Bartholomew's tests by MathJax

The null distribution of the modified Bartholomew's test

Consider $k$ treatment means ${\bar{X}}_{i}$ and an estimate $S^{2}$ of variance:

\begin{aligned} (1) & {\bar{X}}_{i} \sim n . i . i . d . (μ, σ^{2}), (i = 1, 2, \dots, k) \\ (2) & S^{2} \sim σ^{2} χ^{2} (ν) / ν . \end{aligned}

Define a statistic $\bar{B}$ by

\begin{matrix} (3) & \bar{B} = max_{c} \frac{\sum c_{i} {\bar{X}}_{i}}{S \sqrt{\sum c_{i}^{2}}}, \end{matrix}

where the right-hand-side is maximized over all ordered contrasts,

\begin{matrix} (4) & c_{1} \leq c_{2} \leq \dots \leq c_{k}, \sum_{i = 1}^{k} c_{i} = 0. \end{matrix}

The tables give the upper quantiles $\bar{B} (k, ν; α)$ :

\begin{matrix} (5) & Pr {\bar{B} > \bar{B} (k, ν; α)} = α . \end{matrix}