1. FACTORIAL ANOVA

2. Overview of Factorial ANOVA Factorial Designs Types of Effects Assumptions Analyzing the Variance Regression Equation Fixed and Random Effects

3. FACTORIAL DESIGNS All combinations of levels of two or more independent variables (factors) are measured

4. Types of Factorials Between subjects (independent) Within subjects (related) Mixed

5. Between Subjects B 1 2 A 12 Subjects 1-10 Subjects 11-20 Subjects 31-40 Subjects 21-30

6. Within Subjects B 1 2 A 12 Subjects 1-40 Subjects 1-40 Subjects 1-40 Subjects 1-40

7. Mixed (A Between, B Within) B 1 2 A 12 Subjects 1-20 Subjects 1-20 Subjects 21-40 Subjects 21-40

8. Review! I am planning a 2 x 3 factorial design and I want to have 20 participants per condition. How many total participants do I need: For a between subjects design? For a within subjects design? For a mixed design where the between factor has two levels?

9. TYPES OF EFFECTS A main effect is the overall effect of each IV by itself, averaging over the levels of any other IVs. An interaction occurs when the effects of one factor change depending on the level of another factor.

10. Simple Effects An interaction can be understood as a difference in simple effects. A simple effect is the effect of one factor on only one level of another factor. If the simple effects differ, there is an interaction.

11. d.v. 10 20 30 40 50 60 70 0 A 12 B1 B2

12. d.v. 10 20 30 40 50 60 70 0 A 12 B1 B2

13. d.v. 10 20 30 40 50 60 70 0 A 12 B1 B2

14. d.v. 10 20 30 40 50 60 70 0 A 12 B1 B2

15. ASSUMPTIONS Interval/ratio data Normal distribution or N at least 30 Independent observations Homogeneity of variance Proportional or equal cell sizes

16. ANALYZING THE VARIANCE Total Variance = Model + Residual Model Variance is further divided into: Factor A Factor B A x B interaction

17. Comparing Variance F-test for each main effect and for the interaction Each F-test compares variance for the effect to Residual variance

18. REGRESSION EQUATION b o is mean of base group b 1 is the main effect of factor A b 2 is the main effect of factor B b 3 is the A x B interaction

19. FIXED VS. RANDOM EFFECTS Fixed Factor: only the levels of interest are selected for the factor, and there is no intent to generalize to other levels Random Factor: the levels are selected at random from the possible levels, and there is an intent to generalize to other levels

20. APA Format Example The two-way between subjects ANOVA showed a significant main effect of customer type, F(1,1482) = 5.04, p =.025, partial  2 =.00, a non-significant main effect of industry type, F(2,1482) = 0.70, p =.497, partial  2 =.00, and a significant interaction, F(2,1482) = 3.12, p =.044, partial  2 =.00.

21. Choosing Stats Participants respond to a 20-item scale of superstitious beliefs. The researcher would like to determine if there is internal consistency in responses to the 20 items. In other words, are people who believe in some superstitions likely to also believe in other superstitions?