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# Lecture slides stats1.13.l16.air

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### Lecture slides stats1.13.l16.air

1. 1. Statistics One Lecture 16 Analysis of Variance (ANOVA) 1
2. 2. Two segments •  One-way ANOVA •  Post-hoc tests 2
3. 3. Lecture 16 ~ Segment 1 One-way ANOVA 3
4. 4. Analysis of Variance (ANOVA) •  Appropriate when the predictors (IVs) are all categorical and the outcome (DV) is continuous –  Most common application is to analyze data from randomized controlled experiments 4
5. 5. Analysis of Variance (ANOVA) •  More specifically, randomized controlled experiments that generate more than two group means –  If only two group means then use: •  Independent t-test •  Dependent t-test 5
6. 6. Analysis of Variance (ANOVA) •  If more than two group means then use: –  Between groups ANOVA –  Repeated measures ANOVA 6
7. 7. Example •  Working memory training –  Four independent groups (8, 12, 17, 19) •  IV: Number of training sessions •  DV: IQ gain •  Null hypothesis: All groups are equal 7
8. 8. Working memory training 8
9. 9. Analysis of Variance (ANOVA) •  ANOVA typically involves NHST •  The test statistic is the F-test (F-ratio) –  F = (Variance between groups) / (Variance within groups) 9
10. 10. Analysis of Variance (ANOVA) •  Like the t-test and family of t-distributions •  The F-test has a family of F-distributions –  The distribution to assume depends on •  Number of subjects per group •  Number of groups 10
11. 11. Analysis of Variance (ANOVA) 11
12. 12. One-way ANOVA •  F-ratio F = between-groups variance / within-groups variance F = MSBetween / MSWithin F = MSA / MSS/A 12
13. 13. One-way ANOVA •  F = MSA / MSS/A •  MSA = SSA / dfA •  MSS/A = SSS/A/ dfS/A 13
14. 14. One-way ANOVA •  SSA = n Σ(Yj - 2 YT) •  Yj are the group means •  YT is the grand mean 14
15. 15. One-way ANOVA •  SSS/A = Σ(Yij - 2 Yj) •  Yij are individual scores •  Yj are the group means 15
16. 16. One-way ANOVA •  dfA = a - 1 •  dfS/A = a(n - 1) •  dfTOTAL = N - 1 16
17. 17. Summary Table Source SS df MS F MSA /MSS/A A n Σ(Yj - YT)2 a - 1 SSA/dfA S/A Σ(Yij - Yj)2 a(n -1) SSS/A/dfS/A ----- Total Σ(Yij - YT)2 N - 1 ----- ----- 17
18. 18. Effect size •  = (eta-sqaured) 2 = SS / SS •  η A Total 2 R 2 η 18
19. 19. Assumptions •  DV is continuous (interval or ratio variable) •  DV is normally distributed •  Homogeneity of variance •  Within-groups variance is equivalent for all groups –  Levene’s test 19
20. 20. Homogeneity of variance •  If Levene’s test is significant then homogeneity of variance assumption has been violated –  Conduct pairwise comparisons using a restricted error term 20
21. 21. Example •  Working memory training –  Four independent groups (8, 12, 17, 19) •  IV: Number of training sessions •  DV: IQ gain •  Null hypothesis: All groups are equal 21
22. 22. Working memory training 22
23. 23. Working memory training 23
24. 24. Working memory training 24
25. 25. Results from t-test: 12 vs. 17 25
26. 26. Segment summary •  ANOVA is used to compare means, typically in experimental research –  Categorical IV –  Continuous DV 26
27. 27. Segment summary •  ANOVA assumes homogeneity of variance –  Evaluate with Levene’s test 27
28. 28. Segment summary •  Post-hoc tests, such as Tukey’s procedure, allow for multiple pairwise comparisons without an increase in the probability of a Type I error 28
29. 29. END SEGMENT 29
30. 30. Lecture 16 ~ Segment 2 Post-hoc tests 30
31. 31. Post-hoc tests •  Post-hoc tests, such as Tukey’s procedure, allow for multiple pairwise comparisons without an increase in the probability of a Type I error 31
32. 32. Post-hoc tests •  Many procedures are available; the degree to which p-values are adjusted varies according to procedure –  Most liberal: No adjustment –  Most conservative: Bonferroni procedure 32
33. 33. NHST review Experimenter Decision Retain H0 H0 true Truth H0 false Reject H0 Correct Decision Type II error (Miss) Type I error (False alarm) Correct Decision 33
34. 34. NHST review Experimenter Decision Retain H0 H0 true Truth H0 false Reject H0 Correct Decision Type II error (Miss) Type I error p = .05 Correct Decision 34
35. 35. NHST review 35
36. 36. NHST review 36
37. 37. Working memory training 37
38. 38. Tukey’s procedure 38
39. 39. Results from t-test: 12 vs. 17 39
40. 40. Bonferroni procedure 40
41. 41. Comparison of procedures Procedure p-value for 12 vs. 17 Independent t-test 0.0067 Tukey 0.0327 Bonferroni 0.0402 41
42. 42. Post-hoc tests •  Post-hoc tests, such as Tukey’s procedure, allow for multiple pairwise comparisons without an increase in the probability of a Type I error •  Procedures vary from liberal to conservative 42
43. 43. END SEGMENT 43
44. 44. END LECTURE 16 44