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Each factor is then partitioned so that the first partition contains the linear component of the
orthogonal polynomial contrast:

PARTITION(A) /
PARTITION(B) /

Lastly, the design specifies a main effects model along with the linear x linear component of the
interaction:

DESIGN-A, B, A(1) BY B(l)/

The resulting ANOVA table appears in Figure 1.29b.

Figure 1.29b

TESTS OF SIGNIFICANCE FOR Y USING SEGUENTIAL SUMS OF SOUARES

SOURCE OF VARIATION SUM OF SOUARES DF MEAN SOUARE F SIG. OF F
RESIDVAL , 34.33855 5 6.86771

CONSTANT 1452.00000 1 1452.00000 211.42418 -000
A 56.00000 2 28.00000 4.07705 089
B 438.00000 3 146.00000 21.25890 .003
A(1) BY B(l) 17.66145 1 17.66145 2.57166 .l70

The F test for the A(1) BY B(1) interaction is Tukey's test for nonadditivity.
Note that Tukey's test for nonadditivity can be extended to higher-order factorial experiments.

1.30 Simple Effects

The presence of a significant interaction in a two-way design precludes the testing of the main
effects. Instead, the effect of one factor differs at each level of the other factor. Freguently one may
wish to test the significance of these differential effects. Such tests are generally called tests of
simple effects.

Simple effects can be tested in SPSS-MANOVA by using the nesting facility of the DESIGN
subcommand. As an example, consider the data presented in Figure 1.2 for which the ANOVA
table appears in Figure 1.3a. Here the interaction is significant at the 0.006 level. Simple effects
tests are desired to examine the category differences for each of the drugs. The following DESIGN
subcommand accomplishes this:

DESIGN<DRUG, CAT WITHIN DRUG(l

), CAT WITHIN DRUG(2),
CAT WITHIN DRUG(3)/

Here CAT WITHIN DRUG(1) tests the difference in means between category 1 and category 2 for
the first level of drug. Similarly, the two successive effects test for category differences for the
second and third drugs, respectively. Note that DRUG appears first in the design. This eliminates
any confounding effects of CAT. Figure 1.30a presents the output of this design.

Figure 1.30a

TESTS OF SIGNIFICANCE FOR Y USING SEGUENTIAL SUMS OF SOVARES

SOURCE OF VARIATION SUM OF SOUARES DF  NEAN SOUARE F SIG. OF F
WITHIN CELLS 106.00000 12 8.83333

CONSTANT 88200000 Il 882 ..00000 99.84906 0.0
DRUG 48.00000 2 24.00000 2.71698 106
CAT WITHIN DRUG(1) 54.00000 1 54.00000 6.11321 .029
CAT WITHIN DRUG(2) 54.00000 1 54..00000 6.1132. .029
CAT WITHIN DRUG(3) 54.00000 1 54 .00000 6.11321 .029

The simple effects of the three drugs within each category of patients can be tested in the same
manner.

In higher-order designs one may want tests of simple effects for both interactions and main
effects. For example, consider a three-way factorial design with factors A, B, and C, each with two
levels. Should the three-way interaction appear significant then an examination of the second-order
interaction terms at various levels of the third factor would be in order. To accomplish this, the
following DESIGN subcommands would be used:

DESIGN-A, B, C, A BY B, A BY C,
B BY C WITHIN A(l), B BY

BY C,
WITHIN A(2)/

B
loj
DESIGN-A, B, C, A BY B, A BY C, B BY C,
A BY B WITHIN C(l), A BY B WITHIN C(2)/
DESIGN<A, B, C, A BY B, A BY C, B BY C,
A BY C WITHIN B(l), A BY C WITHIN B(2)/