Statistics for the Social Sciences Psychology 340 Fall 2013 Thursday, October 24 Factorial Analysis of Variance (ANOVA)

Homework #9 (handout) due10/29 Homework #10 due10/31 Ch 14 # 1,2,4,5,7,9, SKIP PROBLEM 10, 14,15, 22 (DO NOT use SPSS for #22)

Last Time Brief review of Repeated Measures ANOVA Assumptions in Repeated Measures ANOVA Effect sizes in Repeated Measures ANOVA More practice with SPSS

This Time 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

Figuring a Two-Way ANOVA ANOVA table for two-way ANOVA

Factor B: Arousal Level Low B1 Medium B2 High B3 Factor A: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 13 8 A2 Difficult

Factor B: Arousal Level Low B1 Medium B2 High B3 FactorA: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 13 8 A2 Difficult

Factor B: Arousal Level Low B1 Medium B2 High B3 Factor A: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 13 8 A2 Difficult

Factor B: Arousal Level Low B1 Medium B2 High B3 Factor A: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 13 8 A2 Difficult

Example Source SS df MS F Between A B AB 120 60 1 2 30 24.0 6.0 Within Total 360 24 5 √ √ √

Computational Formulas T = Group (Condition) Total G = Grand Total TRow = Row Total Tcolumn=Column Total n = number of participants in treatment nrow=number of participants in row ncolumn=number participants in col. N = number of participants Same as in one-way ANOVA Same as in one-way ANOVA, but T refers to each treatment or condition (e.g., A1B1 is one treatment) Same as in one-way ANOVA, but each treatment or condition is one cell in the design (e.g., A1B1 is one treatment)

Factor B: Arousal Level Low B1 Medium B2 High B3 Factor A: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 13 8 A2 Difficult TROW1= 90 TROW2= 30 TCOLUMN1= 20 TCOLUMN2= 50 TCOLUMN1= 50 G=120

Computational Formulas T = Group (Condition) Total G = Grand Total TRow = Row Total Tcolumn=Column Total n = number of participants in treatment nrow=number of participants in row ncolumn=number participants in col. N = number of participants

Factor B: Arousal Level Low B1 Medium B2 High B3 Factor A: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 13 8 A2 Difficult Example

Computational Formulas T = Group (Condition) Total G = Grand Total TRow = Row Total Tcolumn=Column Total n = number of participants in treatment nrow=number of participants in row ncolumn=number participants in col. N = number of participants

Factor B: Arousal Level Low B1 Medium B2 High B3 FactorA: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 13 8 A2 Difficult TROW1= 90 TROW2= 30 TCOLUMN1= 20 TCOLUMN2= 50 TCOLUMN1= 50 G=120

Computational Formulas T = Group (Condition) Total G = Grand Total TRow = Row Total Tcolumn=Column Total n = number of participants in treatment nrow=number of participants in row ncolumn=number participants in col. N = number of participants

Factor B: Arousal Level Low B1 Medium B2 High B3 FactorA: Task Difficulty A1 Easy 3 1 6 4 2 5 9 7 13 8 A2 Difficult

Effect Size in Factorial ANOVA    

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

Extensions and Special Cases of the Factorial ANOVA Three-way and higher ANOVA designs Repeated measures ANOVA

Factorial ANOVA in Research Articles A two-factor ANOVA yielded a significant main effect of voice, F(2, 245) = 26.30, p < .001. As expected, participants responded less favorably in the low voice condition (M = 2.93) than in the high voice condition (M = 3.58). The mean rating in the control condition (M = 3.34) fell between these two extremes. Of greater importance, the interaction between culture and voice was also significant, F(2, 245) = 4.11, p < .02.

Factorial ANOVA in SPSS Analyze=>General Linear Model=>Univariate Highlight the column label for the dependent variable in the left box and click on the arrow to move it into the Dependent Variable box. One by one, highlight the column labels for the two factor codes (Independent Variables) and click the arrow to move them into the Fixed Factors box. If you want descriptive statistics for each treatment, click on the Options box, select Descriptives, and click continue. Click OK In the output, look at “corrected model” and ignore “intercept” and look at “corrected total” and ignore “total.”