Statistics for the Social Sciences Psychology 340 Spring 2005 Factorial ANOVA

Outline Basics of factorial ANOVA Interpretations Computations Main effects Interactions Computations Assumptions, effect sizes, and power Other Factorial Designs More than two factors Within factorial ANOVAs

Statistical analysis follows design The factorial (between groups) ANOVA: More than two groups Independent groups More than one Independent variable

Factorial experiments B1 B2 B3 A1 A2 Two or more factors Factors - independent variables Levels - the levels of your independent variables 2 x 3 design means two independent variables, one with 2 levels and one with 3 levels “condition” or “groups” is calculated by multiplying the levels, so a 2x3 design has 6 different conditions

Factorial experiments Two or more factors (cont.) Main effects - the effects of your independent variables ignoring (collapsed across) the other independent variables Interaction effects - how your independent variables affect each other Example: 2x2 design, factors A and B Interaction: At A1, B1 is bigger than B2 At A2, B1 and B2 don’t differ

Results So there are lots of different potential outcomes: A = main effect of factor A B = main effect of factor B AB = interaction of A and B With 2 factors there are 8 basic possible patterns of results: 1) No effects at all 2) A only 3) B only 4) AB only 5) A & B 6) A & AB 7) B & AB 8) A & B & AB

2 x 2 factorial design Interaction of AB A1 A2 B2 B1 Marginal means What’s the effect of A at B1? What’s the effect of A at B2? Condition mean A1B1 Condition mean A2B1 Marginal means B1 mean B2 mean A1 mean A2 mean Main effect of B Condition mean A1B2 Condition mean A2B2 Main effect of A

Examples of outcomes Main effect of A √ Main effect of B Dependent Variable B1 B2 30 60 45 60 45 30 30 60 Main Effect of A Main effect of A √ Main effect of B X Interaction of A x B X

Examples of outcomes Main effect of A Main effect of B √ Dependent Variable B1 B2 60 60 60 30 30 30 45 45 Main Effect of A Main effect of A X Main effect of B √ Interaction of A x B X

Examples of outcomes Main effect of A Main effect of B Dependent Variable B1 B2 60 30 45 60 45 30 45 45 Main Effect of A Main effect of A X Main effect of B X Interaction of A x B √

Examples of outcomes Main effect of A √ Main effect of B √ Dependent Variable B1 B2 30 60 45 30 30 30 30 45 Main Effect of A Main effect of A √ Main effect of B √ Interaction of A x B √

Factorial Designs Benefits of factorial ANOVA (over doing separate 1-way ANOVA experiments) Interaction effects One should always consider the interaction effects before trying to interpret the main effects Adding factors decreases the variability Because you’re controlling more of the variables that influence the dependent variable This increases the statistical Power of the statistical tests

Basic Logic of the Two-Way ANOVA Same basic math as we used before, but now there are additional ways to partition the variance The three F ratios Main effect of Factor A (rows) Main effect of Factor B (columns) Interaction effect of Factors A and B

Partitioning the variance Total variance Stage 1 Between groups variance Within groups variance Stage 2 Factor A variance Factor B variance Interaction variance

Figuring a Two-Way ANOVA Sums of squares

Figuring a Two-Way ANOVA Degrees of freedom Number of levels of B Number of levels of A

Figuring a Two-Way ANOVA Means squares (estimated variances)

Figuring a Two-Way ANOVA F-ratios

Factor B: Arousal Level Low B1 Medium B2 High B3 FactorA: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 11 10 8 A2 Difficult Example

Factor B: Arousal Level Low B1 Medium B2 High B3 FactorA: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 11 10 8 A2 Difficult Example

Factor B: Arousal Level Low B1 Medium B2 High B3 FactorA: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 11 10 8 A2 Difficult Example

Factor B: Arousal Level Low B1 Medium B2 High B3 FactorA: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 11 10 8 A2 Difficult Example

Example: ANOVA table Source SS df MS F Between A B AB 120 60 1 2 30 27.7 6.9 Within Total 104 344 24 4.33 √ √ √

Factorial ANOVA in SPSS What we covered today is a completely between groups Factorial ANOVA Enter your observations in one column, use separate columns to code the levels of each factor Analyze -> General Linear Model -> Univariate Enter your dependent variable (your observations) Enter each of your factors (IVs) Output Ignore the corrected model, intercept, & total (for now) F for each main effect and interaction

Assumptions in Two-Way ANOVA Populations follow a normal curve Populations have equal variances Assumptions apply to the populations that go with each cell