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 , where is the effect of alcohol on the jth ROI,
The Univariate Analysis Approach
For subject i, letdenote the paired difference between baseline and alcohol for the jth ROI, then the (univariate) repeated measures ANOVA model is: , where is the (fixed) effect of ROI j, is the (random) effect of subject i,is the random error independent of . With normality assumptions, we have
and . Let , we have , , where and with and . 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 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 and respectively. That is,. We will reject the null hypothesis at the significance level if .
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 , . Certainly we have more parameters
In this model than the univariate repeated measures ANOVA model, and thus we must have at least subjects. The test statistic is where , , and .
Under the null hypothesis, . The Hotelling’s statistic has the following relationship with the F statistics: . We will reject the null hypothesis at the significance level if .
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 / ROI 41 / 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
;
procanovadata=repeatM;
title'one-way repeated measures ANOVA';
model ROI1-ROI4 = /nouni;
repeated ROI 4 (1234)/printe;
run;
SAS Output: One-way Repeated Measures Analysis of Variance
- 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, 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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