unit 10 lesson 1
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University of North Texas Dr. J. Kyle Roberts © 2004
Unit 10: Repeated Measures ANOVA
Lesson 1: Further Applications of the ANOVA
EDER 6010: Statistics for Educational Research
Dr. J. Kyle Roberts
University of North Texas
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University of North Texas Dr. J. Kyle Roberts © 2004
Paired Samples t-test
Occasion 3121112151614
Occasion 2987898
Occasion 1544565
Person 1Person 2Person 3Person 4Person 5Person 6
t-test t-test
t-test
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University of North Texas Dr. J. Kyle Roberts © 2004
Repeated Measures ANOVA
TXXXXH ...: 3210
K
k
n
iikT N
TXSS
1 1
22
Where Xik is person i’s score in group kT is the sum of all scoresN is the total number of observations
NOTICE: No means!!!
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University of North Texas Dr. J. Kyle Roberts © 2004
Sum of Squares Total
2
1
n
iiT XXSS
ANOVA
K
k
n
iikT N
TXSS
1 1
22
Repeated Measures
SST = 54.667
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University of North Texas Dr. J. Kyle Roberts © 2004
Why Do Repeated Measures?In ANOVA:SST = SSB + SSW
In Repeated Measures ANOVA:1. variation among individuals (SSI)2. variation among occasions (SSO)3. residual variation or error (SSRes)SST SSW
SSB
SST SSRes
SSO
SSI
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University of North Texas Dr. J. Kyle Roberts © 2004
Why Do RM? (cont).
1. The partitioning of the variation in the ANOVA needs to be adjusted so that we are using the correct df (and SS) to compute F-calc based on the corrected MSerror.
2. We may or may not improve our chances of obtaining statistical significance.3. Since we are partitioning out the variation due to individual differences from the residual variation (error), we will most likely note a larger eta-squared (this is an artificial eta-squared, however).
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University of North Texas Dr. J. Kyle Roberts © 2004
Setting Up The DataUse the same example data for repeated measures as is in your book
Mean test1 = 6.1Mean test2 = 10.6Mean test3 = 15.3
3210 : testtesttest XXXH
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University of North Texas Dr. J. Kyle Roberts © 2004
Using SPSS for AnalysisAnalyze General Linear Model Repeated Measures
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University of North Texas Dr. J. Kyle Roberts © 2004
Analyzing the Data
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University of North Texas Dr. J. Kyle Roberts © 2004
The Sphericity Assumption•Put succinctly, the sphericity assumption (also called compound symmetry) states that the variance at each measurement occasion should be equal.•Interpret results the same way we would Levine’s test for homogeneity of variance in ANOVA.
23
22
210 : tttH
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University of North Texas Dr. J. Kyle Roberts © 2004
What if we don’t meet the sphericity assumption?
Use a “correction” for the df:
•Greenhouse-Geisser•Huynh-Feldt•Lower-bound
These all correct the df in an analysis and make it more difficult to find statistically significant results
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University of North Texas Dr. J. Kyle Roberts © 2004
Reading the Results
Occasions
Residual
Individuals
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University of North Texas Dr. J. Kyle Roberts © 2004
The “Correct” Summary Table
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University of North Texas Dr. J. Kyle Roberts © 2004
ANOVA vs. Repeated MeasuresData treated as a One-Way ANOVA with 3 levels
3 Repeated Measurements
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University of North Texas Dr. J. Kyle Roberts © 2004
Eta-squared in Repeated Measures
SST SSRes
SSO
SSI
In ANOVA:
T
B
SS
SS2
In Repeated Measures ANOVA:
RO
O
SSSS
SS
2
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University of North Texas Dr. J. Kyle Roberts © 2004
The Final Summary Table
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University of North Texas Dr. J. Kyle Roberts © 2004
Polynomial Trends
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Linear Trend Quadratic Trend
test1test2test3
Mean6.1
10.615.3
University of North Texas Dr. J. Kyle Roberts © 2004
Polynomial Trends (cont.)
Cubic Trend5 data points
Cubic Trend4 data points
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University of North Texas Dr. J. Kyle Roberts © 2004
Unit 10: Repeated Measures ANOVA
Lesson 1: Further Applications of the ANOVA
EDER 6010: Statistics for Educational Research
Dr. J. Kyle Roberts
University of North Texas
Time
Score