Newman-Keuls (S-N-K)

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The test statistic is:

$t=\frac{\bar x_1-\bar x_2}{\sqrt{\frac{\sum^{J}_{j=1}\sum^{n_j}_{i=1} w_{ij}(x_{ij} - \bar x_j)^2}{v}(\frac{1}{e_1}+\frac{1}{e_1})}}$

where:

$\bar x_1$ and $\bar x_2$ are the means of the two groups being compared and $\bar x_j$ is the mean of the $j$ of $J$ groups,
when applying the test to Repeated Measures, each respondent’s average is initially subtracted from their data and it is this corrected data that constitutes of $x_{ij}$,
$n_j$ is the number of observations in the $j$th of $J$ groups,
$w_{ij}$ is the Calibrated Weight for the $i$th observation in the $j$ group,
$e_j$ is the Effective Sample Size for the $j$ group,
$v = (J - 1)(\sum^J_{j=1} e_j - 1)$ for Repeated Measures and $v = \sum^J_{j=1} e_j - J$ otherwise.
$t$ is evaluated using a Tukey’s Studentized Range distribution with $v$ degrees of freedom for $g$ groups, where $g$ is determined by a step down analysis (see Begun, Janet M.; Gabriel, K. Ruben, 1981, "Closure of the Newman-Keuls Multiple Comparisons Procedure", Journal of the American Statistical Association, 76, 241-245).