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Factorial Designs

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

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Factorial Designs

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  1. 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

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

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

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

  5. 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?

  6. Main Effects & Interactions • Is the impact of sports on aggression different • For Males and females? • IV1: Sports • IV2: Gender • DV: Aggression • IV1: Main Effect • IV2: Main Effect • Interaction: None • Main Effect: Averaged • Across levels of the • Other IV Y M F N

  7. Main Effects & Interactions • Is the impact of sports on aggression different • For Males and females? • IV1: Sports • IV2: Gender • DV: Aggression • IV1: Main Effect • IV2: Main Effect • Interaction: Yes • Main Effect: Averaged • Across levels of the • Other IV Y M Y M F F N N

  8. Main Effects & Interactions • Is the impact of sports on aggression different • For Males and females? • IV1: Sports • IV2: Gender • DV: Aggression • IV1: Main Effect • IV2: Main Effect • Interaction: Yes • Main Effect: Averaged • Across levels of the • Other IV Y M M Y M F F N N F

  9. Main Effects & Interactions • Is the impact of sports on aggression different • For Males and females? • IV1: Sports • IV2: Gender • DV: Aggression • IV1: No Main Effect • IV2: Main Effect • Interaction: Yes • Main Effect: Averaged • Across levels of the • Other IV M Y M M Y M F F F N N F

  10. Main Effects & Interactions • Is the impact of sports on aggression different • For Males and females? • IV1: Sports • IV2: Gender • DV: Aggression • IV1: No Main Effect • IV2: No Main Effect • Interaction: Yes • Main Effect: Averaged • Across levels of the • Other IV M Y M M Y M F F F N N F

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

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

  13. Explaining the Relationship Between the IV and DV • Does Sports (IV1) affect Aggression (DV)? • Yes but more so in males • Yes but only in males • Yes but oppositely in males and females (IV2) • If you have to qualify the relationship with a “But,” then you • Have an Interaction.

  14. Interactions

  15. Statistical Symbols Σ Sum ά Type I Error β Type II Error μ Population Mean ρ Population Correlation σ Population Standard Deviation Interaction

  16. InteractionsArms & 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

  17. Parvulus te Tergum

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

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

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

  21. Step 2: Calculate SS Main Effects For A & B

  22. Step 3: Calculate SS Interaction SSTot-SSA-SSB-SSError Or SSCell-SSA-SSB

  23. Step 4: Calculate Degrees of Freedom

  24. Step 5: Calculate Mean SquaresDivide SS by df

  25. Step 5: Calculate F-ValuesDivide MS by MSError

  26. 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

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

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