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The problem of testing hypotheses with respect to the values of scalar products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m0653501.png" /> of a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m0653502.png" />, the coordinates of which are unknown parameters, with a number of given vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m0653503.png" />. In statistical research the multiple comparison problem often arises in [[Dispersion analysis|dispersion analysis]] where, as a rule, the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m0653504.png" /> are chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m0653505.png" />, and the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m0653506.png" /> itself, in this case, is called a [[Contrast|contrast]]. On the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m0653507.png" /> are unknown mathematical expectations of one-dimensional normal laws, J.W. Tukey and H. Scheffé proposed the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m0653509.png" />-method and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065350/m06535011.png" />-method, respectively, for the simultaneous estimation of contrasts, which are the fundamental methods in the problem of constructing confidence intervals for contrasts.
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The problem of testing hypotheses with respect to the values of scalar products $  \pmb\theta  ^ {T} \cdot \mathbf c $
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of a vector $  \pmb\theta = ( \theta _ {1} \dots \theta _ {k} )  ^ {T} $,  
 +
the coordinates of which are unknown parameters, with a number of given vectors $  \mathbf c = ( c _ {1} \dots c _ {k} )  ^ {T} $.  
 +
In statistical research the multiple comparison problem often arises in [[Dispersion analysis|dispersion analysis]] where, as a rule, the vectors $  \mathbf c $
 +
are chosen so that $  c _ {1} + \dots + c _ {k} = 0 $,  
 +
and the scalar product $  \pmb\theta  ^ {T} \cdot \mathbf c $
 +
itself, in this case, is called a [[Contrast|contrast]]. On the assumption that $  \theta _ {1} \dots \theta _ {k} $
 +
are unknown mathematical expectations of one-dimensional normal laws, J.W. Tukey and H. Scheffé proposed the $  T $-
 +
method and the $  S $-
 +
method, respectively, for the simultaneous estimation of contrasts, which are the fundamental methods in the problem of constructing confidence intervals for contrasts.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Scheffé,  "The analysis of variance" , Wiley  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''3. Design and analysis, and time series''' , Griffin  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Scheffé,  "The analysis of variance" , Wiley  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''3. Design and analysis, and time series''' , Griffin  (1983)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Miller,  "Simultaneous statistical inference" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Miller,  "Simultaneous statistical inference" , McGraw-Hill  (1966)</TD></TR></table>

Revision as of 08:02, 6 June 2020


The problem of testing hypotheses with respect to the values of scalar products $ \pmb\theta ^ {T} \cdot \mathbf c $ of a vector $ \pmb\theta = ( \theta _ {1} \dots \theta _ {k} ) ^ {T} $, the coordinates of which are unknown parameters, with a number of given vectors $ \mathbf c = ( c _ {1} \dots c _ {k} ) ^ {T} $. In statistical research the multiple comparison problem often arises in dispersion analysis where, as a rule, the vectors $ \mathbf c $ are chosen so that $ c _ {1} + \dots + c _ {k} = 0 $, and the scalar product $ \pmb\theta ^ {T} \cdot \mathbf c $ itself, in this case, is called a contrast. On the assumption that $ \theta _ {1} \dots \theta _ {k} $ are unknown mathematical expectations of one-dimensional normal laws, J.W. Tukey and H. Scheffé proposed the $ T $- method and the $ S $- method, respectively, for the simultaneous estimation of contrasts, which are the fundamental methods in the problem of constructing confidence intervals for contrasts.

References

[1] H. Scheffé, "The analysis of variance" , Wiley (1959)
[2] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 3. Design and analysis, and time series , Griffin (1983)

Comments

References

[a1] R. Miller, "Simultaneous statistical inference" , McGraw-Hill (1966)
How to Cite This Entry:
Multiple comparison. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiple_comparison&oldid=11413
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article