1. B AD 6243: Applied Univariate Statistics Repeated Measures ANOVA Professor Laku Chidambaram Price College of Business University of Oklahoma

2. BAD 6243: Applied Univariate Statistics 2 What is a Repeated Measures Design? A repeated measures design involves measuring subjects at different points in time (typically after different treatments) It can be viewed as an extension of the paired-samples t-test (which involved only two related measures) Thus, the measures—unlike in “regular” ANOVA—are correlated, i.e., the observations are not independent In practical terms, we expect the correlations between these variables to be roughly equal In technical terms, we test for “sphericity”

3. BAD 6243: Applied Univariate Statistics 3 Understanding Sphericity Sphericity is a special case of compound symmetry, which requires that variances across conditions be equal and that co-variances between pairs of conditions also be equal Sphericity only requires that the differences between pairs of conditions have equal variance It is measured using Mauchly’s test, which if significant indicates that the assumption of sphericity has not been met In such cases, an appropriate correction—either a conservative one (G) or a liberal one (H)— needs to be applied to the degrees of freedom

4. BAD 6243: Applied Univariate Statistics 4 Points of Departure Interpreting results of simple repeated measures ANOVA is generally similar to the case of between subjects ANOVA, but there are some differences: –Notably, no error term exists for testing between-group effects –In other words, the error term cannot be separated from the subject x time interaction effects; in fact, the error term is the interaction, and is so calculated –Thus, the total variance is divided into three components: (a) variation among subjects; (b) variation across time(s) and (c) residual variation

5. BAD 6243: Applied Univariate Statistics 5 Partitioning Variations SubjectsT1T2T3Mean A2453.67 B5676.00 C3343.33 D1242.33 E2343.00 Mean2.603.604.80 SS Total = SS Time + SS Subjects + SS Residual

6. BAD 6243: Applied Univariate Statistics 6 Interpreting the Results

7. BAD 6243: Applied Univariate Statistics 7 Between-Subjects Effects

8. BAD 6243: Applied Univariate Statistics 8 Within-Subjects Effects (Note that for within group F- values, the appropriate row, based on sphericity tests, should be chosen.)

9. BAD 6243: Applied Univariate Statistics 9 Interaction Effects

10. BAD 6243: Applied Univariate Statistics 10 Multivariate Tests Provides an alternative to the univariate tests May be useful when the sphericity assumption is not valid In some circumstances, can be more powerful