AMS588.
One-way Repeated Measures ANOVA:
The Univariate and the Multivariate Analysis Approaches
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 0 : 1  2   k , where  j is the effect of alcohol on the jth ROI, j  1, , k .
The Univariate Analysis Approach
For subject i, let Y ij denote the paired difference between baseline and alcohol for the jth
ROI, then the (univariate) repeated measures ANOVA model is: Yij   j  Si   ij , where
 j is the (fixed) effect of ROI j, Si is the (random) effect of subject i,  ij is the random
error independent of Si . With normality assumptions, we have
iid
Si
N  0,  s2  and  ij
i  1,
iid
N  0,  2  . Let Yi  Yi1 , Yi 2 ,
, n , where    1 ,  2 ,
, Yik  , we have Yi
'
iid


Nk  ,  ,
,  k  and
'

 s2   2
 s2
 s2 
1 


 1
2
2
2
2
 
s
 s 
s 
 s2
2 

with   2
and






 s   2




2
1
 s2
 s2   2 
 
  s
 2   s2   2 . This particular structure of the 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 twoway 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, F0
H0
Fk 1, n 1 k 1 . We
will reject the null hypothesis at the significance level  if F0  Fk1, n 1 k 1 .
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 Yi
iid


N k  ,  , i  1,
, n . Certainly we have more parameters
In this model than the univariate repeated measures ANOVA model, and thus we must
  CQC  CY 
have at least  k  1 subjects. The test statistic is T02  n  n  1 CY
n
n
i 1
i 1
where Y   Yi , Q  

1 1 0
0 1 1
'
Yi  Y Yi  Y , and C  


0 0 0
Under the null hypothesis, T02

' 1
'
0
0 
.


1 1
0
0

Hotelling ' s Tk21,n1 . The Hotelling’s Tk21,n 1 statistic has
the following relationship with the F statistics: F0  T02
n  k 1
 k 1 n 1
H0
Fk 1,n  k 1 . We will reject
the null hypothesis at the significance level  if F0  Fk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.
Example. One-way Repeated Measures ANOVA (n=4, k=4)
Subject
ROI 1
ROI 2
ROI 3
1
5
9
6
2
7
12
8
3
11
12
10
4
3
8
5
ROI 4
11
9
14
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;
SAS Output: One-way Repeated Measures Analysis of Variance
1. Estimated Error Variance-Covariance Matrix
ROI_1
ROI_2
ROI_3
ROI_1 ROI_2 ROI_3
10.00
8.00
7.00
8.00
16.75
11.75
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
Orthogonal Components
DF Criterion Chi-Square Pr > ChiSq
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
Source
ROI
Error(ROI)
DF
Anova SS
3 50.25000000
9 13.25000000
Adj Pr > F
Mean Square F Value Pr > F G - G H - F
16.75000000
11.38 0.0020 0.0123 0.0020
1.47222222
Greenhouse-Geisser Epsilon
Huynh-Feldt Epsilon
0.5998
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, 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 pvalue).