# Repeated Measures ANOVA

Repeated Measures ANOVA One Factor, Correlated Measures: Same reasoning of Correlated Measures t-test More Power (and more efficient) Pulls out relatively small differences among treatments Relative to Big differences among subjects Removes Differences among subjects from error term Subjects vs. Treatments Large Differences (Error) Small Differences (Effect)

Partitioning The Variance SSTotal = SSSubjects + SSTreatment + SSError SSError Is the Variability which a single subject would have If you repeatedly measured him without changing treatment Partitioning The Variance Common Response to Tx Unique Response to Tx SSTotal = SSSubjects + SSTreatment + SSError SSError Is the Variability which a single subject would have If you repeatedly measured him without changing treatment

The Structure of the ANOVA Partitioning the Total Sum of Squared Deviations From the Grand Mean Spontaneous Variability of Subject Change is not the same for each Subject D.V.: Test Score E.G., Strategies Subjects If you test your subjects repeatedly: Counter Balance for (e.g.) practice effects/fatigue POC: Piece of Cake S1

S2 S3 S4 1st choice 57 71 75 93 Control 60 72 76 92 Notes 64

74 78 96 Step 1: Find The Total SS S1T1 S2T1 S3T1 S4T1 S1T2 S2T2 S3T2 S4T2 S1T3 S2T3 S3T3 S4T3 Grand Mean=

Raw Data 57 71 75 93 60 72 76 92 64 74 78 96 Cell Square

Deviations Deviations -18.7 348.4 -4.7 21.8 -0.7 0.4 17.3 300.4 -15.7 245.4 -3.7 13.4 0.3 0.1 16.3 266.8 -11.7

136.1 -1.7 2.8 2.3 5.4 20.3 413.4 75.67 SS-Total= 1754.7 Step 2: Compute Between Subjects SS Subject Means 60.3 72.3

76.3 93.7 Deviation -15.3 -3.3 0.7 18.0 Sq Dev Sq Dev*3 235.1 705.33 11.1 33.33 0.4 1.33 324 972

SS-Sub= MS-Sub= 1712 "/3 570.7 Step 3: Compute Treatment SS Treatment Means Deviation Sq Dev Sq dev*4 74 -1.7 2.8 11.1 75

-0.7 0.4 1.8 78 2.3 5.4 21.8 SS-Treat= MS-Treat= 34.7 "/2 17.3 Step 4: Compute SS Error SS-Total "1754.7

SS-Sub- SS-Treat "= 1712 34.7 SS-Error 8.0 "/(4-1)(3-1) MS-Error= 1.333 This is the Same as the Interaction Term in a 2-Way ANOVA IV1: Treatment IV2: Subject Interaction: Subject x Treatment

Step 5: Determine Degrees of Freedom Just like Interaction df Step 6: Calculate MS & F F = MS-Treat / " 17.3 / MS-Error 1.333 = 13.003 If ANOVA is Significant Use Tukey Test to compare treatments Nt is Number of Subjects in your

Experiment Caution Caryover effects Counter-balance Vs. Trend Analysis Populations Normally Distributed

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