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Factorial Analysis of Variance: Basics, Interpretations, and Computations

Understand the fundamentals of factorial ANOVA, including main effects, interactions, assumptions, and power. Explore various factorial designs with more than two factors and independent groups. Learn to interpret results and compute F-ratios effectively.

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Factorial Analysis of Variance: Basics, Interpretations, and Computations

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

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

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

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

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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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 √

  13. 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 √

  14. 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

  15. 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

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

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

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

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

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

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

  22. Example √ √ √

  23. 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)

  24. TROW1= 90 TROW2= 30 TCOLUMN1= 20 TCOLUMN2= 50 TCOLUMN1= 50 G=120

  25. 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

  26. Example

  27. 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

  28. TROW1= 90 TROW2= 30 TCOLUMN1= 20 TCOLUMN2= 50 TCOLUMN1= 50 G=120

  29. 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

  30. Effect Size in Factorial ANOVA

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

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

  33. 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.

  34. 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.”

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