1. 1 Psych 5510/6510 Chapter 14 Repeated Measures ANOVA: Models with Nonindependent ERRORs Part 2 (Crossed Designs) Spring, 2009

2. 2 Nonindependence in Crossed Designs Now we are going to look at crossed designs. Example: Each subject is measured once in both conditions (Experimenter Absent and Experimenter Present). Thus the effect of the independent variable is now showing up within-subjects.

3. 3 Design Experimenter AbsentExperimenter Present S1 S2 S3 S4 S5 S6 S7 S8

4. 4 SubjectY 1 : Exp. AbsentY 2 : Exp. Present 178 255 366 479 588 677 756 868 6.3757.125 Data: Note two scores per subject.

5. 5 Inappropriate Analysis SubjectYX (group) S17 S25 S36 S47 S58 S67 S75 S86 S181 S251 S361 S491 S581 S671 S761 S881 Ignore that there are two scores from each subject (one in each group). Contrast code group (X).

6. 6 Inappropriate Analysis Model C: Ŷ i = β o Ŷ i = 6.75 Model A: Ŷ i = β o + β 1 X i Ŷ i = 6.75 +.375X i

7. 7 Inappropriate Analysis (cont.) Ŷ i = 6.75 +.375X i Source bSSdfMSF*PREp SSRRegressionModel (X i ).3752.2531 1.52.10.238 SSE(A)ResidualError20.75141.58 SSE(C)Total 2315

8. 8 Residuals from inappropriate analysis.875-.3758 -1.125-1.3757 -.125.6256.8751.6255 1.875.6254 -1.125-.3753 -2.125-1.3752.875.6251 Exp. PresentExp. AbsentSubject Positive nonindependence

9. 9 Appropriate Approach Due to likely nonindependence among the scores from the same subject, the solution is once again to change the nonindependent scores into one score per person. Remember how we handled this last semester when we learned the t test for dependent groups, we computed a ‘difference’ score for each subject, reflecting how their score differed from the first measure to the second. We then analyzed the difference scores.

10. 10 From t Test for Dependent Groups SubjectY 1 : Exp. AbsentY 2 : Exp. PresentDifference 178 2550 3660 479-2 5880 6770 756 868-2 The ‘difference’ scores measure the effect of the independent variable on each subject, we then test to see whether the mean difference score differ significantly from zero.

11. 11 W 1 Scores We are going to do something very similar using the same formula as before but with different deltas. The deltas come from our contrast code (X=-1 and 1). We plug in the two scores for each subject to arrive at a W 1 score for each subject. The W 1 score for the first subject is shown below.

12. 12 W 1i Scores SubjectY 1 Exp. Absent Y 2 Exp. Present W 1i 178.707 255.0 366 4791.414 588.0 677 756.707 8681.414 Note that when the subject gets the same scores in both Y 1 and Y 2 that W 1i =0

13. 13 SubjectY 1 : Exp. AbsentY 2 : Exp. PresentDifferenceW1W1 178.707 25500 36600 479-2.1414 58800 67700 756.707 868-2.1414 W1 is a measure of the difference between the subjects’ two scores. If the independent variable had no effect the mean value of the W1 scores would be zero The reason the W1 scores have the opposite sign of the difference scores is simply because I used (-1 and 1) for the contrast rather then (1 and –1).

14. 14 Expected Value If we look at the mean value of W 1 across subjects we find it is: Which will equal 0 if there is no difference between the means of the two conditions. So….if the independent variable had no effect we would expect the mean of the W scores to equal zero…consequently…

15. 15 Approach We then do the multiple regression approach (Chapter 5) of testing to see if the mean of the variable we are modeling (i.e. W 1 ) is equal to some value (i.e. zero).

16. 16 The Models and Hypotheses Following the procedures of Chapter 5: Model C: Ŵ i = B o where B o =0 PC=0 Model A: Ŵ i = β o where β o = μ w PA=1 H 0 : β o = B o or μ w = 0 H A : β o  B o or μ w  0

17. 17 Computations p=.0479

18. 18 Appropriate Summary Table In the table above the value of b has been changed back to the metric of the original Y scores by dividing it by the denominator of the W formula (this is a convention). Compare this summary table to the inappropriate analysis, there is a huge drop in SSE(A) and SSE(C) when doing it this way (while SSR is the same in both approaches). Source bSSdfMSF*PREp SSRRegressionModel (X i ).3752.2531 5.73.45.048 SSE(A)ResidualError2.757.39 SSE(C)Total 58

19. 19 Why the Drop in Error? With the original Y scores the variance between the subjects within each group is part of the error that can’t be explained by the independent variable. With the W1 analysis the variance of the W1 scores is part of the error that can’t be explained by the independent variable. Remember that W1 scores measure the effect of the IV on each subject, in our example the IV had a pretty similar effect on everyone, thus the W1 scores didn’t vary much. So what can’t be explained by the independent variable is less with the W1 scores than with the Y scores (see next slide).

