Repeated Measures ANOVA

Repeated-Measures
Analysis of Variance

Types of Within-Subject Designs
PTrue “Within-Subjects” Design
Each subject is measured under each treatment
condition
E.g., effects of amount of background noise on a
memory task (e.g., no background noise, 10 db,
20db)
PRepeated Measures Design
Each subject is measured at two or more points in
time
E.g., effects of exercise on heart rate (rest, after 1
min on treadmill, after 5 min on treadmill)

Types of Within-Subject Designs
PProfile Analysis
Scores on different tests (DVs), which are
comparably scaled, are compared
E.g., comparing scores on scales of the MMPI
PMatched Subjects Designs
A type of within-subject design where instead of
having subjects participate in all levels of the IV,
different subjects, which are matched a priori on
relevant variables, are compared
E.g., do subjects differ in reading speed under
different types of lighting, after matching subjects
on a priori reading speed

The Logical Background for a
Repeated-Measures ANOVA
# The repeated measures analysis of
variance applies to research situations
using within-subject designs
Including repeated measures designs, profile
analysis and matched subject designs
# Some of the logic and formulae for the
repeated measures ANOVA are identical
to the independent measures ANOVA
However, the repeated measures ANOVA
includes a second stage of analysis in which
variability due to individual differences is
removed from the error term.

Individual Differences
Between Subjects
# The repeated measures design
automatically eliminates individual
differences from the between treatments
variability because the same subjects are
used in every condition
# Further, individual differences (which can
be quantified) are eliminated from the
denominator of the F test
# The result is a test statistic similar to the
independent measures F ratio but with all
individual differences removed

Comparing Independent-
Measures and Repeated-
Measures ANOVA
# When individual differences are removed
from the denominator of the F test, what
results is a more ‘powerful’ test of the
research hypothesis (i.e., that the means
differ)
In other words, the denominator is smaller
# This advantage can be very important in
situations where large individual
differences would otherwise obscure the
treatment effect in an independentmeasures
study

Variability in a Within-Subjects /
Repeated Measures Design
PSubjects - variability due to consistent
differences between the subjects
Not usually an important effect to analyze since it
only shows that subjects differ on the DV
PTreatment - variability due to differences
between the levels of the DV
PError - Interaction between the subjects and
levels of the IV
Or in G&W terms, the overall within-subject
variability minus the subject variability

Why is the Treatment X Subjects
Interaction an Appropriate Error
Term?
PMore consistent scores of subjects across
the levels of the IV results in less error (more
confidence that the treatment was effective)
100
80
60
40
20
0
1st Qtr
2nd Qtr
80
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50
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1st Qtr
2nd Qtr
s1 s2 s3 s4
s1 s2 s3 s4
s1 10 40
s2 20 50
s3 50 100
s4 15 40
s1 10 40
s2 20 30
s3 50 30
s4 15 80

Understanding the F-ratio

Null and Alternate Hypotheses
# The null hypothesis is that the means are
all equal
H o : ì 1 = ì 2 = ... = ì k
For example, with three groups: H o : ì 1 = ì 2 = ì 3
# The alternative hypothesis is that at least
one of the means is different from another
Again, H o : ì 1 ì 2 ... ì k would not be an
acceptable way to write the alternate hypothesis

Computation of the F ratio
Total Variability and Degrees of Freedom

Computation of the F ratio
Error Variability and Degrees of Freedom

Computation of the F ratio
Between Treatment Variability and Degrees of Freedom

Computation of the F ratio
Mean Squares and F test
P Note that H o is rejected if:
F F á, df(Between Treatments), df(Error)

Measuring Effect Size for the
Repeated-Measures Analysis of
Variance
# In addition to determining whether the
mean differences are significant 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

Measuring Effect Size for the
Repeated-Measures Analysis of
Variance (cont.)
# In the context of ANOVA this percentage
is identified as ç 2
Before computing ç 2 , however, it is customary to
remove variability due to individual differences
between the subjects

Assumptions of the One-Way
Within-Subjects Design
PSubjects are randomly and independently
selected
PScores in each treatment condition are
normally distributed
PHomogeneity of Variances & Covariances
(Compound Symmetry) / Sphericity

Sphericity
P Sphericity is violated as the treatment
difference variances are unequal

Notes on the Sphericity
Assumption
PLike the homogeneity of variance assumption
for between subjects designs, sphericity is
commonly violated
PViolations of the assumption of sphericity can
severely bias the F statistic
More specifically, when the sphericity assumption
is violated the F test becomes too liberal, or in
other words, the probability of a Type I error
becomes much larger than á

Options for Counteracting the
Effects of Violations of Sphericity
PAdjusted df tests
Reduce the number of numerator and
denominator degrees of freedom to increase the
size of the critical value (or reduce the size of the
p-value) and thus reduce the number of Type I
errors
Greenhouse-Geisser
– Calculates the degree to which the sphericity
assumption is violated and then adjusts the degrees of
freedom accordingly
– The Greenhouse-Geisser adjustment to the ANOVA F
test is included as routine output in SPSS

Psych Grads Example
PDr. White would like to know if the “social life”
of York Psychology students changes as a
function of the number of years they have
been in the program (á=.01)
PH o : ì First Year = ì Second Year = ì Third Year
PH 1 : There are no differences among the
means

Example

Example
Error SS and MS

Example
Between Treatment SS and MS

Example
F test and Conclusions
PF crit (á=.01, df between-treatment = 2, df within-treatment = 16) = 6.23
PTherefore, since F > F crit we reject the null
hypothesis and conclude that there is a
significant difference between 1st, 2nd and
3rd year students in terms of the quality of
their social life

Effect Size
P Therefore, .763 (or 76.3%) of the variability in
social life scores can be attributed to the year
of the program
This would definitely be a large effect
But ... this is made up data so don’t try too hard
to make sense of the results

Post Hoc Tests
P Somehow Gravetter & Wallnau forgot to
mention how you are supposed to know
exactly where differences among the
conditions exist
PIn fact, it is very simple because we just use
multiple paired t-tests to determine exactly
which groups differ
The reason we use a “separate” test (specifically
error term) for each pairwise comparison is that
the problems with sphericity are very severe, but
they do not affect “two group” comparisons

Post Hoc Tests
P1st year vs 2nd year
t (8) = 1.00, p = .347
Social lives do not differ between 1st and 2nd
year students
P1st year vs 3rd year
t (8) = 7.23, p < .001
Social lives are better in 3rd year than in 1st year
P1st year vs 2nd year
t (8) = 6.93, p < .001
Social lives are better in 3rd year than in 2nd year