Research Design - - Topic 7 Split-Plot Factorial Designs (Kirk, Ch. 12) 2008 R.C. Gardner, Ph.D.

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1 Reserch Desig - - Topic 7 plit-plot Fctoril Desigs (Kirk, Ch. ) 00 R.C. Grder, Ph.D. Geerl Descriptio, Purpose, Exmple Uivrite pproch Experimetl Desig Model Multivrite pproch Ruig P GLM REPETED MEURE Tests of Mes s preseted i texts s performed y P Repeted The plit-plot Fctoril Desig cosists of t lest two fctors, where oe fctor is sed o idepedet oservtios d the other is sed o correlted oservtios. It is sometimes referred to s mixed desig, or mixed etwee/withi desig. There re two geerl sources of vritio. Oe is the etwee ujects vritio while the other is the Withi ujects (or Withi locks) vritio. The followig digrm shows the rekdow of the Totl sum of squres ito the etwee d Withi uject compoets chemtic rekdow of plit-plot Exmple etwee s Totl Withi s / / The followig dt set ws dpted from Kirk (995). It cosists of oe etwee ujects fctor () d oe Withi ujects fctor (). The vrile P i / is the me for ech uject (or lock). Mes P i G G 5.75

2 Questios to sk of the dt. Mi Effect of. Do the mes for the etwee ujects fctor () vry more th c e resoly ttriuted to chce? Mi Effect of. Do the mes for the Withi ujects fctor () vry more th c e resoly ttriuted to chce? Iterctio Effects of d. Do the -mes vry from wht you would expect give the vlues of the -Mes d the -mes? 5 Experimetl Desig Model The score for ech idividul is cosidered to e composed of prmeters s follows: X i α π i / β αβ βπ i / i / Note, this is o-dditive model tht ssumes there is iterctio etwee d ujects ested i (i.e., βπ i / ). (We could lso write dditive model y elimitig.) This model c e used to geerte the Corfield Tukey lgorithm (see slide 0). βπ i / The followig slide shows the defiitiol formule for the ummry Tle. etwee s / Withi s / Totl Defiitiol Formule um Of qures ( P ) i / G ( X G) ( ) P X i / ( ) i / X P i X G ( ) ( X X X G) ( X ) i Pi / X X ( G) X i ( ) ( ) ( )( ) ( )( ) / / Me qures ( X G) ( Pi / X ) ( ) ( X G) ( X X X G) ( )( ) ( X Pi / X X ) i ( )( ) 7

3 pplyig the defiitiol formule to the smple dt produces the followig lysis of Vrice ummry Tle. F etwee s / Withi s / Totl Note. This lysis ssumes fixed effects model (i.e., d re fixed fctors). 9 0 etwee s / Withi s Totl / Corfield Tukey lgorithm E () ( ) σ σ σ π π ( ) σ ( ) σ σ σ π π ( ) σ σ π σ π π σ σ Note. refers to the umer of levels of, to the umer of levels of, d to the umer of idividuls i ech level of. The smplig frctios re defied s efore. Forml Expected Me qures d F-rtios for the fixed effects model E() Forml Expected Me qures d F-rtios whe is fixed d is rdom E () etwee s / α ( ) σ π σ π F / etwee s / α π ( ) π σ π π No ovious F-rtio for (ut see ext slide). Withi s / β π ( ) αβ ( )( ) σ π σ π σ F F / / Withi s / σ σ π σ σ π σ π σ F F / /

4 Determie pproprite error term for y ddig d sutrctig vrious Me qures. / σ π π π Therefore: - / Numertor (-) F Deomitor (tterthwite, 9) Computig qusi F-rtios σ π σ σ π π ' / / σ π / (. α π / σ βπ / ) 7.5 / βπ / / / / / /.9 ssumptios Idepedet Rdom mplig. s re rdomly d idepedetly otied from the etwee ujects fctor. Normlity. The oservtios i the popultios re ormlly distriuted. Homogeeity. There re spects:. Homogeeity of vrice of mes for ujects or locks cross (for test of Mi Effects for ).. Equivlece of covrice mtrices for the fctor.. Circulrity of the pooled covrice mtrix. Null Hypotheses: 0 for ll Tests of Homogeeity ox s M test of equivlece of the covrice mtrices. If this test is sigifict, it idictes tht the covrice mtrices re ot equivlet. Ruig P GLM Repeted Mesures Dt Editor Muchly s test of phericity. If this test is sigifict, it idictes tht the pooled covrice mtrix does ot stisfy the ssumptio of circulrity. Geerlly, these tests re ot roust with respect to violtios of ormlity d it is recommeded tht regrdless of the results of these tests, the degrees of freedom for the withi sujects effects e reduced usig epsilo multiplier. Kirk recommeds usig the Greehouse-Geisser estimte. P GLM presets the degrees of freedom for the cse where the ssumptios re stisfied s well s whe the epsilo vlue is pplied. 5

