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

One-way Dependent-Measures ANOVA Really a 2-way ANOVA with Subject as a random factor Factor A SubjectABCDEmean

Partitioning of Sums of Squares Total variation Between subjects (S) Within subjects Between conditions (A) Error (S x A) numerator denominator SS A x S = SS total - SS S - SS A SS within subjects - SS A

Partitioning of Sums of Squares Total variation Between conditions (A)Within conditions Between subjects (S) Error (S x A) numerator denominator SS A x S = SS total - SS S - SS A SS within conditions - SS S

Calculating Error Term Factor A SubjectABCDEmean SS A x S = SS total - SS S - SS A SS within conditions - SS S SS A x S = SS total - SS S - SS A SS within subjects - SS A

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

SPSS

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., MS error 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

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 Factor A SubjectABCDEmean

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

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

Two-Way RM ANOVA

SPSS

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

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

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 MS within 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

Partitioning of Sums of Squares 1 between-subjects factor (Group) 1 within-subjects factor (Condition) Total variation Between subjectsWithin subjects Group  Condition Error Group  Subjects, within groups Condition Subjects, within groups Group

SPSS