16 SPSS UPDATE 7-9

where

Z, — l ifi < 2(level 2 0f A is applied)
0 otherwise

U,x <1 ifj < 2 (level 2 of B is applied)
0 otherwise

If the interaction terms between the covariate and factor variables are added to the model, then

Yix utB(Xjy-X) ta, Ze taUjje tasZi; Uj« —
(GB) X a - K Z (ABA ja JUH ABA Nje Z Zaje jaj

A test of H,: (aB), < (aB), < (aB); < 0 is eguivalent to testing the hypothesis that the regression
slopes are the same for all cells. This test can be performed by specifying the following MANOVA
commands:
MANOVA Y x BY A(1,2),B(1,2)/
ANALYSIS<Y /
DESIGN<X,A,B,A BY B,
X BY A t X BY B t X BY A BY B/

The effects X BY A, X BY B, and X BY A BY Bare lumped together to provide the test of the
parallelism hypothesis. If the test is not significant, the usual analysis of covariance model can be
used to perform the analysis.

If the assumption of the homogeneity of the slope is violated, one of the following three
models might be used:

1. The model of different slopes for each level of factor A. This model can be justified by testing
(aB). < (aB), < 0. The MANOVA specification for the test is

DESIGN-X,A,B,A BY B,
X BY B t X BY A BY B/

If the test is not significant, the following DESIGN specifications can be used for the analysis
of covariance of this model:

DESIGN-X WITHIN A, A, B, A BY B/
DESIGN<A, B, A BY B, X WITHIN A/

The X WITHIN A term represents the regression effects that are separately estimated within
each level of A. The first DESIGN specification reguests the main effects and interaction
adjusted for the covariate effects. The second DESIGN specification gives the regression
effect (last term) adjusted for A, B and AB.

2 The model of different slopes for each level of factor B. The appropriate test for this model is
(aB), — (af); < 0 and is obtained by specifying
DESIGN< X, A, B, A BY B, X BY A 4 X BY A BY B/

The analysis of covariance is obtained by using

DESIGN<X WITHIN B, A, B, A BY B/
DESIGN<A, B, A BY B, X WITHIN B/

3 The model of different slopes for each cell. The MANOVA specifications for this model are

DESIGN<X WITHIN A BY B, A, B, A BY B/
DESIGN<A, B, A BY B, X WITHIN A BY B/

The X WITHIN A BY B term represents the regression slopes, which are different for each
cell.

The same procedure can be simply extended to multiple covariates. For a 2 X 2 design
with covariates Zl and Z2, the X term is replaced by CONTIN(Z1,Z2) throughout the
DESIGN specification discussed above. The keyword CONTIN incorporates Z1 and Z2 into a
single effect. Thus the following specfications may be used for the analysis of covariance for
model 1 with covariates Zl and Z2.

MANOVA Y Z1 Z2 BY A(1,2) B(l,2)/
ANALYSIS-Y /
DESIGN-ZL, Z2, A, B, A BY B,
CONTIN(Zl,Z2) BY B t CONTIN(Z1,Z2)BY A BY B/

DESIGN:CONTIN(Z1,Z2) WITHIN A, A, B, A BY B/
DESIGN<A, B, A BY B, CONTIN(Z1,Z2) WITHIN A/

The first DESIGN specification is used to test the model, while the second and third models
are for the analysis of covariance.