Repeated Measures ANOVA Univariate and Multivariate Approaches Setting

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 • Effects of trt, time and trtxtime interaction of primary interest. – Between Subject Factor: Treatment – Within Subject Factors: Time, Trtx. Time

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

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.

Obtaining Variances of Sums & Means

Variances of Other Means

Analysis of Variance

Expected Values in Analysis of Variance

Expected Mean Squares

Tests for Fixed Effects

Comparing Treatment Means

Comparing Time Means

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

Approximate Degrees of Freedom (Satterthwaite)

Multivariate Approach • Makes use of Multivariate ANOVA • String out each individual’s p measurements into a px 1 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):

Mauchley Test

Adjusted Degrees of Freedom – Within Trt