Psych 610 Handout #31, p. 1Prof. MooreUnequal NA 2 x 2 example to illustrate nonorthogonality of unequal n designsA1A2B1B2B1B2ContrastMain A11-1-1X1Main B1-11-1X2A x B1-1-11X3Assume unequal (and unbalanced) nTotal n = 12AA1A2BB143cell nsB232Regular anova model: Yijk = µ + αj + ßk + αßjk + εijk.X1is the predictor variable which replaces αj in multiple regression.For Ss in cell A1B1, X1 has value 1. For Ss in cell A2B1, X1 = -1, etc.X2is the predictor variable for Bk.X3is the predictor variable for αß jk.Mult. R model:Yij = a + ß1X1 + ß2X2 + ß3X3 + ε.The estimate of intecept, a, will be the grand meanHandout #31, p. 2Main AMain BA x BX1X2X3S1111A1B1S2111S3111S4111S51-1-1A1B2S61-1-1S71-1-1S8-11-1A2B1S9-11-1S10-11-1A2B2S11-1-11S12-1-11Are predictor variables X1, X2, and X3 orthogonal?Find ∑cjck.∑X1X2 = (4 + -3 + -3 + 2) = 0∑X1X3 = (4 + -3 + 3 - 2) = 2∑X2X3 = (4 + 3 - 3 - 2) = 2X1 and X2 are orthogonal, but X1 and X3 are not, nor are X2 and X3.Equal n: imagine 4 Ss/cell∑X1X2 = (4 – 4 – 4 + 4) = 0∑X1X3 = (4 – 4 + 4 - 4) = 0∑X2X3 = (4 + 4 – 4 - 4) =
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