Contrast

comparison

The scalar product $\theta^T.c$ of a vector $\theta = (\theta_1, \ldots, \theta_k)^T$ whose coordinates are unknown parameters, by a given vector $c = (c_1, \ldots, c_k)^T$ such that $c_1+\ldots + c_k = 0$. For example, the difference $\theta_1 - \theta_2 = (\theta_1, \theta_2)(1,-1)^T$ of the unknown mathematical expectations $\theta_1$ and $\theta_2$ of two one-dimensional normal distributions is a contrast. In analysis of variance, the problem of multiple comparison if often considered; this problem is concerned with the testing of hypotheses concerning the numerical values of several contrasts.

#### References[edit]

[1] |  H. Scheffé, "Analysis of variance" , Wiley (1959)

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#### Comments[edit]

Contrasts are invariant under addition of all components of $\theta$ by the same constant, and therefore do not depend on the arbitrary "general level" of the measurements. This can be a great advantage in certain settings.