PSY 1950 Repeated-Measures ANOVA November 3, 2008.
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Transcript of PSY 1950 Repeated-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
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
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