Bartholomew's tests for ordered alternatives
Consider $k$ treatment means $\bar{X}_i$ and an estimate $S^2$ of variance:
\[ \begin{align*} &\bar{X}_i~\sim~\mathrm{n.i.i.d.}(\mu,~\sigma^2),\quad (i=1, 2, \cdots, k) \tag{1}\\ &S^2~\sim~\sigma^2\,\chi^2(\nu)/\nu. \tag{2} \end{align*} \]Define a statistic $\bar{B}$ by
\[ \bar{B} = \max_{\boldsymbol{c}} \frac{\sum c_i\,\bar{X}_i}{S\,\sqrt{\sum c_i^2}}, \tag{3} \]where the right-hand-side is maximized over all ordered contrasts,
\[ c_1 \leq c_2 \leq \cdots \leq c_k,\quad \sum_{i=1}^k c_i = 0. \tag{4} \]The tables give the upper quantiles $\bar{B}(k, \nu; \alpha)$:
\[ \Pr\{\bar{B} \gt \bar{B}(k, \nu; \alpha)\} = \alpha. \tag{5} \]