1. Repeated Measures ANOVA
Univariate and Multivariate Approaches

2. Setting t Treatments/Conditions to compare
N subjects to be included in study (each subject will receive only one treatment)
r subjects receive trt i: tr = N
p time periods of data will be obtained
Effects of trt, time and trtxtime interaction of primary interest.
Between Subject Factor: Treatment
Within Subject Factors: Time, TrtxTime

3. Model
Note the random error term is actually the interaction between subjects (within treatments) and time

4. Mean & Variance Structure
The second assumption (assuming equal covariances among repeated measures on subjects) is not always realistic and can be tested and adjusted for by multivariate approach.

5. Obtaining Variances of Sums & Means

6. Variances of Other Means

7. Analysis of Variance

8. Expected Values in Analysis of Variance

9. Expected Mean Squares

10. Tests for Fixed Effects

11. Comparing Treatment Means

12. Comparing Time Means

13. Comparing Treatment Means @ 1 Time
Approximate degrees of freedom on next slide

14. Approximate Degrees of Freedom (Satterthwaite)

15. Multivariate Approach
Makes use of Multivariate ANOVA
String out each individual’s p measurements into a px1 vector
The Variance-Covariance matrix among measurements on the same subject is assumed to have common variances on the main diagonal and common covariances off-diagonal (Compound Symmetry)
Huynh-Feldt condition (less rigid):

16. Mauchley Test

17. Adjusted Degrees of Freedom – Within Trt