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Factorial Analysis of Variance 46-511 Between Groups Fixed Effects Designs 1 Two-Way ANOVA Example: (Yerkes – Dodson Law) Factor A: Task Difficulty Factor B: Arousal Easy Difficult Low Medium High 3 1 1 6 4 5 5 9 7 7 9 9 13 6 8 0 2 0 0 3 3 8 3 3 3 0 0 0 5 0 2 Partitioning Variance Factor A: Task Difficulty Factor B: Arousal Easy Difficult Low Medium High 3 … 4 5 … 7 9 … 8 0 … 3 3 … 3 0 … 0 Variation among means on B represents effect of B Leftover variation = interaction Variation among means on A represent effect of A Variation among people treated the same = error 3 Partitioning Variance: Interaction Factor A: Task Difficulty Factor B: Arousal Low Medium High Total Easy 3.00 6.00 9.00 6.00 Difficult 1.00 4.00 1.00 2.00 Total 2.00 3.00 5.00 4.00 Dependence of means on levels of both A & B represents the effect of an interaction. 4 Or Graphically… 10 9 8 7 6 Easy 5 Difficult 4 3 2 1 0 Low Medium High 5 In words Types of Effects vs. 1-way Main Effect for A Main Effect for B Interaction (A x B) Structural Model: XIJK = μ++++IJK Partitioning Variance/Sums of Squares First, total variance: Between Groups: Thus Total is: SSTOT SS BG SSW SSBG SS A SSB SS AXB SSTOT SS A SS B SS AXB SSW 6 Sums of Squares Between Definitional Formula ___ SS BG n ( AB ij G ) 2 Variation of cell means around grand mean, weighted by n. Computational Formula 2 SS BG 2 T G n N Computational formulae: •More accurate for hand calculation •Easier to work •Less intuitive 7 Sums of Squares A Definitional Formula _ _ SS A nq ( Ai G ) Computational Formula 2 Variation of row means around grand mean, weighted by n times the number of levels of B, or q. 2 ROW T G2 SS A nROW N 8 Sums of Squares B Definitional Formula _ _ SS B np ( B j G ) 2 Computational Formula Variation of column means around grand mean, weighted by n times the number of levels of A, or p. 2 TCOL G2 SS B nCOL N 9 Sums of Squares AxB Definitional Formula SS AxB n( ABij Ai B j G) 2 Computational Formula SS AB SS Between SS A SS B SSAxB = Variation of cell means around grand mean, that cannot be accounted for by effects of A or B alone. 10 Sums of Squares Within (Error) Definitional Formula SSW ( X ijk ABij ) 2 Computational Formula 2 T SSW X n 2 ijk SSW = Variation of individual scores around cell mean. 11 Numerical Example Easy T Mean SS Effect of Task Difficulty and Anxiety Level on Performance Low Medium High Marginals for B 3 2 9 1 5 9 1 9 13 T 6 7 6 Mean 4 7 8 SS 15 30 45 3 6 9 18 28 26 Difficult T Mean SS Marginal for A T Mean SS 0 2 0 0 3 5 1 8 3 8 3 3 3 20 4 20 20 2 36 50 5 58 0 0 0T 5 Mean 0 SS 5 1 20 50 5 206 90 6 162 30 2 78 12 Degrees of Freedom df between = k – 1; or, (kA x kB – 1) df A = kA – 1 df B = kB – 1 df A x B = dfbetween – dfA – dfB dfW = k(n-1) 13 Source Table Source SS df MS F Between A B AxB Within Total 14 More Digression on Interactions Ways to talk about interactions Scores on the DV depend upon levels of both A and B The effect of A is moderated by B The effect of B is moderated by A There is a multiplicative effect for A and B 15 More Digresions (cont’d) No effect whatsoever… No Significant Effects Interaction Effect: Cell & Marginal Means B: Anxiety A: Task Difficulty Low Medium Easy 4 4 Hard 4 4 Totals 4 4 High 4 4 4 Totals 4 4 4 Deviations: cell mean - row mean - column mean + grand mean Anxiety Task Difficulty Low Medium High Easy 0 0 0 Hard 0 0 0 Interaction Sum of Squares: Main Effect for A Main Effect for B 0 0 0 16 Main effects for A and B… Only Main Effects Significant Interaction Effect: Cell & Marginal Means B: Anxiety A: Task Difficulty Low Medium Easy 3 6 Hard 1 4 Totals 2 5 High 9 7 8 Totals 6 4 5 Deviations: cell mean - row mean - column mean + grand mean Anxiety Task Difficulty Low Medium High Easy 0 0 0 Hard 0 0 0 Interaction Sum of Squares: Main Effect for A Main Effect for B 0 30 180 17 Graphically… 2-Way ANOVA Anxiety by Task