Repeated Measures Analysis of Variance

Transcription

1 Repeated Measures Analysis of Variance

2 Review Univariate Analysis of Variance Group A Group B Group C

3 Repeated Measures Analysis of Variance Condition A Condition B Condition C

4 Repeated Measures Analysis of Variance Day 1 Day 2 Day 3

5 Basic Logic of RM ANOVA Hypothesis Testing H o : υ 1 = υ 2 = υ 3 H 1 : υ 1 υ 2 υ 3 (at least one difference)

6 Basic Logic of RM ANOVA F = MS conditions MS error Variance explained by treatment Variance explained by error

7 Basic Logic of RM ANOVA F = MS conditions MS error Note, Msconditions is the same as Ms between, whats different is the error term.

8 Between Subjects ANOVA Unexplained Variance (Error) Explained Variance Within Subjects ANOVA (Repeated Measures) Subjects Variance Unexplained Variance (Error) Explained Variance

9 Recall, between subjects ANOVA SS total = SS between + SS within Deviations of group means from the grand mean Deviations of subject scores from the cell mean

10 SS total = SS conditions + SS subjects + SS error Deviations of condition MEANS from the grand mean Deviations of subject MEANS from the grand mean

11 An Example

12 Repeated Measures ANOVA Condition P One Two Three

13 Repeated Measures ANOVA Condition P One Two Three x x x

14 Recall SS = (x x GM ) 2

15 SS conditions SS = n (x x ) 2 conditions conditions.. SS conditions = 5[( ) ] SS conditions =10.533

16 SS subjects SS = k (x x ) 2 subjects subjects.. SS subjects = 3[( ) ] SS subjects = 2.266

17 SS error SS error = SS total - SS conditions - SS subjects SS error = SS error = 6.133

18 Recall MS = SS df

19 Degrees of Freedom Condition P One Two Three df total = N - 1

20 Degrees of Freedom Condition P One Two Three df conditions = k - 1

21 Degrees of Freedom Condition P One Two Three df subjects = n - 1

22 Degrees of Freedom Condition P One Two Three df error = df conditions * df subjects

23 Repeated Measures ANOVA Summary Table Source df SS MS F Subjects n-1 SS subjects Conditions k-1 SS conditions SS conditions df conditions MS conditions MS error Error (n-1)*(k-1) SS error SS error df error Total N-1 SS total

24 Repeated Measures ANOVA Summary Table Source df SS MS F Subjects Conditions Error Total

25 Post-Hoc Comparisons: Simple Effects Analysis

26 Repeated Measures ANOVA Condition P One Two Three

27 Repeated Measures ANOVA Condition P One Two Three

28 Repeated Measures ANOVA Condition P One Two Three

29 Assumptions of repeated measures 1. Normality ANOVA

30 Assumptions of repeated measures 1. Normality ANOVA

31 Assumptions of repeated measures ANOVA 2. Homogeneity of Variance σ 1 2 = σ 2 2 = σ 3 2

32 3. The Assumption of Sphericity Correlations among pairs of variables are equal NO!

33 Sphericity Sphericity is the property that the covariance of the difference scores of the IV levels are same Violations generally lead to inflated F statistics (and hence inflated Type I error).

34 Sphericity Maulchy s Test

35 What does it mean? Effect DFn DFd F p ges condition * Mauchly's Test for Sphericity Effect W p condition Sphericity Corrections Effect GGe p[gg] HFe p condition

36 Okay, if the sphericity test is not significant Keep on going

37 Okay, if the sphericity test is significant 1) Check epsilon. The epsilon means the departure from the sphericity, in other words, how far the data is from the ideal sphericity. The epsilon is a number between 0 and 1, if the epsilon is equal to 1, the data have sphericity.

38 Look at $ANOVA Effect DFn DFd F p p<.05 ges 2 condition * $`Mauchly's Test for Sphericity` Effect W p p<.05 2 condition $`Sphericity Corrections` Effect GGe p[gg] p[gg]<.05 HFe p[hf] p[hf]<.05 2 condition * *

39 Which one should I look at? Generally, Greenhouse-Geisser. BUT if GG epsilon > 0.75 USE Huynh-Feldt. WHY? GG tends to be too strict when epsilon is large. ALSO, use Huynh-Feldt when n is small (less than 15)

40 What do I do with it? The test provides you with the corrected p value.

41 Look at $ANOVA Effect DFn DFd F p p<.05 ges 2 condition * $`Mauchly's Test for Sphericity` Effect W p p<.05 2 condition $`Sphericity Corrections` Effect GGe p[gg] p[gg]<.05 HFe p[hf] p[hf]<.05 2 condition * *

42 But you also have to Correct df s (effect and error term) Multiply then by Epsilon: 2 * = 1.885

43 Assumptions of repeated measures ANOVA

44

45 But what is SPERICITY?

46 Variance s 2 = (X X) 2 N 1

47 Covariance The degree to which two variables vary together. COV xy = (x x)(y y) N 1

48 Covariance The degree to which two variables vary together COV = 1.25 COV = 0 COV = COV = 5.875

49 Assumptions of repeated measures 3. Sphericity ANOVA Condition One Two Three Four One S 2 1 Two S 2 2 Three S 2 3 Four S 2 4

50 Assumptions of repeated measures 3. Sphericity ANOVA Condition One Two Three Four One S 1 2 S 12 S 13 S 14 Two S 21 S 2 2 S 23 S 24 Three S 31 S 32 S 2 3 S 34 Four S 41 S 42 S 43 S 2 4

51 Assumptions of repeated measures ANOVA 3. Sphericity: Compound Symmetry Condition One Two Three Four One S 1 2 S 12 S 13 S 14 Two S 21 S 2 2 S 23 S 24 Three S 31 S 32 S 2 3 S 34 Four S 41 S 42 S 43 S 2 4 The variances AND covariances are equal

52 Assumptions of repeated measures ANOVA 3. Sphericity: Difference Scores P Condition C1-C2 C1-C3 C1-C4 1 x 11 -x 12 x 11 -x 13 x 11 -x 14 2 x 21 -x 22 x 21 -x 23 x 21 -x 24 3 x 31 -x 32 x 31 -x 33 x 31 -x 34 4 x 41 -x 42 x 41 -x 43 x 41 -x 44 The variances of the difference scores are equal

53 Assumptions of repeated measures ANOVA 3. Sphericity: Covariance Matrix S 2 x y = S x 2 + S y 2 2S xy The variances of the difference scores are equal

54 Factorial Repeated Measures ANOVA

55 An Example Participants in an experiment are asked to perform a cued reaction time task when they are alert and when they are fatigued. As such, you have participants performing a reaction time task with three conditions (valid cue, no cue, invalid cue) when they are either alert or fatigued.

56 An Example Main Effect: Fatigue (Alert, Fatigued) Main Effect: Condition (Valid Cue, No Cue, Invalid Cue) Interaction: Fatigue x Condition

57 Reaction Time (ms) Alert Condition Fatigued

58 Reaction Time (ms) Valid None Cue Invalid

59 Reaction Time (ms) Valid None Cue Invalid