1. 1 Psych 5510/6510 Chapter 14 Repeated Measures ANOVA: Models with Nonindependent Errors Part 1 (Crossed Designs) Spring, 2009

2. 2 Independence of Residuals All the analyses we have performed so far have assumed that the errors in our predictions (i.e. Y i -Ŷ i ), also known as the residuals, were independent of each other. Residuals are said to be independent if knowing one score’s residual does not help you predict another score’s residual.

3. 3 Independence of Residuals For example, if Fred and Mary were two participants in an experiment, then their residuals are independent if the error in predicting Fred’s score is unrelated to (i.e. can’t be used to predict) the error in predicting Mary’s score. In an earlier lecture on the assumptions of the model comparison approach this was identified as perhaps the most important assumption not to violate.

4. 4 Nonindependence of Residuals There are experimental designs, however, in which the residuals will not be independent. For example, if we measured Fred twice (as if he were two different subjects in the experiment). This is known as a repeated measures (measuring the same person more than once) or a within-subjects design.

5. 5 Nonindependence of Residuals Another example of an experiment that lacks independence of residuals is one where there is some connection between people’s scores. For example, if we are measuring marital satisfaction and Fred and Mary are a married couple, then there is likely to be some connection between Fred’s scores and Mary’s scores.

6. 6 Positive Nonindependence Remember that in a two-group design, the prediction of a person’s score is the mean of the group they are in. In the following chart if one spouse’s rating of marital satisfaction is below the mean then the other spouse’s rating is also likely to be below the mean. If one spouse’s rating is above the mean the other is also likely to be above the mean. This makes the scores within the couples more similar than the scores between the couples, which is called positive nonindependence.

7. 7 Negative Nonindependence In the following chart if one spouse’s rating of housework performed is below the mean then the other spouse’s rating is likely to be above the mean. If one spouse’s rating is above the mean the other is also likely to be below the mean. This makes the scores within the couples more dissimilar than the scores between the couples, which is called negative nonindependence.

8. 8 Nested Design In a nested design scores are nonindependent with other scores in the same group. Note that differences between Group 1 and Group 2 are ‘between subjects’ (i.e. different subjects in each group). s1,s1,s1 represents three scores on the dependent variable from the same person, or from the same family, or from the same committee, or any three scores that are likely to be nonindependent.

9. 9 Crossed Design In a crossed design scores are nonindependent with other scores in a different group. Note that differences between Group 1 and Group 2 are ‘within-subjects’ (i.e. same subjects in each group). s1,s1 represents two scores on Y from the same subject, or from the same couple, or from the same family, or any two scores that are likely to be nonindependent....

10. 10 Example of a Nested Design Dependent variable: number of puzzles a subject can complete in a set period of time. Independent variable: each subject assigned either to a condition where the experimenter is watching or where the experimenter is absent. Repeated measure: each subject is given two sets of puzzles to solve.

11. 11 Design

12. 12 Data Y hi h is which observation for that subject, i is which subject (i.e. Y 25 = 4) n = number of subjects and s = number of scores per subject. n=6, s=2, so the total number of observations = ns = 12

13. 13 Inappropriate Analysis First let’s analyze the data in an inappropriate way by ignoring the lack of independence among scores within each group, and simply analyze the data as a two-group design with six scores in each group. The categorical independent variable X i (experimenter present/absent) will be contrast coded, ‘1’ if present, ‘-1’ if absent.

14. 14 SPSS Setup SubjectYX (group) S18 S17 S29 S29 S35 S36 S431 31 S551 41 S631 41 Note each subject appears twice, violating assumption of independence.

15. 15 Inappropriate Analysis (cont.) Model C: Ŷ i = β o Ŷ i = 5.5 SSE(C) = 57.00 Model A: Ŷ i = β o + β 1 X i Ŷ i = 5.5 + 1.83X i SSE(A) = 16.67 The unweighted mean of the two group means equals 5.5, the difference between the two group means equals 2*1.83=3.66.

16. 16 Inappropriate Analysis (cont.) The difference in mean scores between exp present and exp absent is statistically significant. Source bSSdfMSF*PREp SSRRegressionModel (X i ) 1.8340.331 24.2.71.0006 SSE(A)ResidualError16.67101.67 SSE(C)Total 57.0011

17. 17 Residuals Appears to be positive nonindependence among residuals (as for the most part if one residual is below the mean the other is too, if one residual is above the mean the other is too.