Difficulty: Main Effects, No Interaction 10 9 8 Performance 7 6 Easy Hard 5 4 3 2 1 0 Low Medium High Anxiety Level 18 Interaction significant also… Significant Interaction Interaction Effect: Cell & Marginal Means B: Anxiety A: Task Difficulty Low Medium Easy 3 6 Hard 1 4 Totals 2 5 High 9 1 5 Totals 6 2 4 Deviations: cell mean - row mean - column mean + grand mean Anxiety Task Difficulty Low Medium High Easy -1 -1 2 Hard 1 1 -2 Interaction Sum of Squares: Main Effect for A Main Effect for B 60 120 60 19 Graphically… 2-Way ANOVA Anxiety by Task Difficulty: Main Effects AND Interaction 10 9 8 Performance 7 6 Easy Hard 5 4 3 2 1 0 Low Medium High Anxiety Level 20 Further Analyses on Main Effects Contrasts Planned Comparisons Post-Hoc Methods In the presence of a significant interaction 21 Further Analyses on Interaction What it means Simple (Main) Effects Contrasts Partial Interactions Contrasts Simple Comparisons / Post-Hoc Methods How to get q 22 Simple Main Effects Analysis Low Medium High Total Easy 3.00 6.00 9.00 6.00 Difficult 1.00 4.00 1.00 2.00 Total 2.00 3.00 5.00 4.00 23 Simple Main Effects Sum of Squares Formula: F Ratio: SS J Tij2 at _ rj Fc n T j2 nj MS j MSW df = dfj,dfw: 24 Partial Interaction Analysis Low Medium High Total Easy 3.00 6.00 9.00 6.00 Difficult 1.00 4.00 1.00 2.00 Total 2.00 3.00 5.00 4.00 25 In Class Exercise Drug B1 (no dose) B2 (low dose) B3 (high dose) A1 (no dose) 5 7 5 6 8 9 10 7 6 5 4 8 4 5 9 A2 (low dose) 5 5 4 6 8 7 8 4 5 6 5 4 3 9 7 A3 (high dose) 8 9 7 8 6 12 11 15 13 10 18 17 19 20 20 26 Based on two pieces of information 1) De scri ptive Statistics Dependent Variable: anxiet y FactorA 1.00 2.00 3.00 Total FactorB 1.00 2.00 3.00 Total 1.00 2.00 3.00 Total 1.00 2.00 3.00 Total 1.00 2.00 3.00 Total Mean 6.2000 7.4000 6.0000 6.5333 5.6000 6.0000 5.6000 5.7333 7.6000 12.2000 18.8000 12.8667 6.4667 8.5333 10.1333 8.3778 St d. Deviation 1.30384 2.07364 2.34521 1.92230 1.51658 1.58114 2.40832 1.75119 1.14018 1.92354 1.30384 4.95504 1.50555 3.24844 6.63181 4.51406 N 5 5 5 15 5 5 5 15 5 5 5 15 15 15 15 45 27 Compute simple main effects 2) Tests of Between-Subjects Effects Dependent Variable: anxiety Source Corrected Model Intercept FactorA FactorB FactorA * FactorB Error Total Corrected Total Type III Sum of Squares 781.378b 3158.422 458.178 101.378 221.822 115.200 4055.000 896.578 df 8 1 2 2 4 36 45 44 Mean Square 97.672 3158.422 229.089 50.689 55.456 3.200 F 30.523 987.007 71.590 15.840 17.330 Sig. .000 .000 .000 .000 .000 Partial Eta Squared .872 .965 .799 .468 .658 Noncent. Parameter 244.181 987.007 143.181 31.681 69.319 Observed a Power 1.000 1.000 1.000 .999 1.000 a. Computed using alpha = .05 b. R Squared = .872 (Adjus ted R Squared = .843) 28 3-Way ANOVA Effects A B C AxB AxC BxC AxBxC A Vague Example DV = Treatment Outcome Factor A: Gender Factor B: Age (14 or 17) Factor C: Treatment 29 Results Tests of Between-Subjects Effects Dependent Variable: SCORE Source Corrected Model Intercept SEX AGE TREAT SEX * AGE SEX * TREAT AGE * TREAT SEX * AGE * TREAT Error Total Corrected Total Type III Sum of Squares 221.556 b 2635.111 .111 36.000 24.889 .111 80.889 4.667 74.889 217.333 3074.000 438.889 df 11 1 1 1 2 1 2 2 2 24 36 35 Mean Square 20.141 2635.111 .111 36.000 12.444 .111 40.444 2.333 37.444 9.056 F 2.224 290.994 .012 3.975 1.374 .012 4.466 .258 4.135 Sig. .049 .000 .913 .058 .272 .913 .022 .775 .029 Partial Eta Squared .505 .924 .001 .142 .103 .001 .271 .021 .256 Noncent. Parameter 24.466 290.994 .012 3.975 2.748 .012 8.933 .515 8.270 Obs erved a Power .804 1.000 .051 .482 .266 .051 .710 .086 .674 a. Computed using alpha = .05 b. R Squared = .505 (Adjusted R Squared = .278) 30 Significant Two-Way Interaction 31 Significant Three-Way Interaction 32 Other Stuff Higher order models (4-way, 5-way, etc.) Unequal Cell Sizes and SS Type Use of contrast coefficients Short-Cuts using SPSS Custom Models in SPSS Observed Power 33