PSY 1950Repeated-Measures ANOVA

November 3, 2008

A (Random) x B (Fixed) ANOVA

• 4 College x 3 TestTest is fixed factor, college is random factor

College Test1 Test2 Test3 mean1 28 12 20 202 12 28 20 203 12 28 20 204 12 28 20 20

mean 16 24 20 20

College Test1 Test2 Test3 mean1-250 28 12 20 20

251-500 12 28 20 20mean 20 20 20 20

Ffixed effect = MStest /MSwithin

Frandom effect = MStest /MStest x college

2-way ANOVA: Fixed and Random Effects

Source Error TermA (fixed) S/ABB (fixed) S/ABA x B S/AB

Source Error TermA (random) S/ABB (fixed) A x BA x B S/AB

Source Error TermA (random) A x BB (random) A x BA x B S/AB

One-way Dependent-Measures ANOVA

• Really a 2-way ANOVA with Subject as a random factor

Source Error TermA (random) S/ABB (fixed) A x BA x B S/AB

Source Error TermSubjects (S) S/SAA S x AS x A S/SA

Factor A

Subject A B C D E mean

1 3 0 2 0 0 1

2 4 3 1 1 1 2

3 6 3 4 3 4 4

4 7 6 5 4 3 5

mean 5 3 3 2 2 3

Partitioning of Sums of Squares

Total variation

Between subjects (S)

Within subjects

Between conditions (A)

Error (S x A)numerator denominat

orSSA x S = SStotal - SSS - SSA

SSwithin subjects - SSA

Partitioning of Sums of Squares

Total variation

Between conditions (A)

Within conditions

Between subjects (S) Error (S x A)

numerator

denominatorSSA x S = SStotal - SSS -

SSA

SSwithin conditions - SSS

Calculating Error Term

Factor A

Subject A B C D E mean

1 3 0 2 0 0 1

2 4 3 1 1 1 2

3 6 3 4 3 4 4

4 7 6 5 4 3 5

mean 5 3 3 2 2 3

SSA x S = SStotal - SSS - SSA

SSwithin conditions - SSS

SSA x S = SStotal - SSS - SSA

SSwithin subjects - SSA

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

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., 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

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

Subject A B C D E mean

1 3 0 2 0 0 1

2 4 3 1 1 1 2

3 6 3 4 3 4 4

4 7 6 5 4 3 5

mean 5 3 3 2 2 3

mean

1.5

2.5

5

5

3.5

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

Source Error TermA (random) S/ABCB (fixed) A x BC (fixed) A x CA x B S/ABCA x C S/ABCB x C A x B x CA x B x C S/ABC

Source Error TermSubjects (S) S/ABA S x AB S x BS x A S/ABS x B S/ABB x C S x B x CS x B x C S/AB

If you can calculate SS for three-way independent-measures ANOVA, you can calculate SS for two-way dependent-measures ANOVA

Subject Cold Hot Cold Hot mean1 4 6 6 4 52 2 4 3 3 33 2 4 5 9 54 4 2 2 4 3

mean 3 4 4 5 4

No Robo-chug Robo-chug

Subject No Yes mean Subject Cold Hot mean1 5 5 5 1 5 5 52 3 3 3 2 2.5 3.5 33 3 7 5 3 3.5 6.5 54 3 3 3 4 3 3 3

mean 3.5 4.5 4 mean 3.5 4.5 4

Robochug Tempature

Source Error Term SS df MS FSubjects (S) S/RT 16 3 5.333Robo-Chug (R) S x R 4 1 4 1Temperature (T) S x T 4 1 4 2S x R S/RT 12 3 4S x T S/RT 6 3 2R x T S x R x T 0 1 0 0S x R x T S/RT 10 3 3.333

Two-Way RM ANOVA

Source Error Term SS df MS FSubjects (S) S/RT 16 3 5.333Robo-Chug (R) S x R 4 1 4 1Temperature (T) S x T 4 1 4 2S x R S/RT 12 3 4S x T S/RT 6 3 2R x T S x R x T 0 1 0 0S x R x T S/RT 10 3 3.333

Source SS df MS FBetween Subjects 16 3 5.333Within Subjects

Robo-Chug (R) 4 1 4 1Error 12 3 4

Temperature (T) 4 1 4 2Error 6 3 2R x T 0 1 0 0Error 10 3 3.333

Total 52 15

SPSS

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

Subject Cold Hot Cold Hot mean1 4 6 6 4 52 2 4 3 3 33 2 4 5 9 54 4 2 2 4 3

mean 3 4 4 5 4

No Robo-chug Robo-chug

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 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

Partitioning of Sums of Squares

1 between-subjects factor (Group)1 within-subjects factor (Condition)

Total variation

Between subjects

Within subjects

Group Conditi

on

ErrorGroup Subjects,

within groups

Condition

Subjects, within groups

Group

Subject Cold Hot mean Subject Cold Hot mean1 4 6 5 5 6 4 52 2 4 3 6 3 3 33 2 4 3 7 5 9 74 4 2 3 8 2 4 3

mean 3 4 3.5 4 5 4

No Robo-chug Robo-chug

Source SS df MS FBetween Subjects 32 7

Robo-Chug (R) 4 1 4 0.857Error 28 6 4.667

Within Subjects 20 8Temperature (T) 4 1 4 1.5

R x T 0 1 0Error 16 6 2.667 0

Total 52 15

SPSS