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