View source for 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 $
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====
<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====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Miller,   "Simultaneous statistical inference" , McGraw-Hill  (1966)</TD></TR></table>