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Lecture 4

PSYC 3430 Lecture Notes - Lecture 4: Null Hypothesis, Explained Variation, Repeated Measures Design

Course Code
PSYC 3430
Peter K Papadogiannis

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Instructor Notes - Chapter 13 - page 185
Chapter 13: Repeated-Measures ANOVA
Chapter Outline
13.1 Overview of Repeated-Measures Designs
13.2 The Repeated-Measures ANOVA
Hypotheses for the Repeated-Measures ANOVA
The F-ratio for Repeated-Measures ANOVA
The Logic of the Repeated Measures ANOVA
13.3 Hypothesis Testing and Effect Size with the Repeated-Measures ANOVA
Notation for the Repeated-Measures ANOVA
Stage 1 of the Repeated-Measures Analysis
Stage 2 of the Repeated-Measures Analysis
Calculation of the Variances (MS Values) and the F-Ratio
Measuring Effect Size for the Repeated-Measures ANOVA
In the Literature - Reporting the Results of a Repeated-Measures ANOVA
Post Hoc Tests with Repeated Measures ANOVA
Assumptions of the Repeated-Measures ANOVA
13.3 Advantages of the Repeated-Measures Design
Individual Differences and the Consistency of the Treatment Effects
13.5 Repeated-Measures ANOVA and Repeated-Measures t Test
Learning Objectives and Chapter Summary
1. Students should understand the logic underlying the analysis of variance for a repeated-
measures study.
The denominator of the F-ratio is intended to measure differences (or variance) that occur
simply by chance. These chance differences are the unsystematic, unpredictable
differences that are not caused by a treatment effect or some other systematic and
predictable source. In an independent-measures study where every score comes from a
different person, the naturally occurring differences from one person to another
(individual differences) are unpredictable and are included in the category of chance or
error variance. In a repeated-measures study, however, the differences from one person
to another are often systematic and predictable. For example, one individual may have
scores that are consistently higher than a second person. These consistent differences can
be measured and subtracted out of the variance in the denominator of the F-ratio. Thus,
the repeated-measures ANOVA introduces a second stage to the analysis where the
individual differences are measured and subtracted out of the variance.
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Instructor Notes - Chapter 13 - page 186
2. Students should be able to perform an analysis of variance to evaluate the mean differences
from a single-factor, repeated-measures research study.
The formulas and calculations for the repeated-measures ANOVA are almost identical to
those for the independent-measures analysis; in fact, the first stage of the repeated-
measures analysis is identical to the independent-measures analysis. The second stage of
the analysis involves computing and subtracting the SS and df values that measure the
individual differences between subjects. The calculation of SS and df between-subjects
in the second stage of the analysis follow exactly the same pattern as the calculation of
SS and df between-treatments in the first stage (see Box 13.1).
3. The calculation of Ξ·2 (the percentage of explained variance) to measure effect size is slightly
different for the repeated measures design. Instead of computing a percentage of the total
variability (like we did with the independent-measures ANOVA), we now compute a percentage
of the variability that is otherwise unexplained.
In a repeated-measures ANOVA, part of the variability is explained by the systematic
individual differences. This part can be calculated and subtracted out. We then compute
Ξ·2 as the percentage of the remaining variability that is explained by the treatment effects.
As a result,
SSbetween treatments
Ξ·2 = ──────────────
SStotal - SSbetween subjects
Other Lecture Suggestions
1. The repeated-measures ANOVA builds on the independent-measures analysis in Chapter 12.
The difference between the two methods focuses on removing individual differences. If students
understand the rationale for removing individual differences, then the repeated-measures
ANOVA makes more sense. The logic underlying the repeated-measures analysis is as follows:
The purpose for analysis of variance is to determine whether the data provide evidence
for a significant treatment effect. To accomplish this, the F-ratio is constructed so that the
numerator and the denominator are perfectly balanced if there is no treatment effect (if H0 is
true). Thus, if we obtain an unbalanced, exceptionally large, F-ratio, we have evidence for a
treatment effect. In a repeated-measures design, there are no individual differences between
treatments (the same individuals are in all the treatments), so there are no individual differences
in the numerator of the F-ratio. To keep the F-ratio balanced, we must also remove the
individual differences from the denominator. This is the second stage of the repeated-measures
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Instructor Notes - Chapter 13 - page 187
2. A demonstration that manipulates the data from a repeated-measures ANOVA can help
emphasize two important points:
1. The repeated-measures ANOVA removes individual differences.
2. In situations where individual differences are large, a repeated-measures design can be
much more powerful than an independent-measures design. That is, the repeated-
measures design is much more likely to identify significant differences between
Begin with data similar to the scores in end-of-chapter problem #9 and work through the analysis
of variance. Stop after the first stage of the analysis to demonstrate what would happen if the
scores were from an independent-measures design. Then, increase the individual differences by
adding 5 points to each score for person B and work through both the independent- and repeated-
measures analysis again. Note that the increased individual differences have a large effect on the
independent-measures ANOVA but have no effect at all on the repeated-measures analysis.
Caution: Adding 5 points to each score for person B will make G = 60 and Ξ£X2 = 447.
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