AMS577. Repeated Measures ANOVA: The Univariate and the

12 Pages·2015·0.99 MB·English

AMS577. Repeated Measures ANOVA:
The Univariate and the Multivariate
Analysis Approaches

1. One-way Repeated Measures ANOVA

One-way (one-factor) repeated-measures ANOVA is an extension
of the matched-pairs t-test to designs with more columns of
correlated observations.
Assume that the data used in the computing example for between-
subjects ANOVA represented performance scores of the same 4
respondents under three different task conditions:
Task 1 Task 2 Task 3 Mean
P1 3 5 2 3.33
P2 4 5 3 4.00
P3 5 7 5 5.67
P4 6 7 6 6.33
Mean 4.50 6.00 4.00 4.83

In an analysis of Task effects, any variance between subjects
(variation in the right-hand column) is not of interest. This
variance simply reflects that participants differ overall (e.g. P1
seems to perform less well than P4), but is independent of effects
of the different task conditions on people’s performance.
Between-subjects variance thus is removed before the effects of
the repeated-measures factor are tested. This could be achieved by
subtracting each score from that person’s mean score across the
three tasks, yielding:Between-subjects variance thus is removed
before the effects of the repeated-measures factor are tested. This
could be achieved by subtracting each score from that person’s
mean score across the three tasks, yielding:
Task 1 Task 2 Task 3 Mean
P1 -0.33 1.67 -1.33 0
P2 0.00 1.00 -1.00 0
P3 -0.67 1.33 -0.67 0
P4 -0.33 0.67 -0.33 0
Mean -0.33 1.17 -0.83 0
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These new scores do not reflect differences between subjects any
more, whereas the differences between task conditions are still
reflected in the column means.

The ANOVA for repeated measurements achieves this by first
partitioning the sums of squares into a “between-subjects”
component (i.e. variation between the row means of the first table)
and a “within-subjects” component (all remaining variation).

The “within-subjects” component is further subdivided in variation
“between treatments” (i.e. between the column means in the
second table) and “error” variation (i.e. within the columns of the
second table).

SS
total

SS SS
between-subjects within-subjects

SS SS
between-treatments error

[The between-subjects component can also be further
subdivided if there are between-subjects factors to be
considered; but this will not concern us for the moment.]
The degrees of freedom are divided into the same
components:
df = kn – 1
total
df = n – 1
between-ss
df = n(k – 1)
within-ss
df = k – 1
between-treatments
df = (n – 1)(k – 1)
error
Mean squares are derived as usual by dividing sums of
squares with their associated df.

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The F-ratio of MS and MS thus derived is used to
between-treatments error
test the effect of the within-subjects factor against the null hypo-
thesis that all pairwise differences between treatments are zero.
Note that both MS and MS in this model may
error between-treatments
contain variation that is due to a treatment x subjects interaction,
which itself is not testable (why?). So the fact that different people
may react differently to various treatments cannot be separated
from chance variations or treatment main effects.
The test of treatment effects is not affected by this problem
because the potential interaction term is “hidden” in both the
numerator and the denominator of the F-ratio.
This would not be the case for a test of the between-subjects effect
(for which a conservative bias would be introduced if an
interaction were in fact present). But note that we would not
normally test the between-subjects variation for significance in this
kind of design. (If significant, it would only tell us that people are
different - and didn’t we know this all along?)

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Example: Suppose there are k regions of interest (ROI’s) and n
subjects. Each subject was scanned on baseline (soda) as well as
after drinking alcohol. Our main hypothesis is whether the change
between baseline and alcohol is homogeneous among the ROI’s.
That is H :  , where  is the effect of alcohol on
0 1 2 k j
the jth ROI, j 1, ,k.
Profile Plots Illustrating the
Questions of Interest
Test for Equal Changes in Different ROIs
(That is, whether the two series are parallel.)
le
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F
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ia Baseline
r
B
1 2 3 4 5 … k
Brain Regions of Interest (ROI)
Figure 1. Hypothesis in terms of the original data

Profile Plots Illustrating the
Questions of Interest
Test for Equal Changes in Different ROIs
lev (That is, whether the two series are parallel.)
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la
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itc
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F
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rB
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a
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1 2 3 4 5 … k
Brain Regions of Interest (ROI)

Figure 2. Hypothesis in terms of the paired differences
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The Univariate Analysis Approach

For subject i, let Yij denote the paired difference between baseline
and alcohol for the jth ROI, then the (univariate) repeated
measures ANOVA model is: Y  S  , where  is the
ij j i ij j
(fixed) effect of ROI j,S is the (random) effect of subject i, is
i ij
the random error independent of S . With normality assumptions,
i
we have:

, and are independent to each other.

Let Y Y ,Y , ,Y ', we have , i1, ,n, where
i i1 i2 ik
,, ,' and
1 2 k
2 2 2 2  1  
s  s s
   
2 2 2 2  1 
 s s  s 2  with
   
   
 2 2 2 2   1
s s s 
2
 s and 2 2 2. This particular structure of the
2 2 s 
s 
variance covariance matrix is called “compound symmetry”. For
each subject, it assumes that the variances of the k ROI’s are equal
2and the correlation between each ROI pair is constant ,
which may not be realistic.

