1. Anthony Greene1 Advanced ANOVA 2-Way ANOVA Complex Factorial Designs I.The Factorial Design II.Partitioning The Variance For Multiple Effects III.Independent Main Effects of Factor A and Factor B IV.Interactions

2. Anthony Greene2 The Source Table Keeps track of all data in complex ANOVA designs Source of SS, df, and Variance (MS) –Partitioning the SS, df and MS –All variability is attributable to effect differences or error (all unexplained differences) Total Variability Effect Variability (MS Between) Error Variability (MS Within)

3. Anthony Greene3 Partitioning of Variability for Two-Way ANOVA Total Variability Effect Variability (MS Between) Error Variability (MS Within) Factor A Variability Stage 1 { { Stage 2 Factor B Variability Interaction Variability

4. 4 Source Table for 1-Way ANOVA Effect Variability Error Variability

5. 5 2-Way ANOVA Used when two variables (any number of levels) are crossed in a factorial design Factorial design allows the simultaneous manipulation of variables A1A2A3A4 B1A1B1A2 B1A3 B1A4 B1 B2A1 B2A2 B2A3 B2A4 B2

6. 6 2-Way ANOVA For Example: Consider two treatments for mood disorders 1.This design allows us to consider multiple variables 2.Importantly, it allows us to understand Interactions among variables PlaceboProzacZanexBourbon DepressionA1B1A2 B1A3 B1A4 B1 AnxietyA1 B2A2 B2A3 B2A4 B2

7. 7 2-Way ANOVA Hypothetical Data: 1.You can see that the effects of the drug depend upon the disorder 2.This is referred to as an Interaction PlaceboProzacZanexBourbon Depression-2.30.2-1.1-3.2 Anxiety-2.0-0.11.3-1.6

8. 8 Example of a 2-way ANOVA: Main Effect A Daytime Heart rate Nighttime Heart rate No-Meditation7562 Mediation7463

9. 9 Example of a 2-way ANOVA: Main Effect B Daytime Heart rate Nighttime Heart rate No-Meditation7574 Mediation6463

10. 10 Example of a 2-way ANOVA: Main Effect A & B Daytime Heart rate Nighttime Heart rate No-Meditation8071 Mediation7160

11. 11 Example of a 2-way ANOVA: Interaction Daytime Heart rate Nighttime Heart rate No-Meditation7562 Mediation6563

12. Anthony Greene12 Partitioning of Variability for Two-Way ANOVA Total Variability Effect Variability (MS Between) Error Variability (MS Within) Factor A Variability Stage 1 { { Stage 2 Factor B Variability Interaction Variability

13. Anthony Greene13 Partitioning of Variability for Two-Way ANOVA Total Variability Effect Variability (MS Between) Error Variability (MS Within) Factor A Variability Stage 1 { { Stage 2 Factor B Variability Interaction Variability Numerator for Omnibus F-ratio Denominator for all F-ratios

14. Anthony Greene14 Partitioning of Variability for Two-Way ANOVA Total Variability Effect Variability (MS Between) Error Variability (MS Within) Factor A Variability Stage 1 { { Stage 2 Factor B Variability Interaction Variability Numerator for Factor A F-ratio Denominator for F-ratio

15. 15 Partitioning of Variability for Two-Way ANOVA Total Variability Effect Variability (MS Between) Error Variability (MS Within) Factor A Variability Stage 1 { { Stage 2 Factor B Variability Interaction Variability Numerator for Factor B F-ratio Denominator for F-ratio

16. 16 Partitioning of Variability for Two-Way ANOVA Total Variability Effect Variability (MS Between) Error Variability (MS Within) Factor A Variability Stage 1 { { Stage 2 Factor B Variability Interaction Variability Numerator for Interaction F-ratio Denominator for F-ratio

17. 17 2 Main Types of Interactions

18. Anthony Greene18 Simple Effects of An Interaction

19. Anthony Greene19 Simple Effects of An Interaction

20. Anthony Greene20 Simple Effects of An Interaction

21. Anthony Greene21 Simple Effects of An Interaction

22. Anthony Greene22 Simple Effects of An Interaction

23. Anthony Greene23 Simple Effects of An Interaction

24. Anthony Greene24 Simple Effects of An Interaction

25. Anthony Greene25 Simple Effects of An Interaction

26. Anthony Greene26 Simple Effects of An Interaction

27. Anthony Greene27

28. Anthony Greene28

29. Anthony Greene29

30. Anthony Greene30 +

31. Anthony Greene31 How To Make the Computations A1A2 B1153153 374374 B2254254 324324 A1A2Row Tot B1T SS T B1 B2T SS T B2 Col Tot. T A1 T A2

