Eta Squared, Power, & Factorial ANOVA Computation

1. Eta Squared, Power,&Factorial ANOVA Computation

2. Outline of Today’s Discussion • Eta Squared And Power • Reporting APA-Style Results for Factorial Designs • Between Subjects Factorial ANOVA: SPSS • Between Subjects Factorial ANOVA: Excel

3. Part 1 Eta Squared, and Power

4. Eta Squared, and Power • What does the abbreviation “ANOVA” stand for? • Our job as psychologists is to explain fluctuations in the dependent variable. In other words, we must account for the variability in scores. • Let’s re-visit how we can partition the variability in a factorial experiment…

5. Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

6. Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

7. Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

8. Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

9. Eta Squared, and Power Pie Chart of Total Variability Which effect(s), if any, would have a significant F value?

10. Eta Squared, and Power • To summarize, in a 2-way, between subjects ANOVA, the total variability can be partitioned into two components; Between-Subjects & Within-Subjects. • The Between-Subjects component itself can be sub-divided: Example: Factor A, Factor B, AxB interaction. • Recall, too, that we can partition the within-subjects component, so that consistent individual differences are removed.

11. Eta Squared, and Power • Could someone describe what Eta Squared tells us? http://en.wikiversity.org/wiki/Eta-squared • We could compute an Eta-squared value for each main effect, and for the interaction…

12. Eta Squared, and Power Pie Chart of Total Variability Which factor would likely have the largest Eta-squared?

13. Eta Squared, and Power • One of the “good reporting practices” (discussed later) is providing information effect size… • Effect size – The strength of the relationship between variables, e.g., the proportion of variance explained for each effect (i.e., each main effect and interaction). • NOTE: There is more than one way to measure effect size! (Sorry about that…but that’s the way it is).

14. Eta Squared, and Power Would someone walk us through this?

15. Eta Squared, and Power If we had a large value on the left side of this equation, what might the corresponding pie chart look like?

16. Eta Squared, and Power Here’s another way of saying the same thing r-squared and eta squared provide the same info.

17. Eta Squared, and Power • In Class Exercise: I’ll show you some F-tables here, and you’ll use excel to compute the eta-squared for each effect. • We’ll use an example of a factorial, independent-subjects design. • The DV was the # of hours worked by the employee. The IVs were Gender, Education (MastDoc)….

18. Eta Squared, and Power Let’s Compute Eta Squared for the “Gender” Factor SSGender = 343.066 SSerror = 27,920.02 SStotal = ? Etasquared = ?

19. Eta Squared, and Power Let’s Compute Eta Squared for the “MastDoc” Factor SSMastdoc = 286.686 SSerror = 27,920.02 SStotal = ? Etasquared = ?

20. Eta Squared, and Power Let’s Compute Eta Squared for Gender-By-Mastdoc SSgenderMastdoc = 903.309 SSerror = 27,920.02 SStotal = ? Etasquared = ?

21. Eta Squared, and Power SPSS will automatically compute Eta Squared for us!

22. Eta Squared, and Power • We can have SPSS give us the Eta-squared value for each main effect, and for the interaction. • SPSS can also indicate the amount of “power” that we have when we assess each main effect and interaction….

23. Eta Squared, and Power • Potential Pop Quiz Question: In your own words, explain what POWER is, in a statistical sense. http://en.wikipedia.org/wiki/Statistical_power • Potential Pop Quiz Question: In your own words, explain why it is important to report an estimate of power when an effect is NON-significant. (Your answer requires some critical thinking, and s/b something other than “because APA says so”.  ) • Power estimates range from 0 to 1.

24. Eta Squared, and Power Power for each effect

25. Part 2 Reporting APA-Style Results For Factorial Designs

26. APA-Style Results: Factorial Designs • It is important to establish good reporting habits! • If researchers agree to follow particular reporting standards, then results can be universally understood!

27. APA-Style Results: Factorial Designs • NOTE: In the text of your results, you should report some measure of central tendency and some measure of dispersion. • Usually this will entail reporting the Mean and the S.D. for each condition. • You will very likely be required to do that in your psychology research courses! (In some studies with huge dimensionality, you might use a summary table for means & SDs.)

28. APA-Style Results: Factorial Designs Other APA-Style Reporting Standards From Shaughnessy, Zechmeister & Zechmeister

29. Part 3 Between-Subjects Factorial ANOVA In SPSS

30. Between-Subjects Factorial ANOVA in SPSS • Here’s the sequence of steps in SPSS… • Analyze---> General Linear Model ---->Univariate. • Slide your DV to the Dependent Variable box. • Slide your between-subject IV’s to Fixed Factors box.

31. Between-Subjects Factorial ANOVA in SPSS • Select the Post Hoc button, slide the variables over to the right, then select Scheffe. • Select the Options button, and click on descriptives, estimates of effect size, observed power, and homogeneity tests.

32. Between-Subjects Factorial ANOVA: SPSS As in the single-factor case, we check the equal variance assumption first. Would we retain or reject the equal variance assumption?

33. Between-Subjects Factorial ANOVA: SPSS Evaluate each main effect, and the interaction.

34. Part 4 Between-Subjects Factorial ANOVA In Excel

35. Between-Subjects Factorial ANOVA: Excel Let’s assume we have a 3x2 design: 3 levels of feedback, 2 levels of complexity. From Keppel, Saufley & Tokunaga We can label each ‘cell’ by its A and B coordinates. What are the coordinates of “Praise-Complex”?

36. Between-Subjects Factorial ANOVA: Excel From Keppel, Saufley & Tokunaga To compute the ANOVA in excel, each condition should be in a separate column.

37. Between-Subjects Factorial ANOVA: Excel From Keppel, Saufley & Tokunaga To compute the ANOVA in excel, we need to develop the so-called AB Matrix of sums.

38. Between-Subjects Factorial ANOVA: Excel From Keppel, Saufley & Tokunaga The AB Matrix of sums will have to be squared. This is similar to what we’ve done before.

39. Between-Subjects Factorial ANOVA: Excel From Keppel, Saufley & Tokunaga The Basic Ratios in your hand-out will be based on the squared AB Matrix. What do Basic Ratios do for us?

40. Between-Subjects Factorial ANOVA: Excel • Great news! More basic ratios! • To “manually” compute factorial ANOVAs in excel, we will use the familiar basic ratios [Y] and [T] • We will also have one basic ratio for each of our IVs, and the interaction: [A] [B] [AB] …

41. [Y] [Y] = The sum of the individual squared scores. (Square them first, then sum them.) [AB] [AB] = The sum of the individual cells from the Squared AB Matrix, divided by the number of subjects per condition. Between-Subjects Two-Way ANOVA [A] [A] = The sum of the column totals from the Squared AB Matrix, divided by (number of B levels * subjects per condition). [B] = The sum of the row totals from the Squared AB Matrix, divided by (number of A levels * subjects per condition). [B] [T] [T] = The grand total squared, divided by the total number of scores. The total number of scores (N) equals a * b * n.

42. The F Table using basic ratios Between-Subjects Two-Way ANOVA

43. Analysis of Complex Designs Shaughnessy,JJ, Shaughnessy, EB, and Zechmesiter, JS. Research Methods in Psychology, McGraw Hill. What’s “wrong” with this graph? Keppel, G., Saufley, W., Tokunaga, H., Introduction to Design and Analysis (2nd edition), W.H. Freeman Co.