1. Assignment 1 February 15, 2008Slides by Mark Hancock

2. Factorial (Two-Way) ANOVA February 15, 2008Slides by Mark Hancock

3. You will be able to perform a factorial analysis of variance (ANOVA) and interpret the meaning of the results (both main effects and interactions). February 15, 2008Slides by Mark Hancock

4. Variables (Revisited) Independent Variables – a.k.a. Factors Dependent Variables – a.k.a. Measures February 15, 2008Slides by Mark Hancock

5. Factor e.g., technique: – TreeMap vs. Phylotrees vs. ArcTrees How many levels does the “technique” factor have? February 15, 2008Slides by Mark Hancock

6. Factorial Design Remember the assignment Two factors: – gender (male, female) – technique (TreeMap, Phylotrees, ArcTrees) February 15, 2008Slides by Mark Hancock

7. Factorial Design MaleFemale TreeMap Group 1Group 2 Phylotrees Group 3Group 4 ArcTrees Group 5Group 6 Gender Technique FactorsLevels Cells February 15, 2008Slides by Mark Hancock

8. How did we test these six groups? February 15, 2008Slides by Mark Hancock

9. What did we find out? What could we have found out? Was what we did valid? February 15, 2008Slides by Mark Hancock

10. Factorial ANOVA What is the null hypothesis? February 15, 2008Slides by Mark Hancock

11. Factorial ANOVA What are the null hypotheses? February 15, 2008Slides by Mark Hancock

12. Main Effects February 15, 2008Slides by Mark Hancock

13. Main Effects MaleFemale TreeMap µ1µ1 µ2µ2 Phylotrees µ3µ3 µ4µ4 ArcTrees µ5µ5 µ6µ6 Gender Technique µ TreeMap µ Phylotrees µ ArcTrees µ Male µ Female February 15, 2008Slides by Mark Hancock

14. Null Hypotheses Main effect of gender: µ Male = µ Female Main effect of technique: µ TreeMap = µ Phylotrees = µ ArcTrees February 15, 2008Slides by Mark Hancock

15. Example NightDay Headlights µ1µ1 µ2µ2 No Headlights µ3µ3 µ4µ4 Time of Day Lights Do headlights help see pedestrians? February 15, 2008Slides by Mark Hancock

16. Example NightDay Headlights good No Headlights badgood Time of Day Lights Do headlights help see pedestrians? February 15, 2008Slides by Mark Hancock

17. Results? Main effects – Day is “better” than night – Headlights are “better” than no headlights Is that the real story? February 15, 2008Slides by Mark Hancock

18. What about the cell means? What other null hypothesis could we test? µ 1 = µ 2 = … = µ 4 Why not? February 15, 2008Slides by Mark Hancock

19. Interactions February 15, 2008Slides by Mark Hancock

20. What is our third null hypothesis? e.g. conclusion: – The effect of headlights depends on whether it is day or night. Null hypothesis in words: – The main effect of one factor does not depend on the levels of another factor. February 15, 2008Slides by Mark Hancock

21. Alternative Hypothesis The main effect of factor X depends on the levels of factor Y. February 15, 2008Slides by Mark Hancock

22. Null Hypothesis Level 1Level 2 Level 1 µ 11 µ 12 Level 2 µ 21 µ 22 Level 3 µ 31 µ 32 Factor 1 Factor 2 February 15, 2008Slides by Mark Hancock

23. Null Hypothesis µ 11 – µ 12 = µ 21 – µ 22 = µ 31 – µ 32 In general: µ ij – µ i’j = µ ij’ – µ i’j’ for all combinations of i, i’, j, j’ February 15, 2008Slides by Mark Hancock

24. Null Hypothesis February 15, 2008Slides by Mark Hancock

25. Null Hypotheses Main effects: – row means are equal – column means are equal Interaction: – the pattern of differences in one row/column do not account for the pattern of differences in another row/column February 15, 2008Slides by Mark Hancock

26. Factorial ANOVA Math February 15, 2008Slides by Mark Hancock

27. F-scores Calculate F for each null hypothesis February 15, 2008Slides by Mark Hancock

28. Sum of Squares (revisited) February 15, 2008Slides by Mark Hancock

29. Sum of Squares (revisited) February 15, 2008Slides by Mark Hancock

30. Factorial ANOVA Table Degrees of Freedom Sum of Squares Mean SquareF Factor 1 Factor 2 Interaction Within Groups Total February 15, 2008Slides by Mark Hancock

31. Example Degrees of Freedom Sum of Squares Mean SquareF Gender1712.89 6.38 Condition1462.25 4.14 Gender × Condition 11.21 0.01 Error (Within Groups) 9610,720.82111.68 Total9911,897.17 February 15, 2008Slides by Mark Hancock

