Difference between revisions of "Contrast"
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| − | The scalar product | + | 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 | + | 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. |
| − | {{TEX| | + | {{TEX|done}} |
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.
Contrast. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contrast&oldid=53674