Factorial Designs More than one Independent Variable: Each IV is referred to as a Factor All Levels of Each IV represented in the Other IV

A Two-Way ANOVA Factor A has 3 Levels Factor B Has 2 Levels Each Cell is a COMBINATION Of Treatments for a Group of Subjects

A Two-Way ANOVA Marginal Means For Factor B; do They differ Marginal Means for Factor A; do they differ? Marginal Means average across the Levels of the OTHER Factor

A Two-Way ANOVA A Two-Way ANOVA tells you: 1.What a One-Way ANOVA would find out about Factor A 2.What a One-Way ANOVA would find out about Factor B 3.If there is an Interaction between Factor A and Factor B

A Two-Way Interaction An Interaction is the effect which one IV has on the effect which The other IV has on the DV Is the Difference between subjects who got Treatment B1 and B2 The Same irrespective of whether they got Treatment A1, A2, or A3? The Other Side of the Same Coin: Are the Differences among subjects who got Treatments A1, A2 and A2 The Same irrespective of whether they got Treatment B1 or B2?

Main Effects & Interactions IV 1 : Sports IV 2 : Gender DV: Aggression IV 1 : Main Effect IV 2 : Main Effect Interaction: None Main Effect: Averaged Across levels of the Other IV Is the impact of sports on aggression different For Males and females? M F N Y

Main Effects & Interactions IV 1 : Sports IV 2 : Gender DV: Aggression IV 1 : Main Effect IV 2 : Main Effect Interaction: Yes Main Effect: Averaged Across levels of the Other IV Is the impact of sports on aggression different For Males and females? M F N Y M F N Y

Main Effects & Interactions IV 1 : Sports IV 2 : Gender DV: Aggression IV 1 : Main Effect IV 2 : Main Effect Interaction: Yes Main Effect: Averaged Across levels of the Other IV Is the impact of sports on aggression different For Males and females? M F N Y M F N Y F M

Main Effects & Interactions IV 1 : Sports IV 2 : Gender DV: Aggression IV 1 : No Main Effect IV 2 : Main Effect Interaction: Yes Main Effect: Averaged Across levels of the Other IV Is the impact of sports on aggression different For Males and females? M F N Y M F N Y F M M F

Main Effects & Interactions IV 1 : Sports IV 2 : Gender DV: Aggression IV 1 : No Main Effect IV 2 : No Main Effect Interaction: Yes Main Effect: Averaged Across levels of the Other IV Is the impact of sports on aggression different For Males and females? M F N Y M F N Y F M M F

No Interaction  Main Effect  Differences Same for B1 and B2 And Same as Marginal Means  Main Effect  Differences same For A1, A2, & A3 And Same as Marginal Means

Yummy Interaction B1 and B2 Subjects change differently (across Factor A) from One another and from the Marginal Means A1, A2, & A3 Subjects Change differently (across Factor b) from one Another and from Marginal Means In my opinion: Interactions are more interesting and more important Than main effects

Explaining the Relationship Between the IV and DV Does Sports (IV 1 ) affect Aggression (DV)? Yes but more so in males Yes but only in males Yes but oppositely in males and females (IV 2 ) If you have to qualify the relationship with a “But,” then you Have an Interaction.

Interactions

Statistical Symbols ΣSum άType I Error βType II Error μPopulation Mean ρPopulation Correlation σPopulation Standard Deviation Interaction

Interactions Arms & Legs Not Parallel Yes But more so in males Yes But only in males Yes But in different directions Yes But in different directions, from different directions

Parvulus te Tergum

The Structure of the ANOVA Partitioning the Total Sum of Squared Deviations From the Grand Mean If you must run your reaction time study at 3 different times of day: 1.Counter Balance 2.Use Time as a Second IV to pull Main Effect and Interaction Variance (SS) out of Error Term D.V.: Reaction Time Variation W/I Drug Time Combination E.G., Drug E.G., Time of Day

Partitioning the Sums of Squares Into 4 Parts Variation of cell means From Grand Mean (SS_Between Cell) Variation of Individuals From their cell means (SS-Within Cell) Variation of Individuals from The Grand Mean Sum to SS Cell Every Subjects’ Score is composed of these 4 parts

Do It! Step 1: Calculate SS-Total & SS-Error

Step 2: Calculate SS Main Effects For A & B

Step 3: Calculate SS Interaction SS Tot -SS A -SS B -SS Error Or SS Cell -SS A -SS B

Step 4: Calculate Degrees of Freedom

Step 5: Calculate Mean Squares Divide SS by df

Step 5: Calculate F-Values Divide MS by MS Error

Decision If Interaction is non-significant: Interpret Each Main Effect as if it came from a One-Way ANOVA Do Tukey Post Hoc HSD test for every Significant IV with more than 2 Levels If Interaction is Significant: Do a Simple Effects ANOVA on Each IV For EVERY Level of the other IV A 3x2 design would require 5 One-way ANOVAs

Post Hoc Tests for Each Significant IV (If No Interaction) X-Bars are the Marginal Means N t is the number of scores going into the Marginal Mean N t must be same size for both Marginal Means