A graduate student measured the heart rote5 (tji:cjl:,
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per minute, or bpm) of a set of subjects riding at 5, 7,
and 9 miles per hour [mph)at a minimal work load on
a stationary bicycle. Fifteen subjects rode at increasing speeds for 2 minutes at each of the three rates. The
mean values for the 15 subjects at the end of each 2minute exercise bout were determined to be 1 2 0 bpm
at 5 rnph, 130 bpm at 7 mph, and 1 5 0 bpm at 9
mph. Is there a significant HR difference among the
speeds? Simple analysis of variance assumes that the three groups are
independent (that is, they are separate subjects). However, the data in this
study are not independent because each mean is based on the same 15
subjects. Using the same subjects in repeated-measure research designs
produces a relationship (correlation) among the three scores. To analyze
this type of data, we must account for this relationship. In this chapter we
learn to apply analysis of variance in a repeated-measuredesign.
O n e of the most common research designs in kinesiology involves measuring
subjects before treatment (pretest) and then after treatment (posttest). A dependent
t test with matched or correlated samples is used to analyze such data, because the
same subjects are measured twice (repeated measures). In ANOVA, this type of
design is referred to as a within-subjects design. When three or more tests are
given-for example, if a pre-, mid-, and posttest are given with treatment before
and after the midtest-ANOVA with repeated measures is needed to properly analyze the differences among the three tests.
The simple ANOVA described in chapter 9 assumes that the mean values are
taken from independent groups that have no relationship to each other. In this
independent group design, total variability is the sum of
variability between people in the different groups (interindividual variability),
variability within a person's scores (intraindividual variability),
variability between groups due to treatment effects, and
variability due to error (variability that is unexplained).
When there is only one group of subjects, but they are measured more than
once, the data sets are dependent. The total variability for a single group of subjects measured more than once is expected to be less than if the scores came from
different groups of people (that is, if the scores were independent), because interindividual variability has been eliminated by using a single group. This tends to
reduce the MS, term in the denominator of F in a manner similar to the correction
made to the standard error of the difference in the t test (equation 8.09).
I'll(.; I ~ ~ \ . I I I I . I , Y L0.1 ~ I I CI C ~ I L . ; ~ I L XIncasurc\
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1ictor.s have been controlled, any differences ob\ L . I - \ , C ~I > ~ * ~ w tlic
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must be due to (a) the treatment, (b) variations within
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(intraindividual variability), or (c) error (unexplained variability). Variability bctween subjects (interindividual variability) is no longer a factor. The fornlulas previously presented for simple ANOVA are modified to account for repeated measures.
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Assumptions in Repeated Measures ANOVA
Except for independence of samples, the assumptions for simple ANOVA, (betweensubjects designs), discussed in chapter 9 also hold true for repeated measures ANOVA,
(within-subjects designs). But with repeated measures designs, we must also consider the relationships among the repeated measures. Repeated measures ANOVA
must meet the assumption of sphericity, sometimes referred to as compound symmetry. Sphericity requires that the repeated measures demonstrate homogeneity of
variance (see chapter 8, assumptions for the t test) and homogeneity of covariance.
Homogeneity of covariance means that the relationships, or correlations, on the
dependent variable among all of the three or more repeated measures are equal.
When only two repeated measures are ernployed (such as pre-post measures for a t
test), this assumption is not applicable, because there is only one correlation coeffcient that can be calculated, the correlation between pre- and posttest scores. Methods of dealing with violations of sphericity will be presented later in this chapter.
Calculating Repeated Measures ANOVA
To demonstrate how to calculate ANOVA with repeated measures, we will analyze
a hypothetical study. A graduate student studying motor behavior was interested in
the decrease in balance ability that bicycle racers experience as their fatigue from
the race increases. To measure this, the researcher placed a racing bicycle on a
roller ergometer. The bicycle's front and rear wheels were placed on the tops of
cylindrical rollers so that as the wheels turned, the rollers turned and the rider was
able to ride in place (figure 10.1 ).
A 4-inch-wide stripe was painted in the middle of the front roller, and the rider
was required to keep the front wheel on the stripe. The rear rollers were connected
to a braking system to provide resistance to the rear wheel of the bike. Balance was
indicated by wobble in the front wheel and was measured by counting the number
of times per minute that the front wheel of the bike strayed off the 4" stripe.
As resistance on the rear wheels increased, physiological fatigue increased, and
it became more and more difficult to maintain the front wheel on the stripe. Subjects rode the bicycle for 15 minutes, divided into five 3-minute periods for the
purpose of collecting data. Data were collected on the number of balance errors
\ ~ I I ~ I I C Y M I I I I I I C3 '
Mlnutc 9
MIIIU~C0
Minute 12
Minute 15
CK
-
I
7
7
23
2
12
22
26
3
11
6
9
4
10
18
16
5
6
12
9
6
13
21
30
7
5
0
2
8
15
18
22
9
0
2
0
10
6
8
27
-
Figure 10.1 Roller ergometer.
during the last minute o f each 3-minute period, and resistance was increased at the
end o f each 3-minute period. In this design, the dependent variable is balance
errors and the independent variable is increase in resistance (fatigue).
Table 10.1 presents the raw data in columns and rows. The data (in errors per
minute) for the subjects ( N = 10) are in the rows, and the data for the trials ( k = 5
repeated measures) are in the columns. The sum o f rows (CR)is the total score for
each subject over all five trials, and the sum o f rows squared (CR), is presented at
the right o f the table. The sum o f each column ( C C ) is the total for all 10 subjects
on a given trial; CX,. is presented at the bottom o f the table.
Steps in Calculation
-
-
-
CC
85
114
1 64
Mean =
8.5
11.4
16.4
CX,=85+114+164+311+365=1,039
Table 10.2 Raw Data Squared
p
p
Subject
-
-
Minute 3
-
Minute 6
Minute 9
1
49
49
529
2
144
484
676
3
121
36
81
4
100
324
256
5
36
144
81
6
169
44 1
900
1 . Arrange the raw data ( X ) in tabular form, placing the data for subjects in
rows ( R ) ,and repeated measures in columns ( C )as shown in table 10.1.
7
25
0
4
2. Calculate the row totals ( Z R , ,CR,, ZR,, . . .), the column totals ( C C , ,CC,,
CC,, . . .), and the grand total: CX, = 1,039.
8
225
324
484
9
0
4
0
10
36
3. Square each row total, (CR)', and sum these values: ): (CR),= 129,165.
4. Compute the mean values for columns
5. Square each raw score ( X 2 )and calculate the squared column totals and grand
total: CX' = 34.947. as shown in table 10.2.
Total
64
-
-
905
Minute 12
Minute 15
--
-
-
1,870
729
--
3,740
C X 2 = 905 + 1,870 + 3,740 + 11,091 + 17,341 = 34,947