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Statistics for the Social Sciences

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.

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Statistics for the Social Sciences

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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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