The €; are assumed to be independent and normally distributed with mean 0 and variance ož,

and the B;'s are the unknown parameters to be estimated. The multivariate extension of this model
is

Y;<B, £B,X,t... t£B,Xa te; — B, t B'X; te;

where Y; < (Y;, Yi... Y,)' is a vector of g response variables for observation i, the X; are
independent variables, the B; are g X 1 vectors containing the regression parameters,and the €;
vectors are the errors (assumed to be independent and to have a g-variate normal distribution with
mean 0 and covariance matrix x).

1.40 MANOVA Multivariate Regression Analysis

MANOVA provides estimates of B,, B, and X and tests the hypothesis that B < 0. The constant
vector B, is included in the model unless the subcommand

METHOD<ESTIMATION(NOCONSTANT,) /

is included in the MANOVA run. When NOCONSTANT is specified, the regression line or plane is
forced to pass through the origin (i.e., B, is assumed to be 0 in the eguation). Four test statistics
(described in Section 1.32) are given for testing the hypothesis that B < 0: Pillai's criterion,
Hotelling's trace, Wilks' lambda, and Roy's largest root. All of these are functions of the nonzero
eigenvalues of S,S;!, where S, is the regression SSCP matrix and S, is the error SSCP matrix.

In Figure 1.40a (taken from Finn, 1974), the dependent variable consists of two divergent
measures of achievement, synthesis (SYNTH) and evaluation (EVAL), and the independent
variables are a general intelligence index (INTEL) and three measures of creativity (CONOBV,
CONRMT, and JOB). Three cross products between the creativity measures and INTEL are
formed to represent the interaction terms of the model. Figure 1.40a shows the standard SPSS
command file for this problem. COMPUTE statements are used to create the interaction terms.

Figure 1.40a
RUN NAME MULTIVARIATE MULTIPLE REGRESSION
COMMENT DATA ARE TAKEN FROM FINN(1974) C-3

VARIABLE LIST  SYNTH EVAL CONOBV CONRMT JOB INTEL
INPUT MEDIUM CARD

INPUT FORMAT  FREEFIELD

MISSING VALUES SYNTH TO INTEL(9.9)

N OF CASES UNKNOWN

COMPUTE CIl<CONOBV"INTEL

COMPUTE CI2<CONRMT"INTEL

COMPUTE CI3-JOB"INTEL

MANOVA SYNTH EVAL WITH INTEL CONOBV CONRMT JOB CI1l CI2 CI3/

PRINT-DISCRIM(RAW, STAN, ESTIM, COR) /
READ INPUT DATA

5 l 20.0 5.0 13.0 106.0
0 0 13.0 3.0 10.0 97.0
6 2 9.9 4.0 5.0 90.0
4 2 10.0 3.0 15.0 l2l.0
l 212.0 2.0 4.0 99.0
7 1 25.0 5.0 23.0 120.0
1 2 2l.0 3.0 0 91.0

1l.

3 2 15.0 4.0 12.0 107.0
6 6 22.0 10.0 23.0 143.0
0 0 12.0 2.0 13.0 101.0
4 1 12.0 6.0 10.0 115.0
3 0 10.0 5.0 10.0 97.0
3 l 21.0 3.0 20.0 92.0
END INPUT DATA

Note that no factor variables are specified in the MANOVA procedure card and the keyword
WITH is used to separate the response and independent variables.

The standard output includes muitivariate significance tests and the statistics for parameter
estimation described in Section 1.32. The following tests and statistics are of particular interest:

1 Tests of H,:B—0 and H,:B,<—0. These are automatically printed, along with the multiple R" and
adjusted R" for each response variable regressed on the independent variables. This portion of
the output for Figure 1.40a is given in Figure 1.40b.

MANOVA 43