The univariate approach to one-way repeated measures ANOVA is
equivalent to a two-way mixed effect ANOVA for a randomized
block design with subject as the blocks and ROI’s as the
“treatments”. The degrees of freedom for the ANOVA F-test of
equal treatment effect is k 1 and n1k1 respectively.
That is, . We will reject the null hypothesis at

the significance level  if F  F .
0 k1,n1k1

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The Multivariate Analysis Approach
Alternatively, we can use the multivariate approach where no
structure, other than the usual symmetry and non-negative definite
properties, is imposed on the variance covariance matrix  in
, i1, ,n. Certainly we have more parameters

In this model than the univariate repeated measures ANOVA
model. The test statistic is T2 nn1CY'CQC'1CY
0
where Y n Y , Qn Y YY Y', and
i i i
i1 i1
1 1 0 0 0 
 
0 1 1 0 0
C  .
 
 
0 0 0 1 1

⇔

⇔

Under the null hypothesis, .

Recall that if , then

Therefore the Hotelling’s T2 statistic has the following
k1,n1
relationship with the F statistics:

We will reject the null hypothesis at the significance level  if
F F (upper tail percentile).
0 k1,nk1
When to use what approach?
There are more parameters to be estimated in the multivariate
approach than in the univariate approach. Thus, if the assumption
for univariate analysis is satisfied, one should use the univariate
approach because it is more powerful. Huynh and Feldt (1970)
give a weaker requirement for the validity of the univariate
ANOVA F-test. It is referred to as the “Type H Condition”. A test
for this condition is called the Machly’s Sphericity Test. In SAS,
this test is requested by the “PrintE” option in the repeated
statement.
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Example 1. One-way Repeated Measures ANOVA (n=4,
k=4), Paired Differences in Brain Functional Levels
Subject ROI 1 ROI 2 ROI 3 ROI 4
1 5 9 6 11
2 7 12 8 9
3 11 12 10 14
4 3 8 5 8

SAS Program: One-way Repeated Measures Analysis of
Variance

data repeatM;
input ROI1-ROI4;
datalines;
5 9 6 11
7 12 8 9
11 12 10 14
3 8 5 8
;

proc anova data=repeatM;
title 'one-way repeated measures ANOVA';
model ROI1-ROI4 = /nouni;
repeated ROI 4 (1 2 3 4)/printe;
run;

(Note: SAS Proc GLM and Proc Mixed can also be used for
the repeated measures ANOVA /MANOVA analyses.)
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SAS Output: One-way Repeated Measures Analysis of
Variance

1. Estimated Error Variance-Covariance Matrix

ROI_1 ROI_2 ROI_3
ROI_1 10.00 8.00 7.00
ROI_2 8.00 16.75 11.75
ROI_3 7.00 11.75 8.75

2. Test for Type H Condition --- Mauchly's Sphericity Tests
(Note: p-value for the test is big, so we can use the univariate
approach)

Variables DF Criterion Chi-Square Pr > ChiSq
Orthogonal Components 5 0.0587599 4.8812865 0.4305

3. Multivariate Analysis Approach --- Manova Test Criteria and Exact F Statistics for the
Hypothesis of no drug Effect

Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.00909295 36.33 3 1 0.1212
Pillai's Trace 0.99090705 36.33 3 1 0.1212
Hotelling-Lawley Trace 108.97530864 36.33 3 1 0.1212
Roy's Greatest Root 108.97530864 36.33 3 1 0.1212

4. Univariate Analysis Approach --- Univariate Tests of Hypotheses for Within Subject Effects
Adj Pr > F
Source DF Anova SS Mean Square F Value Pr > F G - G H - F
ROI 3 50.25000000 16.75000000 11.38 0.0020 0.0123 0.0020
Error(ROI) 9 13.25000000 1.47222222

Greenhouse-Geisser Epsilon 0.5998
Huynh-Feldt Epsilon 1.4433

Interpretation

Note that the multivariate F-test has value of 36.33, degrees of
freedom of 3 and 1, and the p-value is 0.1212. While the univariate
F-test has value of 11.38, with degrees of freedom of 3 and 9, and
the p-value is 0.0020. In this case, since the assumption for the
univariate approach is satisfied, we use the univariate approach
which is more powerful (smaller p-value).

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2. Two-way Repeated Measures ANOVA

However, things may not always be so easy.
For example, instead of comparing whether the changes between
two conditions (baseline and alcohol) are constant across several
brain regions of interest (ROI) as we had introduced previously,
now, our situation is:

We have only one region of interest for each subject.
We have two conditions: heavy alcohol, light alcohol
We monitor the brain functional levels in time.
Example 2. Two-way Repeated Measures ANOVA (n=4,
k1=2, k2=4 ) – with repeated measures on both factors
Subject Time 1 Time 2 Time 3 Time 4
Heavy 1 25 29 36 31
Alcohol 2 27 32 38 29
3 31 22 40 34
4 23 28 35 38

Light 1 51 66 69 55
Alcohol 2 47 58 82 78
3 43 71 79 86
4 62 80 78 98

The following 2-factor model, however, has repeated measures
on only one factor (time)
Example 3: Two treatment groups with four
measurements taken over equally spaced time
intervals
(e.g., A = treatment B = placebo)
id group time1 time2 time3 time4
1 A 31 29 15 26
2 A 24 28 20 32
3 A 14 20 28 30
4 B 38 34 30 34
5 B 25 29 25 29
6 B 30 28 16 34
Hypothetical data from Twisk, chapter 3, page 40, table 3.7

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Profile Plots Illustrating the
Questions of Interest
TIME EFFECT ONLY
)ld
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( le
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re treatment
ts
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h
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0 2 4 6 8 10 12
TIME (months)

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