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Difference between revisions of "Multiple comparison"

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The problem of testing hypotheses with respect to the values of scalar products  $  \pmb\theta  ^ {T} \cdot \mathbf c $
 
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} $,  
+
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} $.  
+
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 $
 
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 $,  
 
are chosen so that  $  c _ {1} + \dots + c _ {k} = 0 $,  
 
and the scalar product  $  \pmb\theta  ^ {T} \cdot \mathbf c $
 
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} $
+
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 $-
+
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.
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====

Latest revision as of 01:50, 17 June 2022


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=47930
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article