32. Anthony Greene32 A1A2Row Total B1T SS T B1 B2T SS T B2 Col Total T A1 T A2

33. Anthony Greene33 Higher Level ANOVA N-Way ANOVA: Any number of factorial variables may be crossed; for example, if you wanted to assess the effects of sleep deprivation: 1. Hours of sleep per night: 4, 5, 6, 7, 8 2. Age: 20-30, 30-40, 40-50, 50-60, 60-70 3. Gender: M, F You would need fifty samples

34. Anthony Greene34 Higher Level ANOVA Mixed ANOVA: Any number of between subjects and repeated measures variables may be crossed For example, if you wanted to assess the effects of sleep deprivation using sleep per night as the repeated measure: 1. Hours of sleep per night: 4, 5, 6, 7, 8 2. Age: 20-30, 30-40, 40-50, 50-60, 60-70 3. Gender: M, F You would need 10 samples

35. Anthony Greene35 How to Do a Mixed Factorial Design Total Variability Effect Variability (MS Between) MS Within Individual Variability Error Variability Stage 1 { { Stage 2 Factor A Variability Interaction Variability Factor B Variability

36. Anthony Greene36 Two-Way ANOVA An experimenter wants to assess the simultaneous effects of having breakfast and enough sleep on academic performance. Factor A is a breakfast vs. no breakfast condition. Factor B is three sleep conditions: 4 hours, 6 hours & 8 hours of sleep. Each condition has 5 subjects.

37. Anthony Greene37 Two-Way ANOVA Sourced.f.SSMSF Between60 Main A5 Main B A x B30 Within2 Total Factor A has 2 levels, Factor B has 3 levels, and n = 5 (i.e., six conditions are required and each has five subjects). Fill in the missing values.

38. Anthony Greene38 Two-Way ANOVA Sourced.f.SSMSF Between560 Main A15 Main B2 A x B30 Within2 Total First the obvious: The degrees freedom for A and B are the number of levels minus 1. The degrees freedom Between is the number of conditions (6 = 2x3) minus 1.

39. Anthony Greene39 Two-Way ANOVA Sourced.f.SSMSF Between560 Main A15 Main B2 A x B230 Within2 Total The interaction (AxB) is then computed: d.f. Between = d.f. A + d.f. B + d.f. AxB. OR d.f. AxB = d.f. A  d.f. B

40. Anthony Greene40 Two-Way ANOVA Sourced.f.SSMSF Between560 Main A15 Main B2 A x B230 Within242 Total29 d.f. Within = Σd.f. each cell d.f. Total = N-1 = 29. d.f. Total = d.f. Between + d.f. Within

41. Anthony Greene41 Two-Way ANOVA Sourced.f.SSMSF Between56012 Main A15 Main B2 A x B230 Within242 Total29 Now you can compute MS Between by dividing SS by d.f.

42. Anthony Greene42 Two-Way ANOVA Sourced.f.SSMSF Between56012 Main A110 5 Main B2 A x B230 Within242 Total29 You can compute MS A by remembering that F A = MS A MS Within, so 5 = ?/2. SS A is then found by remembering that MS = SS df, so 10 = ?/1

43. Anthony Greene43 Two-Way ANOVA Sourced.f.SSMSF Between56012 Main A110 5 Main B22010 A x B23015 Within242 Total29 Now SS B is computed by SS A + SS B + SS AxB = SS Between MS B = SS B /df B and MS AxB = SS AxB /df AxB

44. Anthony Greene44 Two-Way ANOVA Sourced.f.SSMSF Between56012 Main A110 5 Main B22010 A x B23015 Within24482 Total29 MS Within =SS Within /df Within, solve for SS.