18. 18 Basic Strategy for Repeated Measures If you have scores that are nonindependent (e.g. two or more scores from the same person), then turn them into one score. For example, we could make everyone’s score on the dependent variable be their mean score on the two puzzle sets (turning their two scores into just one...their mean score).

19. 19 Mean score as D.V. Analyzing the mean score of each subject.

20. 20 Setup SubjectYMYM X (group) S17.5 S29 S35.5 S431 S54.51 S63.51 Model C: Ŷ Mi = β o Model A: Ŷ Mi = β o + β 1 X i

21. 21 Summary Table: Analysis of Mean Score Ŷ Mi =5.5 +1.83X i Note the value of p has changed compare to that of the inappropriate analysis (which had a p=.0006) Source (text) Source (SPSS) SourcebSSdfMSF*PREp SSRRegressionModel (X i ) 1.8320.171 11.73.0295 SSE(A)ResidualError7.3341.83 SSE(C)Total 27.55

22. 22 A Better Way A similar, but better, way to combine the nonindependent scores into one score is to use: Where δ h (delta) is the weight you give an individual score for that subject.

23. 23 I know......the formula looks terrible but it is quite simple. For nested designs we will use δ h = 1 for each score from the subject. When we do that we will symbolize it as W 0i Thus the single score for subject one would be:

24. 24 More Examples If we had 3 scores per subject it would like like this (say the 3 rd score for subject one was 10): If we had an a priori reason for having the score on the second set count twice as much as the score on the first set (perhaps the second set had twice as many puzzles to solve) we could use:

25. 25 Note This formula is pretty close to what we would do to get the mean score for the subject, the difference lies in the denominator. The value of doing it as ‘W’ will become clear later.

26. 26 W oi as our Dependent Variable Note we only have 6 scores now, not 12, and they are independent. The W oi Scores

27. 27 Setup SubjectW oi X (group) S110.61 S212.76 S37.78 S44.241 S56.361 S64.951 Model C: Ŵ i = β o Model A: Ŵ i = β o + β 1 X i Rather than modeling each person’s two Y scores, we are modeling their W 0 score which combines their performance on both puzzle sets.

28. 28 Analysis using W 0i If we regress W oi on X i we get: Ŵ oi = 7.78 + 2.59X i b 0 =7.78 is the unweighted mean value of Ŵ oi b 1 =2.59, (2.59)(2)=5.18 is the difference between the mean values of Ŵ oi in the two groups. If we divide the coefficients by the denominator of W oi, then we get back to meaningful metrics in terms of Y. Ŷi=5.5 +1.83Xi, which is the same regression formula we arrived at through the inappropriate analysis of regression Y on X. The difference from doing it appropriately has to do with getting the correct SS and p value (see next slide).

29. 29 Summary Table Analysis of W oi (compare to inappropriate analysis..repeated on next slide) The convention is to translate the b values in the table back into the original measure (Y), thus the value for b 1 given above is 1.83 rather than 2.59 (see the previous slide for how to translate). Source (text) Source (SPSS) SourcebSSdfMSF*PREp SSRRegressionModel (X i ) 1.8340.331 11.73.0295 SSE(A)ResidualError14.6743.67 SSE(C)Total 555

30. 30 Inappropriate Analysis (Repeated) Source bSSdfMSF*PREp SSRRegressionModel (X i ) 1.8340.331 24.2.71.0006 SSE(A)ResidualError16.67101.67 SSE(C)Total 57.0011

31. 31 Where’d it go? The SS error and SS total dropped by 2 when we went from the inappropriate analysis (with all 12 Y i scores) to the appropriate analysis (with 6 w 0i scores). This was the within-subjects variance (e.g. the variance of the two scores for subject 1, and the variance of the two scores for subject 2...) that was lost when we went from two scores per person to one score per person.

32. 32 Full Summary Table for the Nested Design The shaded area is the area of interest, the full table expresses the analysis of ALL of the data, including the nonindependent repeated measures. SS Total is the SS of ALL of the scores (just compute SS of all the Y scores) and SS WithinS is the SS within the repeated measures (computed by finding SS Total – SS BetweenS ). See next slide.

33. 33 SS’s SS total is SS of all the Y scores (8.....4) SS within-subjects is SS within each subject (8 and 7, 9 and 9, 5 and 6, etc.). SS between-subjects is the SS of the W 0 scores (10.61...4.95). SS model is how much the W 0 scores can be modeled by knowing which group the subject was in (X). SS error is how much the W 0 scores can not be modeled by knowing which group the subject was in (X).