1. Intro to Statistics for the Behavioral Sciences PSYC 1900
Lecture 15: Interactions in Factorial ANOVA

2. Factorial ANOVA Review
An example: Effects of Temperature and Gender on Aggression
Each Factor has marginal means (i.e., means averaged across the other iv)
Main Effect
The effect of one iv averaged across the levels of the other
Simply a one-way ANOVA on the marginal means
Here, there are two main effects.
They tell us if gender and/or temperature affects aggression.

3. Factorial ANOVA Calculations
As with one-ways, we calculate SS for each effect.
SStotal again captures all variance of scores around grand mean.
Each main effect captures variance of cell means around grand mean
Differences among all cell means is captured by SScells.

4. Main Effects

5. Testing Significance of Main Effects
We convert each relevant SS into Mean Squares (MS).
We divide SS by associated df’s.
dftotal=N-1
dffactor= (#conds)-1
dferror=(a)(b)(n-1)
F is the ratio of these two estimates of population variance.
Critical values of F are found using F(dfgroup, dferror),

6. Interactions
The effects of one independent variable depend on the level of another independent variable.
Pattern of means cannot be captured based solely on deviations form grand mean due to main effects.

7. Interaction Calculations
Interactions represent variation among cell means that cannot be captured by main effects AND is not likely due to simple sampling error.
Differences among all cell means is captured by SScells.
Therefore, interaction can be defined as variation among cell means not explained by main effects.
Degrees of freedom for interactions are: (a-1)(b-1)
Applet

8. Types of Interactions Ordinal Disordinal
Ordinal positions of group differences remain constant
Main effects may be interpreted
Disordinal
Group differences reverse their signs at some level of the other variable
Main effects usually not interpretable

9. Interpreting Interactions
When an interaction is present, it simply means that the effects of one variable on the dv depend on levels of the other.
This implies that the “simple effects” of one iv differ across levels of the other iv.
Simple effect is the effect of one iv at one specific level of the other.

10. Analysis of Simple Effects
•Analysis of simple effects proceeds by dividing the factorial design into a series of single-factor experiments.
•Simple effects are based upon the cell means within a given level of one of the iv’s. Main effects are based upon marginal means.
•If an interaction is present, at least one of the simple effects must differ from at least one of the others.
•Multiple comparison techniques are used to determine the relations among the means at each level of one of the iv’s.
•Here, one might use Fisher’s LSD Tests to compare 3 means at each level of A.

11. An Example
A researcher conducts a study examining the effects of stress and sex (i.e., gender) on aggression.
Men and women receive either a stressful or simple task to perform and then have their performance insulted by a confederate. Aggression is then measured.
Prediction was that stress should increase male aggression but decrease female aggression.

12. The pattern of means looks to support the claim, but we must use a factorial ANOVA to verify it.

14. Simple Effects of Stress

15. Simple Effects of Sex