MANOVA - EquationsLecture 11Psy524Andrew AinsworthData design for MANOVAIV1 IV2 DV1 DV2 DVM S01(11) S01(11) S01(11) S02(11) S02(11) S02(11) 1 S03(11) S03(11) S03(11) S04(12) S04(12) S04(12) S05(12) S05(12) S05(12) 1 2 S06(12) S06(12) S06(12) S07(21) S07(21) S07(21) S08(21) S08(21) S08(21) 1 S09(21) S09(21) S09(21) S10(22) S10(22) S10(22) S11(22) S11(22) S11(22) 2 2 S12(22) S12(22) S12(22)Data design for MANOVASteps to MANOVA MANOVA is a multivariate generalization of ANOVA, so there are analogous parts to the simpler ANOVA equationsSteps to MANOVA ANOVA –()222() ()() ()ij y j y ij jij j ijYGM n YGM YY−=−+ −∑∑ ∑ ∑∑() () ()Total y bg y wg ySS SS SS=+Steps to MANOVA When you have more than one IV the interaction looks something like this:()()()()()222() () ()222() () ()()km km DT k k D m m Tkm k mkm km DT k k D m m Tkm k mnDTGMnDGMnTGMnDTGMnDGMnTGM−=−+−+−−−−−∑∑ ∑ ∑∑∑ ∑ ∑bg D T DTSS SS SS SS=++Steps to MANOVA The full factorial design is: ()()()()()222() () ()222() () ()2()()ik m ik m k k D m m Tikm k mkm km DT k k D m m Tkm k mik m k mikmYGM n DGM n TGMnDTGMnDGMnTGMYDT−=−+−+−−−−−+−∑∑∑ ∑ ∑∑∑ ∑ ∑∑∑∑Steps to MANOVA MANOVA - You need to think in terms of matrices Each subject now has multiple scores, there is a matrix of responses in each cell Matrices of difference scores are calculated and the matrix squared When the squared differences are summed you get a sum-of-squares-and-cross-products-matrix (S) which is the matrix counterpart to the sums of squares. The determinants of the various S matrices are found and ratios between them are used to test hypotheses about the effects of the IVs on linear combination(s) of the DVs In MANCOVA the S matrices are adjusted for by one or more covariatesMatrix Equations If you take the three subjects in the treatment/mild disability cell:11115 98 107108 105 98iY=Matrix Equations You can get the means for disability by averaging over subjects and treatments12395.83 88.83 82.6796.50 88.00 77.17DDD===Matrix Equations Means for treatment by averaging over subjects and disabilities1299.89 78.3396.33 78.11TT==Matrix Equations The grand mean is found by averaging over subjects, disabilities and treatments.89.1187.22GM=Matrix Equations Differences are found by subtracting the matrices, for the first child in the mild/treatment group:111115 89.11 25.89()108 87.22 20.75YGM  −= − =    Matrix Equations Instead of squaring the matrices you simply multiply the matrix by its transpose, for the first child in the mild/treatment group:[]111 11125.89 670.29 537.99()()' 25.8920.7520.75 537.99 431.81YGMYGM−−= =Matrix Equations This is done on the whole data set at the same time, reorganaizing the data to get the appropriate S matrices. The full break of sums of squares is:()()'()()'()()'[()()'()()'()()']()()'ikm ikm k k kikm kmm m km km kmmkmkk k mm mkmikm km ikm kmikmYGMYGMn DGMDGMn T GM T GM n DT GM DT GMnDGMDGMnTGMTGMYDTYDT−−=−−+−−+ − −−−−−−−+−−∑∑∑ ∑∑∑∑∑∑∑∑Matrix Equations If you go through this for every combination in the data you will get four S matrices (not including the S total matrix):/570.29 761.72 2090.89 1767.56761.72 1126.78 1767.56 1494.222.11 5.28 544.00 31.005.28 52.78 31.00 539.33DTDT s DTSSSS ==   ==  Test Statistic – Wilk’s Lambda Once the S matrices are found the diagonals represent the sums of squares and the off diagonals are the cross products The determinant of each matrix represents the generalized variance for that matrix Using the determinants we can test for significance, there are many ways to this and one of the most is Wilk’s LambdaTest Statistic – Wilk’s Lambda this can be seen as the percent of non-overlap between the effect and the DVs. erroreffect errorSSSΛ=+Test Statistic – Wilk’s Lambda For the interaction this would be:///292436.522.11 5.28 544.00 31.00 546.11 36.285.28 52.78 31.00 539.33 36.28 529.11322040.95292436.52.908068322040.95sDTDT s DTDT s DTSSSSS=  += + =    +=Λ= =Test Statistic – Wilk’s Lambda Approximate Multivariate F for Wilk’sLambda is 2121221/22121(, )()4 , ,()5 , ( )1()2()22effectseffecteffecteffect effecterrordfyFdf dfydfpdfwhere y spdfp number of DVs df p dfpdf pdfdf s df−=−=Λ =+−==−+−=− −Test Statistic – Wilk’s Lambda So in the example for the interaction:22221/ 212(2) (2) 42(2) (2) 5.908068 .9529262(2) 4(1)3(2)(31)12221 2(2)2212 2222.047074 22F(4,22)= 0.2717.952926 4errorsydfdf dt ndfApproximate−==+−=====−= −=−+ −=− − ==Eta Squared21η=−ΛPartial Eta