32. Break: 15 Minutes February 15, 2008Slides by Mark Hancock

33. Example February 15, 2008Slides by Mark Hancock

34. Example February 15, 2008Slides by Mark Hancock

35. Example SourceSS df MSFP between groups273.391 rows153.271 17.67<.01 columns120.061 13.84<.01 interaction0.061 0.01ns within groups (error) 312.31368.68 TOTAL585.7039 February 15, 2008Slides by Mark Hancock

36. Three-Way ANOVA February 15, 2008Slides by Mark Hancock

37. What’s different about a three-way ANOVA? February 15, 2008Slides by Mark Hancock

38. Example Factors: – Gender (Male, Female) – Screen Orientation (Horizontal, Vertical) – Screen Size (Small, Medium, Large) February 15, 2008Slides by Mark Hancock

39. Null Hypotheses Main effects – gender, screen orientation, screen size (no diff) Interactions (2-way) – gender × screen orientation – gender × screen size – screen orientation × screen size (no pattern) Interactions (3-way) – gender × screen orientation × screen size (?) February 15, 2008Slides by Mark Hancock

40. Three-way Alternate Hypothesis Interpretation #1: – The main effect of a factor depends on the levels of both of the other two factors Interpretation #2: – The interaction effect between two factors depends on the level of another February 15, 2008Slides by Mark Hancock

41. Level 1Level 2 Level 1 µ 111 µ 121 Level 2 µ 211 µ 221 Level 3 µ 311 µ 321 Factor 1 Factor 2 Level 1Level 2 Level 1 µ 112 µ 122 Level 2 µ 212 µ 222 Level 3 µ 312 µ 322 Factor 1 Factor 2 Level 1Level 2 Factor 3 February 15, 2008Slides by Mark Hancock

42. Factorial ANOVA Table Degrees of Freedom Sum of Squares Mean Square F Factor 1 Factor 2 Factor 3 Factor 1 × Factor 2 Factor 1 × Factor 3 Factor 2 × Factor 3 Factor 1 × Factor 2 × Factor 3 Within Groups Total February 15, 2008Slides by Mark Hancock

43. Between-Participants vs. Within-Participants February 15, 2008Slides by Mark Hancock

44. Participant Assignment Level 1Level 2 Level 1 N = 10 Level 2 N = 10 Level 3 N = 10 Factor 1 Factor 2 February 15, 2008Slides by Mark Hancock

45. Between Participants Level 1Level 2 Level 1 Group 1Group 2 Level 2 Group 3Group 4 Level 3 Group 5Group 6 Factor 1 Factor 2 February 15, 2008Slides by Mark Hancock

46. Within Participants Level 1Level 2 Level 1 Group 1 Level 2 Group 1 Level 3 Group 1 Factor 1 Factor 2 February 15, 2008Slides by Mark Hancock

47. Mixed Design Level 1Level 2 Level 1 Group 1Group 2 Level 2 Group 1Group 2 Level 3 Group 1Group 2 Factor 1 Factor 2 February 15, 2008Slides by Mark Hancock

48. How does this change the math? February 15, 2008Slides by Mark Hancock

49. T-test Independent Variance Paired Variance February 15, 2008Slides by Mark Hancock

50. One-Way ANOVA Independent Repeated Measures Degrees of Freedom Sum of Squares Mean SquareF Between Groups Within Groups Total Degrees of Freedom Sum of Squares Mean Square F Factor Subjects Error (Factor × Subjects) Total February 15, 2008Slides by Mark Hancock

51. Two-Way ANOVA Degrees of Freedom Sum of Squares Mean Square F Subjects Factor 1 Factor 1 × Subjects Factor 2 Factor 2 × Subjects Factor 1 × Factor 2 Factor 1 × Factor 2 × Subjects Total February 15, 2008Slides by Mark Hancock

52. Post-hoc Analysis February 15, 2008Slides by Mark Hancock

53. Main Effects Main effect for factor with two levels – No need to do post-hoc Main effect for factor with >2 levels – Same as one-way ANOVA – Pairwise t-tests February 15, 2008Slides by Mark Hancock

54. How do we (correctly) interpret the results when there’s an interaction effect? February 15, 2008Slides by Mark Hancock

55. Example There is a significant interaction between gender and technique. example answer: men were quicker with TreeMaps than with Phylotrees and ArcTrees, but women were quicker with ArcTrees than with Phylotrees and TreeMaps. February 15, 2008Slides by Mark Hancock

56. Example February 15, 2008Slides by Mark Hancock

57. Post-hoc Tests For each level of one factor – pairwise comparisons of each level of the other Hold level of one factor constant February 15, 2008Slides by Mark Hancock

58. Three-Way Interactions Much more difficult to interpret Same strategy: hold levels of two factors constant and perform pairwise comparisons Alternate strategy: don’t bother (use graphs) February 15, 2008Slides by Mark Hancock