45. Anthony Greene45 Two-Way ANOVA Sourced.f.SSMSF Between560126 Main A110 5 Main B220105 A x B230157.5 Within24482 Total29 Now Solve for the missing F’s (Between, B, AxB). F=MS/MS Within

46. Anthony Greene46 Two-Way ANOVA An experimenter is interested in the effects of efficacy on self-esteem. She theorizes that lack of efficacy will result in lower self-esteem. She also wants to find out if there is a different effect for females than for males. She conducts an experiment on a sample of college students, half male and half female. She then puts them through one of three experimental conditions: no efficacy, moderate efficacy, and high efficacy. Then she measures level of self-esteem. Her results are below. Conduct a two-way ANOVA. Report all significant findings with α= 0.05.

47. Anthony Greene47 Data No Moderate High EfficacyEfficacyEfficacy 147 Males 3 88 0710 Females210 16 57 13 4 8 15

48. Anthony Greene48 No ModerateHigh Efficacy Efficacy Efficacy 1 T=44 T=197 T=25 Males3 SS=4.68 SS=8.68 SS=4.7 0710T m = 48 Females2 T=1110 T=25 16 T=44 5 SS=4.6 7 SS=4.713 SS=4.7 T f = 80 4815 T ne =15 T me =44 T he =69 n=3 k=6 N=18 G=128 ∑x 2 =1260

49. Anthony Greene49 SS between SS btw = ∑T 2 /n – G 2 /N SS btw = (4 2 + 19 2 + 25 2 + 11 2 + 25 2 + 44 2 )/3 –28 2 /18 SS btw = 1228-910.2=317.8

50. Anthony Greene50 SS sex, SS efficacy, SS interaction SS sex = ∑T 2 sex/nsex – G 2 /N SS sex = (48 2 + 80 2 )/9 – 910.2 SS sex = 56.9 SS efficacy = ∑T 2 e/ne– G2/N SS efficacy = (15 2 + 44 2 + 69 2 )/6 – 910.2 SS efficacy = 243.47 SS interaction = SS between – SS sex – SS efficacy SS interaction = 317.8-56.9-243.47 SS interaction = 17.43

51. Anthony Greene51 SS within and SS total SS within = ∑SS SS within =4.6+8.6+4.7+4.6+4.7+4.7=31.9 SS total = ∑x2 – (∑x)2/N SS total = 1260 – 910.2 SS total = 349.8

52. Anthony Greene52 Degrees Freedom df btw = cells – 1 = k-1 df sex = rows - 1 df eff = columns - 1 df int = df btw – df sex - df eff df win = Σdf each cell = df tot -df btw df tot = N-1 = nk-1

53. Anthony Greene53 Degrees Freedom df btw = cells – 1 = k-1 = 5 df sex = rows – 1 = 1 df eff = columns – 1 = 2 df int = df btw – df sex – df eff = df sex  df eff = 2 df win = Σdf each cell = df tot -df btw = 12 df tot = N-1= nk-1= 17

54. 54 Source Table Source SS df MS F Fcrit Between 317.8 Sex 56.9 Efficacy 243.5 Int. 17. 4 Within 31. 9 Total 349.8

55. 55 Source Table Source SS df MS F Fcrit Between 317.8 5 Sex 56.9 1 Efficacy 243.5 2 Int. 17. 4 2 Within 31. 9 12 Total 349.8 17

56. 56 Source Table Source SS df MS F Fcrit Between 317.8 5 63.6 Sex 56.9 1 56.9 Efficacy 243.5 2 121.7 Int. 17. 4 2 8.7 Within 31. 9 12 2.7 Total 349.8 17

57. 57 Source Table Source SS df MS F Fcrit Between 317.8 5 63.6 23.5 F(5,12)=3.11 Sex 56.9 1 56.9 21.4 F(1,12)=4.75 Efficacy 243.5 2 121.7 45.8 F(2,12)=3.88 Int. 17. 4 2 8.7 3.3 F(2,12)=3.88 Within 31. 9 12 2.7 Total 349.8 17

58. 58 Source Table Source SS df MS F Fcrit Between 317.8 5 63.6 23.5 F(5,12)=3.11 Sex 56.9 1 56.9 21.4 F(1,12)=4.75 Efficacy 243.5 2 121.7 45.8 F(2,12)=3.88 Int. 17. 4 2 8.7 3.3 F(2,12)=3.88 Within 31. 9 12 2.7 Total 349.8 17 1 main effect for sex 2 main effect for efficacy 3 no significant interaction