Modeling and Multivariate Methods - SAS

168 Fitting Multiple Response Models Chapter 5
Multivariate Tests
Univariate Tests and the Test for Sphericity
There are cases, such as a repeated measures model, that allow transformation of a multivariate problem into
a univariate problem (Huynh and Feldt 1970). Using univariate tests in a multivariate context is valid in the
following situations:
• If the response design matrix M is orthonormal (M´M = Identity).
• If M yields more than one response the coefficients of each transformation sum to zero.
• If the sphericity condition is met. The sphericity condition means that the M-transformed responses are
uncorrelated and have the same variance. M´ΣM is proportional to an identity matrix, where Σ is the
covariance of the Ys.
If these conditions hold, the diagonal elements of the E and H test matrices sum to make a univariate sums
of squares for the denominator and numerator of an F test. Note that if the above conditions do not hold,
then an error message appears. In the case of Golf Balls.jmp, an identity matrix is specified as the M-matrix.
Identity matrices cannot be transformed to a full rank matrix after centralization of column vectors and
orthonormalization. So the univariate request is ignored.
Example of Univariate and Sphericity Test
1. Open the Dogs.jmp sample data table.
2. Select Analyze > Fit Model.
3. Select LogHist0, LogHist1, LogHist3, and LogHist5 and click Y.
4. Select drug and dep1 and click Add.
5. In the Construct Model Effects panel, select drug. In the Select Columns panel, select dep1. Click
Cross.
6. For Personality, select Manova.
7. Click Run.
8. Select the check box next to Univariate Tests Also.
9. In the Choose Response menu, select Repeated Measures.
Time should be entered for YName, and Univariate Tests Also should be selected.
10. Click OK.
Figure 5.8 Sphericity Test