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PSY 1950 Repeated-Measures ANOVA November 3, 2008

PSY 1950 Repeated-Measures ANOVA November 3, 2008. A (Random) x B (Fixed) ANOVA. 4 College x 3 Test Test is fixed factor, college is random factor. F fixed effect = MS test / MS within. F random effect = MS test / MS test x college. 2-way ANOVA: Fixed and Random Effects.

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PSY 1950 Repeated-Measures ANOVA November 3, 2008

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  1. PSY 1950 Repeated-Measures ANOVA November 3, 2008

  2. A (Random) x B (Fixed) ANOVA • 4 College x 3 Test Test is fixed factor, college is random factor Ffixed effect = MStest /MSwithin Frandom effect = MStest /MStest x college

  3. 2-way ANOVA: Fixed and Random Effects

  4. One-way Dependent-Measures ANOVA • Really a 2-way ANOVA with Subject as a random factor

  5. Partitioning of Sums of Squares Total variation Between subjects (S) Within subjects Error (S x A) Between conditions (A) numerator denominator SSA x S = SStotal - SSS - SSA SSwithin subjects - SSA

  6. Partitioning of Sums of Squares Total variation Between conditions (A) Within conditions numerator Between subjects (S) Error (S x A) denominator SSA x S = SStotal - SSS - SSA SSwithin conditions - SSS

  7. Calculating Error Term SSA x S = SStotal - SSS - SSA SSwithin subjects - SSA SSA x S = SStotal - SSS - SSA SSwithin conditions - SSS

  8. Violations of Sphericity • Three different estimates of  • Lower-bound • 1/(k - 1) ≤  ≤ 1 • Always too conservative, never too liberal • Greenhouse-Geisser • Too conservative when > .75 • Huynh-Feldt • Too liberal when < .75 • Take home message • When G-G estimate > .75, use H-F correction • When G-G estimate < .75, use G-G correction

  9. SPSS

  10. SPSS

  11. One-way RM ANOVA: Contrast Effects • Same for dependent-measures ANOVA as independent-measures ANOVA, provided all conditions have non-zero weights • Use pooled error term, i.e., MSerror from whole analysis • Exactly the same as what you already know • If any conditions have zero weights, calculate a new error term by excluding those conditions with zero weights

  12. Multiple Comparisons • Use Bonferroni/Sidak correction • Do not use pooled error term • Any violation of sphericity will wreak havoc on corrected p-values unless separate errors terms are calculated for each pairwise comparison • Same as dependent-measures t-test

  13. Higher Level RM ANOVA • Think of a n-dimensional RM ANOVA as a (n+1)-dimensional ANOVA with n fixed factors and subject as random factor • Different error terms for each fixed effect, based upon the interaction of that effect with the subject factor • Different sphericity assumptions/tests for each effect

  14. Two-way RM ANOVA If you can calculate SS for three-way independent-measures ANOVA, you can calculate SS for two-way dependent-measures ANOVA

  15. Two-Way RM ANOVA

  16. SPSS

  17. Simple Effects in RM ANOVA • Same as simple effect analysis for independent-measures ANOVA, except you calculate a new error term

  18. Interaction Contrasts in RM ANOVA • Provided there are no non-zero weights, interaction contrasts for dependent-measures ANOVA is same as for independent-measures ANOVA • If there are zero weights, recalculate error term by omitting conditions with zero-weights

  19. Mixed-Design ANOVA • At least one between-subjects factor, at least one within-subjects factor • Different error terms for between-subjects and within-subjects effects • For between-subjects effects, use the the MSwithin you know and love • For within-subjects effects, use the same error terms as RM ANOVA • Interaction effects between within- and between-factors are within-subject effects

  20. Partitioning of Sums of Squares1 between-subjects factor (Group)1 within-subjects factor (Condition) Total variation Between subjects Within subjects ErrorGroup  Subjects, within groups Group  Condition Subjects, within groups Condition Group

  21. SPSS

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