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Statistics for the Social Sciences

Statistics for the Social Sciences. Psychology 340 Spring 2005. Factorial ANOVA. 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.

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Statistics for the Social Sciences

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  1. Statistics for the Social Sciences Psychology 340 Spring 2005 Factorial ANOVA

  2. 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

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

  4. 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

  5. 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

  6. 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

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

  8. A Main Effect A2 A1 of B B1 30 60 B1 B Dependent Variable B2 B2 30 60 Main Effect A1 A2 of A A Examples of outcomes 45 45 30 60 Main effect of A √ Main effect of B X Interaction of A x B X

  9. A Main Effect A2 A1 of B B1 60 60 B1 B Dependent Variable B2 B2 30 30 Main Effect A1 A2 of A A Examples of outcomes 60 30 45 45 Main effect of A X Main effect of B √ Interaction of A x B X

  10. A Main Effect A2 A1 of B B1 60 30 B1 B Dependent Variable B2 B2 60 30 Main Effect A1 A2 of A A Examples of outcomes 45 45 45 45 Main effect of A X Main effect of B X Interaction of A x B √

  11. A Main Effect A2 A1 of B B1 30 60 B1 B Dependent Variable B2 B2 30 30 Main Effect A1 A2 of A A Examples of outcomes 45 30 30 45 √ Main effect of A √ Main effect of B Interaction of A x B √

  12. 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

  13. 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

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

  15. Figuring a Two-Way ANOVA • Sums of squares

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

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

  18. Figuring a Two-Way ANOVA • F-ratios

  19. Figuring a Two-Way ANOVA • ANOVA table for two-way ANOVA

  20. Example

  21. Example

  22. Example

  23. Example

  24. Example √ √ √

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

  26. Effect Size in Factorial ANOVA

  27. Approximate Sample Size Needed in Each Cell for 80% Power (.05 significance level)

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

  29. 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.

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