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''comparison''
 
''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.
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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|multiple comparison]] if often considered; this problem is concerned with the testing of hypotheses concerning the numerical values of several contrasts.
  
 
====References====
 
====References====
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====Comments====
 
====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.
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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.
  
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Latest revision as of 08:11, 13 February 2024

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

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


Comments

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.

How to Cite This Entry:
Contrast. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contrast&oldid=53674
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article