PSY 1950 Repeated-Measures ANOVA October 29, 2008
Jan 24, 2016
PSY 1950Repeated-Measures ANOVA
October 29, 2008
Partitioning of Sums of Squares
Total variation
Between subjects
Within subjects
Between conditions
Errornumerator
denominator
Partitioning of Sums of Squares
Total variation
Between conditions
Within conditions
Between subjects
Error
numerator
denominator
Error as Interaction• Why does the interaction between Subject and Condition equal the error?– Interaction reveals how much the treatment effect varies across subjects
– Small interaction means that the treatment has a reliable effect
– Big interaction means that the treatment has an unreliable effect
0
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14
Time1 Time2
Score
Subject Time 1 Time 21 1 42 2 53 3 54 4 75 5 86 6 9
Mean 3.50 6.33r = 0.98
Source df SS MS F pBetween 5 36.4Within 6 24.5
Time 1 24.1 24.1 289 <.001Error (TxS) 5 0.4 0.08
Total 11 60.9
Subject Time 1 Time 21 0 32 3 63 7 94 1 45 1 46 9 12
Mean 3.50 6.33r = 0.99
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Time1 Time2
Score
Source df SS MS F pBetween 5 128.4Within 6 24.5
Time 1 24.1 24.1 289 <.001Error (TxS) 5 0.4 0.08
Total 11 152.9
Source df SS MS F pBetween 5 17.4Within 6 43.5
Time 1 24.1 24.1 6.2 0.06Error (TxS) 5 19.4 3.9
Total 11 60.9
0
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Time1 Time2
Score
Subject Time 1 Time 21 1 52 2 73 3 94 4 45 5 86 6 5
Mean 3.50 6.33r = -0.05
Assumptions• Independence (of errors)
– from scores from different sampling units, not of measurements
• Normality (of errors)• Sphericity ()
– AKA circularity– Equal dependency among conditions – Homogeneity of the variances of the differences between conditions
– Only when > 2 levels of repeated measure
SphericityDrug
Subject A B C D E
1 3 0 2 0 0
2 4 3 1 1 1
3 6 3 4 3 4
4 7 6 5 4 3
Subject A-B A-C A-D A-E B-C B-D B-E C-D C-E D-E1 3 1 3 3 -2 0 0 2 2 02 1 3 3 3 2 2 2 0 0 03 3 2 3 2 -1 0 -1 1 0 -14 1 2 3 4 1 2 3 1 2 1
Variance 1.333 0.667 0 0.667 3.333 1.333 3.333 0.667 1.333 0.667
Mauchly’s test determines whether population variances differ significantly
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John Mauchly
ENAIC– The first Turing-complete purpose digital computer
– Built to calculate artillery firing tables
– A “giant brain”– 5 million hand-soldered joints
Violations of Sphericity• Correct degrees of freedom
– Only to convert F to p, not to calculate MS
• Multiply numerator and denominator dfs by
• 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
SPSS
SPSS