Statistics for the Social Sciences Psychology 340 Spring 2005 Factorial ANOVA

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

Statistics for the Social Sciences Statistical analysis follows design The factorial (between groups) ANOVA: –More than two groups –Independent groups –More than one Independent variable

Statistics for the Social Sciences Factorial experiments 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 B1B2B3 A1 A2

Statistics for the Social Sciences 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

Statistics for the Social Sciences 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: 5) A & B 6) A & AB 7) B & AB 8) A & B & AB 1) No effects at all 2) A only 3) B only 4) AB only

Statistics for the Social Sciences 2 x 2 factorial design Condition mean A1B1 Condition mean A2B1 Condition mean A1B2 Condition mean A2B2 A1A2 B2 B1 Marginal means B1 mean B2 mean A1 meanA2 mean Main effect of B Main effect of A Interaction of AB What’s the effect of A at B1? What’s the effect of A at B2?

Statistics for the Social Sciences Main effect of A Main effect of B Interaction of A x B A B A1 A2 B1 B2 Main Effect of A Main Effect of B A A1 A2 Dependent Variable B1 B2 √ X X Examples of outcomes

Statistics for the Social Sciences Main effect of A Main effect of B Interaction of A x B A B A1 A2 B1 B2 Main Effect of A Main Effect of B A A1 A2 Dependent Variable B1 B2 √ X X Examples of outcomes

Statistics for the Social Sciences Main effect of A Main effect of B Interaction of A x B A B A1 A2 B1 B2 Main Effect of A Main Effect of B A A1 A2 Dependent Variable B1 B2 √ X X Examples of outcomes

Statistics for the Social Sciences Main effect of A Main effect of B Interaction of A x B A B A1 A2 B1 B2 Main Effect of A Main Effect of B A A1 A2 Dependent Variable B1 B2 √ √ √ Examples of outcomes

Statistics for the Social Sciences 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

Statistics for the Social Sciences 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

Statistics for the Social Sciences Partitioning the variance Total variance Stage 1 Between groups variance Within groups variance Stage 2 Factor A varianceFactor B varianceInteraction variance

Statistics for the Social Sciences Figuring a Two-Way ANOVA Sums of squares

Statistics for the Social Sciences Figuring a Two-Way ANOVA Degrees of freedom Number of levels of A Number of levels of B

Statistics for the Social Sciences Figuring a Two-Way ANOVA Means squares (estimated variances)

Statistics for the Social Sciences Figuring a Two-Way ANOVA F-ratios

Statistics for the Social Sciences Figuring a Two-Way ANOVA ANOVA table for two-way ANOVA

Statistics for the Social Sciences Example Factor B: Arousal Level Low B 1 Medium B 2 High B 3 FactorA: Task Difficulty A 1 Easy A 2 Difficul t

Statistics for the Social Sciences Example Factor B: Arousal Level Low B 1 Medium B 2 High B 3 FactorA: Task Difficulty A 1 Easy A 2 Difficul t

Statistics for the Social Sciences Example Factor B: Arousal Level Low B 1 Medium B 2 High B 3 FactorA: Task Difficulty A 1 Easy A 2 Difficul t

Statistics for the Social Sciences Example Factor B: Arousal Level Low B 1 Medium B 2 High B 3 FactorA: Task Difficulty A 1 Easy A 2 Difficul t

Statistics for the Social Sciences Example SourceSSdfMSF Between A B AB Within Total √ √ √

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

Statistics for the Social Sciences Effect Size in Factorial ANOVA

Statistics for the Social Sciences Approximate Sample Size Needed in Each Cell for 80% Power (.05 significance level)

Statistics for the Social Sciences Extensions and Special Cases of the Factorial ANOVA Three-way and higher ANOVA designs Repeated measures ANOVA

Statistics for the Social Sciences 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.