1. Factorial Design One Between-Subject Variable One Within-Subject Variable SS Total SS between subjects SS within subjects Treatments by Groups Treatments Treatments by Subjects within groups Subjects within groups Groups Differences Between Subjects Differences Within Subjects Groups – differences between groups of subjects SS w/in Groups – differences between subjects w/in a group Treatment – differences between subject’s scores across treatments Treat x Groups – interaction between Treatments and Groups Treats x Ss w/in Groups – interaction between Subjects and Treatments hold Groups factor constant

3. Sub #5152535 177435.25 288566.75 397435.75 486335.00 T rc 32281615 8.07.04.03.755.69 1105214.5 2106325.25 395425.00 4116325.5 T rc 4022127 105.531.755.06 TcTc 72502822172 96.253.52.755.38 =GT =GM Speed (Repeated Measure) Example Group1Group1 Group2Group2

4. Divide SS by appropriate df SS bs by #Ss - 1 SS grp by #Grps - 1 SS ss w/in grps by (#S ingrp -1) x (# of grps) SS ws by #Ss (# Treatments – 1) SS treat by # Treatments - 1 SS TxG by (#grp – 1) (#Treats -1) SS TxS w/in grps by (#Treats -1) x (n-1) x (# of grps) Calculate MS

5. Prepare Summary Table SourceSSDfMSFP Btwn S12.57 Grp3.1251 2n.s. Ss w/in Grp 9.37561.563 Within S22324 Treat194.5364.833127.89< 0.01 TxG19.37536.45812.74< 0.01 TxS w/in grp 9.12518.507 Total235.531 What are the appropriate error terms? (the denominators for the Fratios) Interpolation? 7 8

6. Repeated Measures Assumptions normality1) 2)homogeneity of variance 3)compound symmetry - constant variances on diagonal - constant covariances off diagonal A variance / covariance matrix for each group and overall T X Ss interactions are constant across groups4) - test with Fmax

7. Example No STRATVar/Covar Matrix 5152535 5 0.66000 15 0.66 1.0 25 0.661.0 35 2.25 Speed The assumption of compound symmetry is usually replaced by the assumption of sphericity =.574 =.853 = a constant across all pairs of conditions

8. Simple Effects One B-S variable One W-S variable Factorial Design The W-S variable - Separate One-Way ANOVAs (repeated measures) ∙ Error terms pooled = MS T X Ss w/in groups ∙ Or, use the MS T X Ss for each separate analysis No STRATSTRAT SS Total = 67.44 SS bs = 7.14 SS Treat = 54.69 SS error = 5.56 SS Total = 67.44SS Total = 168.04 SS bs = 5.29 SS Treat = 159.19 SS error = 3.56

9. STRATNo STRAT SS Total SS bs SS Treat SS error 67.44 + 54.69 7.19 5.56 + + + 168.04 = 235.48 = = = 12.45 213.88 9.12 5.29 159.19 3.56 SS Total (overall) SS bs (overall) SS Treat SS T X G + (overall) SS T X S w/in group (overall) Why?

10. Between-Subjects Simple Effects We could do a separate analysis of each level - unnecessary loss of df SS grp at 5 SS grp at 15 SS grp at 25 SS grp at 35 = = = = = = = = 8.0 4.5 2.0 8.0 MS all 1 df SS error term = SS w/cells = SS Ss w/in grp + SS T X Ss w/in grps MS error = Why?