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10_2way_2 - Detailed Analyses of Interaction Effect When...

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1 Detailed Analyses of Interaction Effect When the interaction effect is significant, we might be interested in what exactly causes the interaction effect. There are several different ways to look at the interaction effect. For example: 1. Simple Main Effect 2. Simple Effect Comparisons 3. Partial Interaction Etc. # 1. Simple Main Effect • Compare cell means at a specific treatment level. (e.g.) • Simple main effect of B (length of course) at a 3 H 0 : β k at α 3 = 0 for all k or H 0 : µ 31 = µ 32 = µ 33 • Simple main effect of A (type of beat) at b 1 H 0 : α k at β 1 = 0 for all j or H 0 : µ 11 = µ 21 = µ 31 Estimated Marginal Means of Y Length of Course 15 10 5 Mean Attitude 60 50 40 30 20 10 Type of Beat 1.00 2.00 3.00 Estimated Marginal Means of Y Type of Beat 3.00 2.00 1.00 Estimated Marginal Means 60 50 40 30 20 10 Length 1.00 2.00 3.00 B at a 3 A at b 1
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2 MANOVA attitude BY a(1,3) b(1,3) /error=within /design=b within a(1), b within a(2), b within a(3). (SPSS syntax) • Dependent variable = “attitude” • First factor = “a”, with levels from 1 to 3, and • Second factor = “b” with levels from 1 to 3. Error term = “withincell” Specifies the models to be analyzed. B at a 1 B at a 2 B at a 3 * * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * * Tests of Significance for ATTUDE using UNIQUE sums of squares Source of Variation SS DF MS F Sig of F WITHIN CELLS 2250.00 36 62.50 B WITHIN A(1) 63.33 2 31.67 .51 .607 B WITHIN A(2) 103.33 2 51.67 .83 .446 B WITHIN A(3) 2613.33 2 1306.67 20.91 .000 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (Output) SSB at a 1 + SSB at a 2 + SSB at a 3 = SSB + SSAB = 2780 • Effect of (time) is significant at only a 3 (inner city). • Partitioning sum of squares SSA b SSA SSAB k k q at = = + 1 1 at p j j SSB a SSB SSAB = = + Exactly the same MS error as the omnibus 2-way ANOVA
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3 2. Simple Effect Comparisons • Contrast of cell means at a specific treatment level. (e.g.) • A contrast of factor B (time) → ψ Β 1 = (0.5, 0.5, -1) • Then, look at the contrast at a specific level of factor A (beat). • For example, ψ Β 1 at a 1 , ψ Β 1 at a 2 , and ψ Β 1 at a 3 . Estimated Marginal Means of Y Length of Course 15 10 5 Mean Attitude 60 50 40 30 20 10 Type of Beat 1.00 2.00 3.00 Estimated Marginal Means of Y Type of Beat 3.00 2.00 1.00 Estimated Marginal Means 60 50 40 30 20 10 Length 1.00 2.00 3.00 MANOVA attitude BY a(1,3) b(1,3) /error=within /contrast (b) = special (1 1 1, 0.5 0.5 -1,-1 1 0) /partition (b) = (1,1) /design=b(1) within a(1), b(1) within a(2), b(1) within a(3).
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