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34 SPSS UPDATE ?-y

For the dental calculus data, the multivariate tests of the hypothesis that there is no TR efiect
(adjusted for the YEAR effect) are presented in Figure 1.32c.

Figure 1.32c

EFFECT .. TR

MULTIVARIATE TESTS OF SIGNIFICANCE (S < 3, M <— 0, N - 48)

TEST NAME VALUE APPROX. F HYPOTH. DF ERROR DF SIG. OF F
PILLAIS 20122 1.79739 12.00 300.00 -D4B
HOTELLINGS 22813 1.83769 12.00 290.00 .D42
WILKS 80733 1.82255 12.00 259.58 045
ROYS 14402

The name of the test statistic is given under TEST NAME and its value listed under
VALUE. For Pillai's criterion, Hotelling's trace, and Wilks lambda, approximate F statistics
are given, with the degrees of freedom under HYPOTH. DE and ERROR DF and the
p-values under SIG, OF F. A comparison (with references) of the powers of these four tests
can be found in Morrison (1976).
£igenvalues and canonical correlations. The nonzero eigenvalues of S,S,' and the correspond-
ing canonical correlations for each effect in the model are given. For example, the results for
the effect TR are shown in Figure 1.32d.

Figure 1.32d

EIGENVALUES AND CANONICAL. CORRELATIONS

ROOT NO. EIGENVALUE PCT. CUM. PCT. CANON. COR.
l -16825 73.75366 73.75366 37950
2 05255 23.02709 96..78075 22340
3 00734 3.21925 100.00000 .08538

The canonical correlation coefficients p; are calculated as p? < X/(1 f X); they are the
canonical correlations between the response variables and the effect. p; also measures the
correlation between the ith canonical variate of the response variables and the tested effect (in
certain linear combinations). The canonical correlations in this example can also be obtained
by using the following dummy variables to represent the YEAR and TR effects.

X, < | ifYEAR<2

0 otherwise
Y, < | ifTR:-2
0 otherwise
Y, < 1 ifTR<S
0 otherwise

If X, is already in the regression eguation (since TR is adjusted for YEAR) and the
within-cells SSCP matrix is used as the error matrix, then the p;'s above are the canonical
correlations between RCAN, RLI and RCI, and Y,, Y;, Y;, and Y,.
Dimension reduction analysis. Dimension reduction analysis, based on Wilks' lambda, is used
to assess the dimensionality of a significant relationship between the response variables and
the tested effect. The first test is based on all the eigenvalues and is eguivalent to the overall
Wilks' lambda test; the second test is performed on all the eigenvalues except the largest, and
so on. Hence the value of Wilks' lambda for testing roots n, to n, is found by calculating the
product from i < n, to i < n,of1/(1:x).

MANOVA also prints the approximate F statistic for each of these Wilks' lambda
statistics. For the effect TR, the output in Figure 1.32e is obtained.

Figure 1.326
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DIMENSION REDUCTION ANALYSIS

ROOTS WILKS LAMBDA F HYPOTH. DF ERROR DF SIG. OF F
l TO 3 .B0733 1.82255 12.00 259.58 . 045
2 TO 3 94316 .9T402 6.00 240.16 443

3 TO 3 99271 36286 2.00 198.00 . 696