Decomposition of Treatment Sums of Squares using
prior information on the structure of the treatments
and/or treatment groups
Contrasts, notation….
For a Oneway ANOVA, a contrast is a specific
comparison of Treatment group means. Contrast
constants are composed to test a specific hypothesis
related to Treatment means based upon some prior
information about the Treatment groups. For k
treatment groups, contrast constants are a sequence of
numbers c1 , c2, ......, ck
such that
k
 ci  0
i 1
Contrasts and Hypothesis testing
A given contrast will test a specific set of hypotheses:
k
H 0 :  ci i  0
and
i 1
k
H a :  ci i  0
i 1
using
k
C   ci Yi.
i 1
to create an F-statistic with one numerator df.
Example 1: Control and two equivalent
treatments
Suppose we have two treatments which are supposed
to be equivalent. For example, each of two drugs is
supposed to work by binding to the receptor for
adrenalin. Propanolol is such a drug sometimes used
for hypertension or anxiety.
We may think that:
 the two drugs are equivalent, and
 they are different from Control
The Layout of the experiment:
The two contrasts:
Contrast 1
Contrast 2
Control
-1
0
Drug A
½
-1
Drug B
½
+1
Contrast 1 tests whether or not the Control group
differs from the groups which block the adrenalin
receptors.
Contrast 2 tests whether or not the two drugs differ in
their effect.
Orthogonal Contrasts
 The contrasts in the last example were orthogonal.
 Two contrasts are orthogonal if the pairwise products
of the terms sum to zero.
 The formal definition is that two contrasts
c1, c2 ,..., ck
and
c1,' c2' ,..., ck'
k
'
c
c
are orthogonal if:  i i  0
i 1
Orthogonal Contrasts allow the Trt Sums of
Squares to be decomposed
The Trt Sums of Squares can be written as a sum of two
Statistically independent terms:
SSTrt  SSC1  SSC2
Which can be used to test the hypotheses in the
example. The a priori structure in the Treatments can
be tested for significance in a more powerful way.
Why?
If all of the differences in the means are described by
one of the contrasts, say the first contrast, then
F  SSC1 MSE
is more likely to be significant than
F  SSTrt MSE
Since the signal in the numerator is not combined
with “noise”.
Example 2: Two-way ANOVA
Because there is structure to the Treatment
groups involving Drugs and Gender
We can look into the Main Effects of Drug and Gender
and Interaction via Orthogonal Contrasts
Drug
A
A
B
B
Gender
M
F
M
F
Contrast 1 +1/2 +1/2 -1/2 -1/2
Contrast 2 +1/2 -1/2 +1/2 -1/2
Contrast 3 +1/2 -1/2 -1/2 +1/2
The Contrasts correspond to the Main
Effects and Interaction terms
 Contrast 1 is the Main effect for Drug
 Contrast 2 is the Main effect for Gender
 Contrast 3 is the Interaction term
 The Sums of Squares for these Contrasts adds up to the
Sums of Squares Model in the Two-way ANOVA since
each pair of Contrasts is orthogonal