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# ANOVA approaches to Repeated Measures

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ANOVA approaches to Repeated Measures univariate repeated measures ANOVA chapter 2 repeated measures MANOVA chapter 3 Assumptions Interval measurement and normally distributed errors homogeneous across groups transformation may help Group comparisons estimation and comparison of group means not informative about individual growth Fixed time points time treated as classification variable timepoints can be evenly or unevenly spaced 1 Better suited for short term or moderate change esp true for trend or growth curve analysis LS techniques adversely affected by outliers Missing data ANOVA can handle some unbalanced ANOVA MANOVA can t handle any Variance covariance structure for y i compound symmetry for ANOVA unstructured for MANOVA 2 Univariate Repeated Measures ANOVA One sample case randomized block ANOVA yij i j eij i 1 N j 1 n grand mean i individual difference component for subject i constant over time j effect of time same for all subjects eij error for subject i and time j Random components i N 0 2 subjects random between subjects variance eij N 0 e2 within subjects variance 3 Data layout timepoint 2 subject 1 1 y11 y12 y1n 2 y21 y22 y2n N yN 1 yN 2 yN n n one observation per cell similar to randomized block design with subject as block identical to paired t test when n 2 4 Model assumptions n X j 1 j 0 n 1 df for Time E yij j V yij V j i eij 2 e2 C yij yi0j 0 for i 6 i0 C yij yij 0 2 for j 6 j 0 5 different subjects same subject Intraclass correlation 2 Corr yij yij 0 2 e2 same correlation for all longitudinal pairs average correlation of y from any two timepoints ranges from 0 to 1 so more like a proportion of variance proportion of total variance that is attributable to subjects that is not accounted for by Time 6 Compound symmetry structure of variance covariance matrix y i 2 2 e 2 2 2 2 2 2 e2 2 2 2 e2 2 2 variance homogeneous across time e2 2 covariances homogenous across time 2 correlation 2 e2 2 7 2 2 2 e2 2 Not too realistic variances often change over

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