k
E
H
:
i
:
H
'
H

Figure 1.32a
RUN NAME DENTAL CALULUS DATA FROM FINN(1974) PAGE C-56
FILE NAME DATA FOR ANTI-CALCULUS AGENT

NUUNIN
MSO. -

VARIABLE LIST 'YEAR,TR,RCAN,RLI,RCI,LCI,LLI,LCAN
INPUT FORMAT  FIXED(2Fl.0,6F2.0)

N OF CASES 107
MISSING VALUES YEAR TO LCAN(BLANK) d
MANOVA RCAN,RLI,RCI BY YEAR(1,2),TR(1,5)/

READ INPUT DATA
lil22il22l

oo210
I1004400
222322

Hoor-o...
ZM OH, o , -
NOHU. . O.
AOOA...
-OLO: -
oooo...

Since no DESIGN specifications are given in Figure 1.32a, a full factorial model is assumed.

The standard output (without the PRINT subcommand) includes

1

General information about the design. This includes the number of observations, the number
of levels of each effect, and the redundant effects (if any) in the model. This output is given in
Figure 1.32b for the dental calculus data. (Three degrees of freedom are lost in the interaction
effect because of empty cells.)

Figure 1.32b

l07 CASES ACCEPTED.
0 CASES REJECTED BECAUSE OF OUT-OF-RANGE FACTOR VALUES.
0 CASES REJECTED BECAUSE OF MISSING DATA.
7 NON-EMPTY CELLS.

CORRESPONDENCE BETWEEN EFFECTS AND COLUMNS OF BETWEEN-SUBJECTS DESIGN

STARTING  ENDING
COLUMN  COLUMN EFFECT NAME

1 l CONSTANT

2 2 YEAR

3 6 TR

T 10 YEAR BY TR

REDUNDANCIES IN DESIGN MATRIX

COLUMN EFFECT
8 YEAR BY TR
9 (SAME)
lo (SAME )

2 Multivariate tests of the significance of each effect in the model. The four test statistics

previously mentioned are given. Each of these statistics is a function of the nonzero
eigenvalues A, of the matrix S,S,!. The number of nonzero eigenvalues, s, is egual to the
minimum of the number of dependent variables, g, and the degrees of freedom for the tested
effect, n,. The distributions of these statistics, under the null hypothesis, depend on g, n,, and
n, (the error degrees of freedom).

Pillai's criterion. This test statistic, sum of X/(1X;), can be approximated by an F variate (see
Pillai, 1960). (The degrees of freedom are a function of g, n,, and n,.) :

Hotelling's trace. This is the statistic T < sum of A;, which is egual to the trace of S,S«!. The
critical points of the distribution of T have been tabulated by Pillai (1960) and depend on S <
min(p,g), M < ([n, - g| — 1)/2, and N < (n, — g — 1)/2. (The values of S, M, and N for each
effect are printed by MANOVA.) MANOVA also gives an approximate F statistic based on T,
where the degrees of freedom depend on g, n, and n,.

Wilks' lambda. his test statistic, product of 1/(1--A,), can be transformed, using Rao's
formula (Rao, 1973), into an approximate F statistic with degrees of freedom determined by
g, n,, and n,.

Roy's largest root criterion. Upper percentage points of the distribution of this test statistic,
N,/(1A;), where A, is the largest eigenvalue of S,S!, can be found in Heck (1960), Pillai
(1967), and Morrison (1976). This distribution, like that of Hotelling's trace, depends on S, M,
and N.,

MANOVA — 33