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

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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

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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