The second example is adapted from Cochran and Cox (1957, p. 46). The experiment was
conducted to compare the effectiveness of four soil fumigants in keeping down the number of
eelworms in the soil. The fumigants were CN, CS, CM, and CK. Each fumigant was tested both in
a single and double dose. The control was used as another treatment. The nine treatments are
denoted as C00 (control), CNI (CN with single dose), CS1, CMI, CKI1, CN2 (CN with double
dose), CS2, CM2, and CK2. There were four replications for each dose of each fumigant and 16
replications of the control. The desired subdivisions of the treatment sum of sguares are as follows:

1 Ifthe effect of the fumigants is proportional to the dose, then both CNi and CN2/2 are the
estimate of the effect of CN per unit dose. The pooled estimate of this effect is (CN14
2(CN2))/5. The differences in the linear responses to the four fumigants can be measured by
the following three contrasts:

(0 1-1 0 0 2-2 0 0. .
(0 1 1-2 0 2 2 -4 0)
(0 1 1 1-3 2 2 2 -6)

2 The curvature of the treatment CN is measured by C00 — (2CNI) 4 CN2. The differences in
curvature are compared by the guantities CN2 -2(CNI), (the C00 term cancelled out in the
comparison) or by the following three contrasts:

(0 2-2 0 0-1 1 0 0)
(0 2 2-4 0-i-i 2 0)
(0 2 2 2 -6 -i -i -l1 3)

3 The sum of sguares between levels (control: 0 level; treatments with single dose: level 1;
treatments with double level: level 2) can be partitioned into a component due to the linearity
between levels and one representing the curvature between levels. The former is given by the
comparison of — l(level 0) -£O(level 1) -1(level 2), or the contrast (-40000111 1). The
curvature between levels is measured by l(level0) —2(level 1) 4-1(level 2), or the contrast (—4
—2-2-2-21111).

The above partitions can be summarized by the following MANOVA CONTRAST subcommand:

CONTRAST ( TREATMNT)<SPECIAL(1l 1 1 1 l
-l 0 0

VONDueEE-e-

U

a]

o

o

U

-
-rucobooc

akooooco
paj

PARTITION( TREATMNT)<(3 3 l l)/
DESIGN<BLOCK TREATMNT(l1) TREATMNT(2)
TREATMNT(3) TREATMNT(4)/

Note that TREATMNT(1), TREATMNT(2), TREATMNT(3), and TREATMNTGO) are the
effects of the differences in linear response, in curvature, linear response between levels, and
curvature between levels, respectively. Also, it can be verified that the effects are orthogonal.

1.17 Analysis of Covariance

SPSS MANOVA can perform an analysis of covariance in which interval-scaled independent
variables (covariates) are used in conjunction with categorical variables (factors). Analysis of
covariance is a technigue that combines the features of analysis of variance and regression. A
two-way analysis of covariance model with two covariates can be described as follows:

Yi 5 prtačtyj H(oy)ijHB(Xnja—X) Bel Kaje Ža) eije

where Y;, is the dependent variable, a;, y; are the main effects, and (a-y); is the interaction effect.
X,, X, are the covariates, and X,, X, are the means for the two covariates.

In the covariance model, Y has a (multiple) linear regression (see Section 1.38) on X, and X,
with regression coefficients B, and B;. The regression procedure is used to remove the variation in
the dependent variable due to covariates.

From the standpoint of the analysis of variance model, the covariate model is essentially an
analysis of variance model on the corrected scores or

Y ir Bel Xrijx- X))- Bl Neja — X2) < prat yyHaY)teja

which is the analysis of variance model for Y adjusted for the two covariates.

The following illustrative example is taken from Snedecor and Cochran (1967, p. 422). The
model is a one-way analysis of covariance with one covariate. The experiment was conducted to
evaluate the effect of three drugs on the treatment of leprosy. For each patient, six sites were
selected. The variate X, based on laboratory tests, is a score representing the abundance of leprosy

MANOVA

3