5 Clopig the pproprite choices produces the ytx file. GET FILE'C:\PYCH50\kirk5dt.sv'. DTET NME Dtet WINDOWFRONT. GLM Y /WFCTOR Polyomil /METHOD TYPE() /PRINT ETQ OPOWER HOMOGENEITY /CRITERI LPH(.05) /WDEIGN /DEIGN. The followig slides preset the mjor output. 7 Homogeeity Tests. ox s M test of Equivlece of Covrice mtrices. This test cot e produced for this exmple ecuse there re fewer th two o-sigulr covrice mtrices.. Levee s test. This tests whether the vrices for ech level of re heterogeeous over the groups. I this exmple, oly is sigifict (p<.5). Levee's Test of Equlity of Error Vrices F ig Tests the ull hypothesis tht the error vrice of the depedet vrile is equl cross groups.. Desig: Itercept Withi ujects Desig:.Muchly s Test of phericity of the Pooled Covrice Mtrix Mesure: MEURE_ Muchly's Test of phericity Uivrite Tests of the Withi ujects Effects Tests of Withi-ujects Effects Epsilo pprox. Greehous Withi ujects Effect Muchly's W Chi-qure ig. e-geisser Huyh-Feldt Lower-oud Tests the ull hypothesis tht the error covrice mtrix of the orthoormlized trsformed depedet vriles is proportiol to idetity mtrix.. My e used to djust the degrees of freedom for the verged tests of sigificce. Corrected tests re displyed i the Tests of Withi-ujects Effects tle.. Desig: Itercept Withi ujects Desig: Muchly s test is ot sigifict, idictig tht the ssumptio of circulrity is stisfied. Noetheless, it is customry to djust the degrees of freedom for the repeted mesures F-rtios y multiplyig them y Mesure: MEURE_ phericity ssumed Greehouse-Geisser Huyh-Feldt Lower-oud phericity ssumed Greehouse-Geisser Huyh-Feldt Lower-oud phericity ssumed Greehouse-Geisser Huyh-Feldt Lower-oud Type III um Prtil Et Nocet. Oserved Power of qures Me qure F ig. qured Prmeter epsilo multiplier. 9 0 * Error(). Computed usig lph.05 s idicted erlier, it is typicl to iterpret the results usig the Greehouse-Geisser djustmet. Thus, the results for would e writte s F(,)7.9, p<.000), roudig the degrees of freedom to the ext highest iteger; those for would e writte s F(,).7, p<.00. 5

6 Mesure: MEURE_ Trsformed Vrile: verge Itercept Uivrite Tests of the etwee uject Effects Error Tests of etwee-ujects Effects Type III um Prtil Et Nocet. Oserved Power of qures Me qure F ig. qured Prmeter Computed usig lph The test of the Itercept is test tht the grd me devites sigifictly from 0, F(,) 59., p<.000. The test of idictes tht the effects due to re ot sigifict, F(,).00, s. The Multivrite pproch The Withi ujects compoets of the plit-plot Fctoril desig c lso e ivestigted from multivrite perspective, where the dt re cosidered to e set of vriles dmiistered to groups of sujects. There re cosequetly two clsses of effects. The mi effects of. This test is comprle to the test of effects for the sigle fctor repeted mesures desig ut i this cse the groups re collpsed so tht there is oly oe group with oservtios i ech me. The iterctio of d. This tests the equivlece of the cotrsts etwee the mes t ech level of the fctor. The tests of sigificce. I P GLM Repeted, sttistics re give, Pilli s Trce, Wilks Lmd, Hotellig s Trce, d Roy s Lrgest Root. Whe the umer of levels of the etwee ujects fctor is, the F-rtio correspods to Hotellig s T² for oth the mi d iterctio effects. For more th levels, the sttistics produce differet F-rtios d degrees of freedom for the iterctio. ssumptios Multivrite Null Hypotheses for Withi ujects Effects ssumptios for the repeted mesures effects re: Idepedet rdom smplig. s re idepedetly d rdomly smpled from the etwee ujects fctor Mi Effect for Multivrite Normlity. This ssumptio pplies to ech level of the etwee ujects fctor. Equivlece of the Covrice Mtrices. The covrice mtrices for the etwee ujects fctor re the sme i the popultio. Degrees of freedom: v - v N-(-) -(-) Iterctio Effect for Degrees of freedom: v (-)(-) v N -

