Contrasts in ANOVA Decomposition of Treatment Sums of Squares using

Contrasts in ANOVA 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 c1 , c2,groups, ., ck contrast constants are a k treatment sequence of numbers k such that ci 0 i 1 2

Contrasts and Hypothesis testing A given contrast will test a specific set of hypotheses: k H 0 : ci i 0 i 1 and k H a : ci i 0 i 1 k using C ci Yi. i 1 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 Contrast 2 0 -1 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. 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 definition is that two contrasts c1, c2 ,.,formal ck ' ' ' cand c ,., c 1, 2 k k ' c c i i 0 1 are orthogonali 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: 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. 8

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”. 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