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

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

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

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

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

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.

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.

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

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

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

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

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

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

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

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

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

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.

Take-Home Points Factorial ANOVA allows us to test for interactions. Most things are affected by multiple factors that do not work independently of each other.