinary least squares regression, the only parameter to estimate is ... Problem: this estimate can be negative! One of the difficulties in random effects model. Today’s class Two-way ANOVA Random vs. fixed effects When to use random effects?

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1 - p. 1/19 Statistics203 s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p. 2/19 Today s mixed& s s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p.

2 3/19 Two-way ANOVA model:observations:(Yijk);1 i r;1 j m;1 k nij:rgroupsin firstgroupingvariable,mgroupsinssecondan dnijsamplesin(i; j)- cell :Yijk= + i+ j+ ( )ij+"ijk;"ijk N(0; 2):nConstraints:uPri=1 i= 0uPmj=1 j= 0uPmj=1( )ij= 0;1 i ruPri=1( )ij= 0;1 j m:lToday s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p. 4 ANOVA exampleswe have seensofar, thecategoricalvariablesarewell-definedca tegories:below averagefitness,longduration, somedesigns, thecategoricalvariable is subject.

3 NSimplestexample:repeatedmeasures, wheremorethanone(identical)measurementis thiscase, the group effect iis bestthoughtof asrandombecausewe onlysamplea subsetof theentirepopulationof s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p. 5/19 Whentouserandomeffects?nA group effectis randomif we canthinkofthelevelsweobserve in thatgroupto besamplesfroma :if collectingdatafromdifferentmedicalcenter s, center :if surveyingstudentsondifferentcampuses, campus may bea s classlTwo-way ANOVAlRandomvs.

4 FixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p. 6/19 Example:sodiumcontentinbeernHow muchsodiumis therein North Americanbeer?Howmuchdoesthisvary by brand?nObservations:for6 brandsof beer, researchersrecordedthesodiumcontentof8 interest:whatis the grandmean sodiumcontent?How muchvariabilityis therefrombrandto brand?n Individuals in thiscasearebrands, repeatedmeasuresarethe8 s classlTwo-way ANOVAlRandomvs.

5 FixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p. 7/19 One-way randomeffectsmodelnSupposewe + i+"ij;1 i r;1 j nn"ij N(0; 2);1 i r;1 j nn i N(0; 2 );1 i r:nWe mightbeinterestedin thepopulationmean, : CIs, is itzero? , we mightbeinterestedin thevariabilityacrosssubjects, 2 : CIs, is it zero?lToday s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p.

6 8/19 ImplicationsformodelnIn randomeffectsmodel,theobservationsarenol ongerindependent(evenif" s areindependent).In factCov(Yij; Yi0j0) = 2 i;i0+ 2 j;j0:nIn morecomplicatedmixedeffectsmodels, thismakesMLEmorecomplicated:notonlyareth ereparametersin themean,but in ordinary leastsquaresregression,theonlyparametert oestimateis 2becausethecovariancematrix is s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p.

7 9/19 One-way randomANOVA tableSourceSSdfE(MS)TreatmentsSST R=Pri=1n Yi Y 2r 1 2+n 2 ErrorSSE=Pri=1 Pnj=1(Yij Yi )2(n 1)r 2nOnlychangehereis theexpectationofSST Rwhichreflectsrandomnessof i table is stillusefulto setuptests:thesameFstatisticsforfixedorr andomwillwork : 2 = 0, it is easyto seethatM ST RM SE Fr 1;(n 1)r:lToday s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p.

8 10/19 Inferencefor nWe know thatE(Y ) = , andcanshow thatVar(Y ) =n 2 + 2rn:nTherefore,Y qSST R(r 1)rn tr 1nWhyr 1degreesof freedom?Imaginewe couldrecordaninfinitenumberof observationsforeachindividual,sothatYi ! learn anythingabout we stillonlyhaverobservations( 1; : : : ; r).nSamplingmorewithinanindividualcannot narrow theCIfor .lToday s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p.

9 11/19 Estimating 2 nFromtheANOVA table 2 =E(SST R=(r 1)) E(SSE=((n 1)r))n:nNatural estimate:S2 =SST R=(r 1) SSE=((n 1)r)nnProblem:thisestimatecanbenegative! Oneofthedifficultiesin s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p. 12/19 Example:productivitystudynImaginea studyontheproductivityofemployeesin a wantsto getanideaofdailyproductivity, andhowit :takememployeesandrmachines, havingeachemployeework oneachmachinefora , andthesemachinesarenotallmachinesit makessenseto thinkofboththeeffectsof machineandemployees(andinteractions) s classlTwo-way ANOVAlRandomvs.

10 FixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p. 13/19 Two-way randomeffectsmodelnYijk + i+ j+ ( )ij+"ij;1 i r;1 j m;1 k nn"ijk N(0; 2);1 i r;1 j m;1 k nn i N(0; 2 );1 i r:n j N(0; 2 );1 j m:n( )ij N(0; 2 );1 j m;1 i r:nCov(Yijk; Yi0j0k0) = ii0 2 + jj0 2 + ii0 jj0 2 + ii0 jj0 kk0 2:lToday s classlTwo-way ANOVAlRandomvs. fixedeffectslWhentouserandomeffects?lExa mple:sodiumcontentinbeerlOne-way randomeffectsmodellImplicationsformodell One-way randomANOVA tablelInferencefor lEstimating 2 lExample:productivitystudylTwo-way randomeffectsmodellANOVA tables:Two-way(random)lMixedeffectsmodel lTwo-way mixedeffectsmodellANOVA tables:Two-way(mixed)lConfidenceinterval sforvarianceslSattherwaite s procedure- p.

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