1. Factorial ANOVA 11/15

2. Multiple Independent Variables Simple ANOVA tells whether groups differ – Compares levels of a single independent variable Sometimes we have multiple IVs – Factors – Subjects divided in multiple ways Training type & testing type – Not always true independent variables Undergrad major & sex – Some or all can be within-subjects (gets more complicated) Memory drug & stimulus type Dependent variable measured for all combinations of values Factorial ANOVA – How does each factor affect the outcome? – Extends ANOVA in same way regression extends correlation

3. Basic Approach Calculate sum of squares for each factor – Variability explained by that factor – Essentially by averaging all data for each level of that factor Separate hypothesis test for each factor – Convert SS to mean square – Divide by MS residual to get F Testing TrainingDominantNon-dominantMean Dominant[3,7,5,4,6][14,15,11,13,12]9 Non-dominant[11,7,10,8,9][10,12,13,11,9]10 Mean7129.5

4. Interactions Testing TrainingDominantNon-dominant DominantM = 5M = 13 Non-dominantM = 9M = 11 Difference-4+2 Effect of one factor may depend on level of another – Pick any two levels of Factor A, find difference of means, compare across levels of Factor B Testable in same way as main effect of each factor – SS interaction, MS interaction, F, p Can have higher-order interactions – Interaction between Factors A and B depends on C Partitioning variability – SS total = SS A + SS B + SS C + SS A:B + SS A:C + SS B:C + SS A:B:C + SS residual

5. Brain Injury DelayNoneMTLOccipitalMean Short78%73%65%72% Long66%61%53%52% Difference12% Example: Memory and Brain Injury 37% 36% EffectSSdfMSF p Delay6000 1 12.91.0007 Injury3160 215803.40.041 Delay:Injury1920 29602.07.136 Residual2509454464.7 Mean 72% 55% 59% 62%

6. Logic of Sum of Squares Total sum of squares: Null hypothesis assumes all data are from same population – Expected value of is  2 for each raw score – No matter how we break up SS total, every individual square has expected value  2 – SS treatment, SS interaction, SS residual are all sums of numbers with expected value  2 MS has expected value  2 – Average of many numbers that all have expected value  2 – E(MS treatment ), E(MS interaction ), E(MS residual ) all equal  2, according to H 0 If H 0 false, then MS treatment and MS interaction tend to be larger – F is sensitive to such an increase