View source for Contrast

''comparison''

The scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025960/c0259601.png" /> of a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025960/c0259602.png" /> whose coordinates are unknown parameters, by a given vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025960/c0259603.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025960/c0259604.png" />. For example, the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025960/c0259605.png" /> of the unknown mathematical expectations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025960/c0259606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025960/c0259607.png" /> of two one-dimensional normal distributions is a contrast. In analysis of variance, the problem of [[Multiple comparison|multiple comparison]] if often considered; this problem is concerned with the testing of hypotheses concerning the numerical values of several contrasts.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Scheffé,   "Analysis of variance" , Wiley  (1959)</TD></TR></table>



====Comments====
Contrasts are invariant under addition of all components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025960/c0259608.png" /> 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.