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) |
Multiple comparison. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiple_comparison&oldid=47930