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FACTORIAL ANOVA

FACTORIAL ANOVA. Overview of Factorial ANOVA. Factorial Designs Types of Effects Assumptions Analyzing the Variance Regression Equation Fixed and Random Effects. FACTORIAL DESIGNS. All combinations of levels of two or more independent variables (factors) are measured.

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FACTORIAL ANOVA

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  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 A 1 2 Subjects 1-10 Subjects 21-30 1 B Subjects 11-20 Subjects 31-40 2

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

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

  8. 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

  9. 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

  10. 70 60 50 B2 40 d.v. 30 B1 20 10 0 1 2 A

  11. 70 B2 60 50 40 B1 d.v. 30 20 10 0 1 2 A

  12. B2 70 60 50 40 d.v. 30 B1 20 10 0 1 2 A

  13. 70 60 B2 50 40 d.v. 30 20 B1 10 0 1 2 A

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

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

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

  17. REGRESSION EQUATION • bo is mean of base group • b1 is the main effect of factor A • b2 is the main effect of factor B • b3 is the A x B interaction

  18. 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

  19. 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 h2 = .00, a non-significant main effect of industry type, F(2,1482) = 0.70, p = .497, partial h2 = .00, and a significant interaction, F(2,1482) = 3.12, p = .044, partial h2 = .00.

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