1 G Lect 14a G Lecture 14a Examples of repeated measures A simple example: One group measured twice The general mixed model Independence of analysis units Adjustments to degrees of freedom Within subjects contrasts

2 G Lect 14a Examples of repeated measures Marsh et al. –Source monitoring at baseline vs 1 week followup Andersen’s transference paradigm –Within-subject contrasts of reactions to stimuli that are either generated using subject-specific information or using another subject’s information Adolescent pathways project –Self esteem of youth before, during, and after transition to middle school

3 G Lect 14a A simple example: One group measured twice Suppose we were interested in the tendency of subjects to attribute ideas to themselves vs others Let Y be a measure of source confusion such that Y 0 for attribution bias to self. Suppose Y is measured twice on a sample of 15 persons, once at baseline and once a week later. Using what we knew last week, how could we answer these two questions: –Is there attribution bias overall? –Does attribution bias change over a week?

4 G Lect 14a Example, continued Is there attribution bias overall? Does attribution bias change ?

5 G Lect 14a Example, continued Howell points out that only the F ratio for time is technically correct in this table. It is a fixed effect and it uses an error term that includes the interaction of subject by time The F ratio for subjects includes an ST interaction term in the denominator, but not in the numerator. –Thus, the MS/F/p values for subjects are only valid if there is no ST interaction; otherwise they are conservative (less likely to reject than they should be).

6 G Lect 14a The general mixed model The mixed model analysis generalizes to problems with n subjects and J time points –where µ is the grand mean –  i is the tendency for subject i to have either high or low scores on the average –  j is the tendency for measurement point j to have high or low scores –  ij is subject-time interaction –  ij is the residual Time is usually fixed and subjects are random, hence the mixed model Testing the variation of subjects is less of interest than assessing its magnitude

7 G Lect 14a Independence of analysis units Note that the “unit of analysis” in the repeated measures mixed model is not subjects, but subject-measurements. These are clearly not independent, since the same subject affects J different measurements. The ANOVA model assumes that the residuals are independent and identically distributed. This assumption has implications: –homoscedasticity The residuals will be independent if the only source of subject correlation is the level of response. –implies compound symmetry, whereby the covariances of the measures are equal.

8 G Lect 14a Patterns of correlation If the subjects differ in their course over time, or in their reactions to specific stimuli, then the inclusion of  i in the model will not account for correlation of the residuals. Often a pattern like the following is observed: Adjacent observations are more correlated. There is variability in the covariances as a result.

9 G Lect 14a Adjustments to degrees of freedom If residual degrees of freedom are to be pooled across different within-subject contrasts, then our assessment of how much new information we have may be biased. Greenhouse and Geisser presented a way of adjusting the degrees of freedom with a multiplicative factor, where. The  calculation depends on the variation of covariance elements. Instead of testing Time on –(J-1) and (n-1)(J-1) degrees of freedom. –Use (J-1) and (n-1)(J-1) At the extreme, it would be reduced to a test with df of the paired t test (1,n-1).

10 G Lect 14a Example: J=4, n=15 Using SPSS we calculate –Greenhouse-Geisser  =.79 df=(2.4, 33.4), –the less conservative Huynh-Feldt factor df=(2.9, 40.7),

11 G Lect 14a Within subjects contrasts When we compute a paired t-test with repeated measures data, we are calculating a planned contrast on the within-subjects design. Just as the degrees of freedom in a factorial between-subjects design are partitioned, the degrees of freedom on the within subjects design can also be partitioned. Possible contrasts: –Deviation: each level is compared to the average of all the others. (I-1 allowed) –Simple: Each level is compared to a standard reference group (either the first or the last) –Polynomial: Sequence of levels are decomposed into trends: linear, quadratic, cubic and so on