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.
2
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.
3
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
4
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.
5
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.
6
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)
7
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.
8
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
9
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.
12
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.
14
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.
15
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.
16
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
17