SPSS UPDATE 7-9

Figure 1.9

TESTS OF SIGNIFICANCE FOR Y USING SEOUENTIAL SUMS OF SOUARES

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

CONSTANT 882 .00000 l 882 ..00000 99 .84906 0.0
CAT 18.00006 l 18.00000 2.03774 .179
DRUG(1) 36.00000 l 36. 00000 4.07547 066
DRUGI 2) 12.00000 l 12.00000 1.35849 .266
CAT BY DRUGI) > 144.00000 1 144.00000 16.30189 .002
CAT BY DRUG(2) 0.0 l 0.0 0.0 1.000

ON OOODOEOO

The above discussion of orthogonal contrasts assumeS that the cell freguencies are egual. For
the use of the orthogonal contrasts in unbalanced designs, see Section 1.16.

1.10 Designs with Unegual Cell Freguencies

In many experiments, it may not be possible to have egual numbers of observations for each cell.
Such designs are termed unbalanced or nonorthogonal. In nonorthogonal designs the effects are
correlated with each other and cannot be estimated independently of one another. That is, the
component sum of sguares will not add up to the total sum of sguares because the main effects will
usually not be independent of each other and the interaction effects wili not be independent of the
main effects. Different ANOVA solutions can be obtained for the same design depending on the
"type" of sum of sguares calculated. For example, in an unbalanced design with two factors A and
B. the sum of sguares for main effect A differs depending on whether effect A is the only one in the
model or whether it is added to a model already containing effect B.

1.11 Seguential Sums of Sguares (Fitting Constants)

Seguential sums of sguares are the default type calculated by MANOVA. The sums of sguares for
each effect are ''adjusted'" for all effects previously entered into the model. That is, the sum of
sguares for an effect is adjusted only for all terms to the left of it in the DESIGN subcommand. All
terms to the right are ignored. Therefore the order in which terms are specified on the DESIGN
subcommand, or the MANOVA command if a DESIGN subcommand is not present, is important.

Different orders may produce different results. For the two-factor design specified using
DESIGN<A,B/

the B main effect is adjusted for A and the overali mean, while A is adjusted only for the mean. If
the model is specified as .

DESIGN-B,A/

the A main effect is adjusted for B and the mean, while the B effect is adjusted only for the mean.

Since several DESIGN subcommands can be used in one invocation of the MANOVA
procedure, it is possible to obtain easily various sums of sguares. For example, in a two-factor
model, to obtain the main effect sum of sguares adjusted for other main effects and the interaction
effect adjusted for main effects, specify both

DESIGN<A,B,A BY B/
DESIGN<B,A, A BY B/

The first ANOVA table will contain B adjusted for A, and A BY Baadjusted for both main effects.

The second ANOVA table will contain A adjusted for B and the interaction adjusted for both main
effects.

1.12 Regression Model Sum of Sguares (Weighted Sguares of Means)
It is possible to obtain sums of sguares adjusted for all effects listed on the DESIGN subcommand,
by specifying

METHOD-SSTYPE(UNJOVE) /

For the two-factor model this results in main effect A being adjusted for both B and the A BY B
interaction. Similarly B is adjusted for A and the interaction, while the interaction is adjusted for
main effects A and B.

1.13 Decomposition and Bias Matrices
If the design is unbalanced and the default seguential sums of sguares are used, the decomposition
and bias matrices may be of interest. They are obtained by specifying

PRINT-DESIGN( DECOMP , BIAS) /