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Repeated Measures ANOVA

Repeated Measures ANOVA. Univariate and Multivariate Approaches. 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

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Repeated Measures ANOVA

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  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

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