Contrasts in ANOVADecomposition 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

such that

1 2,, ......, kc c c

1

0k

ii

c

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.

.1

k

i ii

C c Y

01

: 0k

i ii

H c

1

: 0k

a i ii

H c

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, andthey are different from Control

The Layout of the experiment:

The two contrasts:Control Drug A Drug B

Contrast 1 -1 ½ ½Contrast 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.

Orthogonal ContrastsThe 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:

1, 2 ,..., kc c c

' ' '1, 2 ,..., kc c c

'

1

0k

i ii

c c

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.

1 2Trt C CSS SS SS

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”.

TrtF SS MSE

1CF SS MSE

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 termsContrast 1 is the Main effect for DrugContrast 2 is the Main effect for GenderContrast 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