a
H

Dimension reduction analysis can be interpreted as follows: If the roots from n, to s are
not significant (in other words, if the s - n, t£ 1 smallest canonical correlations are not
significantly different from. zero), we may say that the data do not provide evidence of
association in more than n, — 1 dimensions (only n, — 1 discriminant functions are significant).
In the dental calculus example, only one canonical correlation is significant at the 0.05 level for
the TR effect.

5 Univariate analysis of variance results for each of the g response variables. In our example,
Figure 1.32f gives the results obtained for the effect TR.

Figure 1.32f

UNIVARIATE F-TESTS WITH (4,100) D. F.

VARIABLE HYPOTH. SS ERROR SS HYPOTH. MS ERROR MS F SIG, OF F
RCAN 6.18306 137.89515 1.54577 — 1.37895 1.12097 351
RLI 28.07315 261.87433 1.01829 2.61874 2.68002 036
RCI 69.55358 423. 98046 17.38839 4.23980 4.10123 004

The sum of sguares for the tested efiect (HYPOTH. SS) and for the error (ERROR SS)
of each response variable are the appropriate diagonal elements of S, and S, respectively.
Output for the YEAR effect and the YEAR BY TR interaction is given in Figure 1.32g.

Figure 1.32g

EFFECT .. YEAR BY TR
MULTIVARIATE TESTS OF SIGNIFICANCE (S < 1, M < 1/2, N < 48)

TEST NAME VALUE APPROX. F HYPOTH. DF ERROR DF SIG. OF F
PILLAIS 02445 81881 3.00 98.00 487
HOTELLINGS 02507 81881 3.00 98.00 487
WILKS .97555 -B1881 3.00 98.00 487
ROYS . 02445

EIGENVALUES AND CANONICAL CORRELATIONS
ROOT NO. EIGENVALVE PCT. CUM. PCT. CANON. COR.
1 02507 100.00000 100.00000 15637
DIMENSION REDUCTION ANALYSIS
ROOTS WILKS LAMBDA F HYPOTH. DF ERROR DF SIG. OF F
1 T0 1 97555 -B1881 3.00 98.00 487

UNIVARIATE F-TESTS WITH (1.100) D. F.

VARIABLE HYPOTH. SS ERROR SS HYPOTH. MS ERROR MS F SIG. OF F
RCAN - 09862 137.89515 -09862 1.37895 07152 -T90
RLI 1.08877 261.87433 1.08877 2.61874 -41576 521
RCI 9.73563 423.98046 9.73563 4.23980 2.29625 133

EFFECT .. YEAR
MULTIVARIATE TESTS OF SIGNIFICANCE (S < 1, M - 1/2, N - 48)

TEST NAME VALVE APPROX. F HYPOTH. DF ERROR DF SIG. OF F
PILLAIS . 04077 1.38843 3.00 98.00 251
HOTELLINGS 04250 1.38843 3.00 98.00 251
WILKS 95923 1. 38843 3.00 98.00 251
ROYS 04077

EIGENVALUES AND CANONICAL CORRELATIONS
ROOT NO. EIGENVALUE PCT. CUM. PCT. CANON. COR.
1 - 04250 100.00000 100.00000 20192
DIMENSION REDUCTION ANALYSIS
ROOTS WILKS LAMBDA F HYPOTH. DF ERROR DF SIG. OF F
1T01 . 95923 1. 38843 3.00 98.00 .251

UNIVARIATE F-TESTS WITH (1,100) D. F.

VARIABLE HYPOTH. SS ERROR SS ERROR MS F SIG. OF F
RCAN 3.54279 137 .89515 1.37R95 2.5869]9 112
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