Statistics One
Lecture 16
Analysis of Variance (ANOVA)
1
Two segments
•  One-way ANOVA
•  Post-hoc tests

2
Lecture 16 ~ Segment 1
One-way ANOVA

3
Analysis of Variance (ANOVA)
•  Appropriate when the predictors (IVs) are all
categorical and the outcome (DV) is
continuo...
Analysis of Variance (ANOVA)
•  More specifically, randomized controlled
experiments that generate more than two
group mea...
Analysis of Variance (ANOVA)
•  If more than two group means then use:
–  Between groups ANOVA
–  Repeated measures ANOVA
...
Example
•  Working memory training
–  Four independent groups (8, 12, 17, 19)
•  IV: Number of training sessions
•  DV: IQ...
Working memory training

8
Analysis of Variance (ANOVA)
•  ANOVA typically involves NHST
•  The test statistic is the F-test (F-ratio)
–  F = (Varian...
Analysis of Variance (ANOVA)
•  Like the t-test and family of t-distributions
•  The F-test has a family of F-distribution...
Analysis of Variance (ANOVA)

11
One-way ANOVA
•  F-ratio

	

F = between-groups variance / within-groups variance	

	

F = MSBetween / MSWithin 	

	

F = ...
One-way ANOVA
•  F = MSA / MSS/A
•  MSA = SSA / dfA
•  MSS/A = SSS/A/ dfS/A
13
One-way ANOVA
•  SSA = n Σ(Yj -

2
YT)

•  Yj are the group means
•  YT is the grand mean

14
One-way ANOVA
•  SSS/A = Σ(Yij -

2
Yj)

•  Yij are individual scores
•  Yj are the group means

15
One-way ANOVA
•  dfA = a - 1
•  dfS/A = a(n - 1)
•  dfTOTAL = N - 1

16
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	...
Effect size
• 
= (eta-sqaured)
2 = SS / SS
•  η
A
Total
2
R

2
η

18
Assumptions
•  DV is continuous (interval or ratio variable)
•  DV is normally distributed
•  Homogeneity of variance
•  W...
Homogeneity of variance
•  If Levene’s test is significant then
homogeneity of variance assumption has
been violated
–  Co...
Example
•  Working memory training
–  Four independent groups (8, 12, 17, 19)
•  IV: Number of training sessions
•  DV: IQ...
Working memory training

22
Working memory training

23
Working memory training

24
Results from t-test: 12 vs. 17

25
Segment summary
•  ANOVA is used to compare means, typically
in experimental research
–  Categorical IV
–  Continuous DV

...
Segment summary
•  ANOVA assumes homogeneity of variance
–  Evaluate with Levene’s test

27
Segment summary
•  Post-hoc tests, such as Tukey’s procedure,
allow for multiple pairwise comparisons
without an increase ...
END SEGMENT

29
Lecture 16 ~ Segment 2
Post-hoc tests

30
Post-hoc tests
•  Post-hoc tests, such as Tukey’s procedure,
allow for multiple pairwise comparisons
without an increase i...
Post-hoc tests
•  Many procedures are available; the degree
to which p-values are adjusted varies
according to procedure
–...
NHST review
Experimenter Decision	

Retain H0	

H0 true	


Truth	

H0 false	


Reject H0	


Correct	

Decision	

Type II e...
NHST review
Experimenter Decision	

Retain H0	

H0 true	


Truth	

H0 false	


Reject H0	


Correct	

Decision	

Type II e...
NHST review

35
NHST review

36
Working memory training

37
Tukey’s procedure

38
Results from t-test: 12 vs. 17

39
Bonferroni procedure

40
Comparison of procedures
Procedure

p-value for 12 vs. 17

Independent t-test

0.0067

Tukey

0.0327

Bonferroni

0.0402

...
Post-hoc tests
•  Post-hoc tests, such as Tukey’s procedure,
allow for multiple pairwise comparisons
without an increase i...
END SEGMENT

43
END LECTURE 16

44
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  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
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