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Statistics for the Social Sciences

This outline covers basics of factorial ANOVA, interpretations, main effects, interactions, computations, assumptions, effect sizes, power, other designs, within-factorial ANOVAs, interactions of independent variables, statistical analyses, main and interaction effects, and examples.

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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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