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:
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If M yields more than one response the coefficients of each transformation sum to zero.
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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 Y variables.
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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.
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Select Analyze > Fit Model.
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In the Construct Model Effects panel, select drug. In the Select Columns panel, select dep1. Click Cross.
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For Personality, select Manova.
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Click Run.
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Select the check box next to Univariate Tests Also.
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Time should be entered for YName, and Univariate Tests Also should be selected.
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Click OK.
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Figure 8.8 Sphericity Test
The sphericity test checks the appropriateness of an unadjusted univariate F test for the within-subject effects using the Mauchly criterion to test the sphericity assumption (Anderson 1958). The sphericity test and the univariate tests are always done using an orthonormalized M matrix. You interpret the sphericity test as follows:
The univariate F statistic has an approximate F-distribution even without sphericity, but the degrees of freedom for numerator and denominator are reduced by some fraction epsilon (ε). Box (1954), Greenhouse and Geisser (1959), and Huynh-Feldt (1976) offer techniques for estimating the epsilon degrees-of-freedom adjustment. Muller and Barton (1989) recommend the Greenhouse-Geisser version, based on a study of power.
The epsilon adjusted tests in the multivariate report are labeled G-G (Greenhouse-Geisser) or H-F (Huynh-Feldt), with the epsilon adjustment shown in the value column.