7 The multivrite tests re pproprite for the Withi ujects Effects. The etwee ujects Effects re ssessed usig the uivrite pproch s preseted i lide. Effect * Pilli's Trce Wilks' Lmd Hotellig's Trce Roy's Lrgest Root Pilli's Trce Wilks' Lmd Hotellig's Trce Roy's Lrgest Root. Computed usig lph.05. Exct sttistic c. Desig: Itercept Withi ujects Desig: Multivrite Tests c Prtil Et Nocet. Oserved Power Vlue F Hypothesis Error ig. qured Prmeter The mi effect for is sigifict, F(,) 7.9, p<.00 idictig tht the -mes vry more th c e resoly ttriuted to chce. The iterctio is sigifict, F(,) 7.9, p<.07 idictig tht some cotrsts i differ from the correspodig oes i 5. This is equivlet to uivrite iterctio. Tests of Mes s preseted i most textooks (cf., Kirk, 995) The formule re writte for uequl s for the geerl cse. With equl s (s is the cse here), the deomitors c e writte more simply s times oe of the elemets. Mi Effects of Mi Effects of X X t. t.5.5. / / X X t.0 t / / Tests of Cell Mes. Tests of imple mi Effects. Tests of Iterctio Effects. imple Mi Effects To determie the pproprite error term, sk questios:. Wht is the error term for the iterctio?. Wht is the error term for the fctor eig vried? If the swer is the sme, use tht oe Me qure s the error term. If the swer is ot the sme clculte pooled error term y ddig the sums of squres for the two error terms d dividig y the sum of their degrees of freedom. 7 imple Mi Effects of t ech level of. The swer to ech questio is /. Therefore: X X t.97 t / / imple Mi Effects of t ech level of. The swer to questio is / while tht for questio is /. Therefore: pooled error pooled error / ( / / / / ) ( ) / / / / / X X t. t pooled error pooled error 7

8 . Iterctio Effects Tests of Mes i P GLM Repeted Mesures. Tretmet/Cotrst Iterctios. Cotrst/Cotrst Iterctios These re coducted s discussed i Topic 5. Note tht the error term i ech cse would e the error term for the iterctio ecuse these re pure iterctio effects d ot cofouds of mi d iterctio effects s is the cse whe performig tests of simple mi effects. 9 Mi Effects of. P uses the formul descried erlier (see slide ) for this test. Mi Effects of. Compute pooled estimte of the vrice of the differece for ech level of over ll levels of. This requires computig the vrice of the differece (i.e., -) i ech group, d poolig them s follows: ( ) ) ( ) ( ( ) (.9) (.5) pooled ( ) t X X pooled ( ) t Note. This is differet from tht preseted i slide.5 0 imple Mi Effects of t ech level of. Use the pooled error term descried o slide 0. P Output for Tests of Mes X t X pooled ( ) t Mi Effect -Mes Estimtes Mesure: MEURE_ 95% Cofidece Itervl Me td. Error Lower oud Upper oud imple Mi Effects of t ech level of. Compute pooled error term for ech level of (over ech level of ). Compute t. t X pooled X ( ) ( ) (.5 ) (.5 ) ( ) t pooled ( ) pooled ( ) Tests of Mi Effects for -Mes Mesure: MEURE_ (I).00 (J).00 Pirwise Comprisos 95% Cofidece Itervl for Me Differece Differece (I-J) td. Error ig. Lower oud Upper oud sed o estimted mrgil mes. djustmet for multiple comprisos: oferroi. Note. These tests re the sme s descried o slide.

9 Mi Effect -Mes Estimtes Mesure: MEURE_ 95% Cofidece Itervl Me td. Error Lower oud Upper oud Exmiig the Iterctio Effect Mes Estimtes Mesure: MEURE_ 95% Cofidece Itervl Me td. Error Lower oud Upper oud Pirwise Comprisos Mesure: MEURE_ Tests of Mi Effects of (I) (J) 95% Cofidece Itervl for Me Differece Differece (I-J) td. Error ig. Lower oud Upper oud * * * * * * * * * * sed o estimted mrgil mes *. The me differece is sigifict t the.05 level.. djustmet for multiple comprisos: oferroi. Note. These tests re differet from those descried o slide, ut re the sme s those descried o slide 0. Tests of imple Mi Effects of t ech level of. Pirwise Comprisos Tests of imple Mi Effects of t. Mesure: MEURE_ Me 95% Cofidece Itervl for Differece Differece (I) (J) (I-J) td. Error ig. Lower oud Upper oud * * * * * * * * * * * * * * * * * * sed o estimted mrgil mes *. The me differece is sigifict t the.05 level.. djustmet for multiple comprisos: oferroi. Note. These tests differ from those o slide ut gree with slide. 5 Mesure: MEURE_ (I) (J) Me Differece Pirwise Comprisos sed o estimted mrgil mes *. The me differece is sigifict t the.05 level.. djustmet for multiple comprisos: oferroi. 95% Cofidece Itervl for Differece (I-J) td. Error ig. Lower oud Upper oud.000* * * * * * Note. These tests differ from those descried o slide ut re the sme s those o slide. 9

10 Refereces tterthwite, F.E. (9). pproximte distriutio of estimtes of vrice compoets. iometrics ulleti,,