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Chapter 14: Repeated-Measures
Analysis of Variance
1
The Logical Background for a
Repeated-Measures ANOVA
• Chapter 14 extends analysis of variance to
research situations using repeated-measures (or
related-samples) research designs.
• Much of the logic and many of the formulas for
repeated-measures ANOVA are identical to the
independent-measures analysis introduced in
Chapter 13.
• However, the repeated-measures ANOVA
includes a second stage of analysis in which
variability due to individual differences is
subtracted out of the error term.
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The Logical Background for a
Repeated-Measures ANOVA (cont.)
• The repeated-measures design eliminates
individual differences from the
between-treatments variability because the
same subjects are used in every treatment
condition.
• To balance the F-ratio the calculations require
that individual differences also be eliminated
from the denominator of the F-ratio.
• The result is a test statistic similar to the
independent-measures F-ratio but with all
individual differences removed.
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Comparing Independent-Measures
and Repeated-Measures ANOVA
• The independent-measures analysis is used in
research situations for which there is a separate
sample for each treatment condition.
• The analysis compares the mean square (MS)
between treatments to the mean square within
treatments in the form of a ratio
MS between treatments
F = ───────────
MS within treatments
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Comparing Independent-Measures and
Repeated-Measures ANOVA (cont.)
• What makes the repeated-measures analysis
different from the independent-measures analysis
is the treatment of variability from individual
differences. The independent-measures F ratio
(Chapter 13) has the following structure:
MS between treatments
differences)
F = ──────────── =
MS within
treatment effect + error (including individual
───────────────────────────────────
error (including individual differences)
• In this formula, when the treatment effect is zero
(H0 true), the expected F ratio is one.
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Comparing Independent-Measures and
Repeated-Measures ANOVA (cont.)
• In the repeated-measures study, there are no
individual differences between treatments
because the same individuals are tested in
every treatment.
• This means that variability due to individual
differences is not a component of the numerator
of the F ratio.
• Therefore, the individual differences must also
be removed from the denominator of the F ratio
to maintain a balanced ratio with an expected
value of 1.00 when there is no treatment effect.
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Comparing Independent-Measures and
Repeated-Measures ANOVA (cont.)
• That is, we want the repeated-measures
F-ratio to have the following structure:
treatment effect + error (without individual differences)
F = ────────────────────────────────────
error (with individual differences removed)
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Comparing Independent-Measures and
Repeated-Measures ANOVA (cont.)
• This is accomplished by a two-stage analysis. In
the first stage, total variability (SS total) is
partitioned into the between-treatments SS and
within-treatments SS.
• The components for between-treatments
variability are the treatment effect (if any) and
error.
• Individual differences do not appear here
because the same sample of subjects serves in
every treatment. On the other hand, individual
differences do play a role in SS within because the
sample contains different subjects.
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Comparing Independent-Measures and
Repeated-Measures ANOVA (cont.)
• In the second stage of the analysis, we
measure the individual differences by
computing the variability between
subjects, or SS between subjects
• This value is subtracted from SS within
leaving a remainder, variability due to
experimental error, SS error
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Comparing Independent-Measures and
Repeated-Measures ANOVA (cont.)
• A similar two-stage process is used to analyze
the degrees of freedom. For the repeatedmeasures analysis, the mean square values and
the F-ratio are as follows:
SS between treatments
MS between treatments = ───────────
df between treatments
SS error
MS error = ─────
df error
MS between treatments
F = ──────────
MS error
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Comparing Independent-Measures and
Repeated-Measures ANOVA (cont.)
• One of the main advantages of the
repeated-measures design is that the role
of individual differences can be eliminated
from the study.
• This advantage can be very important in
situations where large individual
differences would otherwise obscure the
treatment effect in an independentmeasures study.
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Measuring Effect Size for the RepeatedMeasures Analysis of Variance
• In addition to determining the significance of the
sample mean differences with a hypothesis test,
it is also recommended that you determine the
size of the mean differences by computing a
measure of effect size.
• The common technique for measuring effect size
for an analysis of variance is to compute the
percentage of variance that is accounted for by
the treatment effects.
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Measuring Effect Size for the RepeatedMeasures Analysis of Variance (cont.)
• In the context of ANOVA this percentage is
identified as η2 (the Greek letter eta, squared).
• Before computing η2, however, it is customary to
remove any variability that is accounted for by
factors other than the treatment effect.
• In the case of a repeated-measures design, part
of the variability is accounted for by individual
differences and can be measured with
SS between subjects.
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Measuring Effect Size for the RepeatedMeasures Analysis of Variance (cont.)
• When the variability due to individual differences
is subtracted out, the value for η2 then
determines how much of the remaining,
unexplained variability is accounted for by the
treatment effects.
• Because the individual differences are removed
from the total SS before eta squared is
computed, the resulting value is often called a
partial eta squared.
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Measuring Effect Size for the RepeatedMeasures Analysis of Variance (cont.)
• The formula for computing effect size for a
repeated-measures ANOVA is:
SS between treatments
SS between treatments
η2 = ───────────── = ──────────────
SS total - SS between subjects
SS error + SS between treatments
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