20. 20 SubjectY 1 : Exp. AbsentY 2 : Exp. PresentW1W1 178.707 2550 3660 479.1414 5880 6770 756.707 868.1414 The scores within Y1 and Y2 vary more than the scores within W1, thus the analysis of the W1 scores will be more powerful. This is common in repeated measures designs, that the effect of the independent variable (measured by W1) shows less variability than the differences between subjects (as reflected in their Y scores)

21. 21 The Error Term What is MS error in the summary table? 1) Model A is using the mean of W to predict each W score. 2) W measures the effect of the IV on each individual. 3) If the W scores differ from each other (i.e. differ from mean of W) then that is due to the IV having different effects on each individual, and there will be error in the model... Source bSSdfMSF*PREp SSRRegressionModel (X i ).3752.2531 5.73.45.048 SSE(A)ResidualError2.757.39 SSE(C)Total 58

22. 22 Thus... Thus the error of Model A reflects a difference in how the strength of the IV varies across various individuals, or in other words, the error of the model is the interaction between the treatment (IV) and the individual subjects.

23. 23 Full Summary Table for the Crossed Design The gray cells represent the analysis within subjects, what we just accomplished by using W scores, which is what we are really interested in. The white cells represent what we lost when we moved to W scores, they are included just to be complete. SS Total is the SS of all of the Y scores (including two per subject), SS BetweenS is found by SS Total – SS WithinS. The same goes for the df.

24. 24 More on Crossed Designs What if we have three levels to our independent variable and subjects are crossed with this variable? Group: a1Group: a2Group: a3 S1 S2 S3 S4 S5 S6

25. 25 Data SubjectGroup: a1Group: a2Group: a3 S1572 S2111410 S3293022 S4881 S5364238 S6151712 Note large within group variance.

26. 26 With three levels in our independent variable we are going to need two contrasts to completely code it. Let’s say we select: Contrast 1: (first group vs. other two groups combined) λ 11 = -2 λ 12 = 1 λ 13 = 1 Contrast 2: (second group vs. third group) λ 21 = 0 λ 22 = -1 λ 23 = 1

27. 27 Analyzing Contrast 1 Contrast 1: λ 11 = -2 λ 12 = 1 λ 13 = 1 Using SPSS you have it compute W1 scores, then analyze them to see if the mean of the W1 scores differs significantly from zero.

28. 28 Data SubjectGroup: a1Group: a2Group: a3W1 S1572-.41 S2111410.82 S3293022-2.45 S4881-2.86 S53642383.27 S6151712-.41 Does mean of W1 differ from zero?

29. 29 Contrast 1

30. 30 Contrast 1 Source SSdfMSF*PREp SSRRegressionModel (X i ) 0.691.137.027.726 SSE(A)ResidualError25.1455.028 SSE(C)Total 25.836 You could simply say PRE (or R²)=.027, p=.726, or you could express it in a summary table as seen below.

31. 31 Analyzing Contrast 2 Contrast 2: λ 11 = 0 λ 12 = -1 λ 13 = 1 Using SPSS you have it compute W2 scores, then analyze them to see if the mean of the W2 scores differs significantly from zero.

32. 32 Data SubjectGroup: a1Group: a2Group: a3W2 S1572-3.54 S2111410-2.83 S3293022-5.66 S4881-4.95 S5364238-2.83 S6151712-3.54 Does mean of W2 differ from zero?

33. 33 Contrast 2

34. 34 Contrast 2 Source SSdfMSF*PREp SSRRegressionModel (X i ) 90.791 67.25.93.0004 SSE(A)ResidualError6.7551.35 SSE(C)Total 97.546 You could simply say PRE (or R²)=.931, p=.0004, or you could express it in a summary table as seen below.

35. 35 Biases in Ignoring Nonindependence All these are taken care of by changing the data until you get just one score per person.

36. 36 Summary W 0 is used to come up with one score that represents (more or less) that subject’s average score. It is used to see how much the subjects differed from each other. Use in nested designs. W 1, W 2, etc., are used to measure the difference in the subject’s score across various contrasts (i.e. to see how the subject’s scores differed across various levels of the independent variable). Use in crossed designs.