Decomposition of Treatment Sums of Squares
Decomposition of Treatment Sums of Squares using
prior information on the structure of the treatments
and/or treatment groups
1
Contrasts, notation….
For a One-way 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
such that
2
Contrasts and Hypothesis testing
A given contrast will test a specific set of hypotheses:
and
using
to create an F-statistic with one numerator df.
3
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
4
The Layout of the experiment:
5
The two contrasts:
Control Drug A Drug B
Contrast 1 -1 ½ ½
Contrast 2 0 -1 +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.
6
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
and
are orthogonal if:
7
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:
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.
8
Why?
If all of the differences in the means are described by
one of the contrasts, say the first contrast, then
is more likely to be significant than
Since the signal in the numerator is not combined
with “noise”.
9
Example 2: Two-way ANOVA
10
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
11
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
12
Decomposition of treatment sums of squares involves utilizing prior information about treatment structure to analyze treatment group means through contrasts and hypothesis testing. This process allows for the testing of specific hypotheses and the creation of F-statistics. In an example scenario with control and two equivalent treatments, orthogonal contrasts play a crucial role in decomposing treatment sums of squares for more powerful significance testing.
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Presentation Transcript
Decomposition of Treatment Sums of Squares using prior information on the structure of the treatments and/or treatment groups 1
Contrasts, notation. For a One-way 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 such that 0 i i = , ......, c c c 1 2, k k = c 1 2
Contrasts and Hypothesis testing A given contrast will test a specific set of hypotheses: k c = : 0 H 0 i i = 1 i and k c : 0 H a i i = 1 i k = using C c Y . i i = 1 i to create an F-statistic with one numerator df. 3
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 4
The two contrasts: Control Drug A Drug B Contrast 1 -1 Contrast 2 0 -1 +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. 6
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 c c c 1, 2,..., k and c c ' ' ' k 1, 2,..., c k = ' 0 cc are orthogonal if: i i = 1 i 7
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: = + SS SS SS Trt C C 1 2 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. 8
Why? If all of the differences in the means are described by one of the contrasts, say the first contrast, then = F SS MSE C 1 is more likely to be significant than = F SS MSE Trt Since the signal in the numerator is not combined with noise . 9
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 11
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 12
