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Contrasts

Posted February 26th, 2009 by mrychlik

A contrast is a linear functional of the sample. Contrasts are a central notion of ANOVA. They are more closely related to the research hypothesis rather than the statistical hypothesis.

A contrast between the means is a functional of the means:

Thus, a contrast is a function $ c:\mathbb{R}^t\to \mathbb{R} $, where $ t $ is the number of groups. More precisely:

\[ c(X) = \sum_{j=1}^t c_j\cdot\mu_j \]

where $ \mu_j $ is the $ j $-th treatment mean. Additionally, we require that $ \sum_{j=1}^t c_j = 0 $. Thus, the total mean $ \mu = \bar{x}_{\cdot\cdot} $ is not a contrast. Moreover, all contrasts are orthogonal to the mean. Hence, under the assumption of normality, contrasts are independent from the total mean.