Multiple comparison

The problem of testing hypotheses with respect to the values of scalar products $\theta {\theta }^T\cdot \mathbf{c}$ of a vector $\theta \theta =({\theta }_1,\dots ,{\theta }_k{)}^T$, 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 $c_1+\cdots +c_k=0$, and the scalar product $\theta {\theta }^T\cdot \mathbf{c}$ 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.

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