Multiple comparison

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)

References

[a1]	R. Miller, "Simultaneous statistical inference" , McGraw-Hill (1966)