1. Factorial ANOVA
11/10

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
Testing
Training
Dominant
Non-dominant
Mean
[3,7,5,4,6]
[14,15,11,13,12] 9
[11,7,10,8,9]
[10,12,13,11,9] 10 7 12
9.5
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 MSresidual to get F

4. Interactions Can have higher-order interactions
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
SSinteraction, MSinteraction, F, p
Can have higher-order interactions
Interaction between Factors A and B depends on C
Partitioning variability
SStotal = SSA + SSB + SSC
+ SSA:B + SSA:C + SSB:C + SSA:B:C
+ SSresidual
Testing
Training
Dominant
Non-dominant
M = 5
M = 13
M = 9
M = 11
Difference -4 +2

5. Logic of Sum of Squares Total sum of squares:
Null hypothesis assumes all data are from same population
Expected value of is s2 for each raw score
No matter how we break up SStotal, every individual square has expected value s2
SStreatment, SSinteraction, SSresidual are all sums of numbers with expected value s2
MS has expected value s2
Average of many numbers that all have expected value s2
E(MStreatment), E(MSinteraction), E(MSresidual) all equal s2, according to H0
If H0 false, then MStreatment and MSinteraction tend to be larger
F is sensitive to such an increase