# Newman-Keuls (S-N-K)

The test statistic is:

${\displaystyle t={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{\sqrt {{\frac {\sum _{j=1}^{J}\sum _{i=1}^{n_{j}}w_{ij}(x_{ij}-{\bar {x}}_{j})^{2}}{v}}({\frac {1}{e_{1}}}+{\frac {1}{e_{1}}})}}}}$

where:

${\displaystyle {\bar {x}}_{1}}$ and ${\displaystyle {\bar {x}}_{2}}$ are the means of the two groups being compared and ${\displaystyle {\bar {x}}_{j}}$ is the mean of the ${\displaystyle j}$ of ${\displaystyle J}$ 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 ${\displaystyle x_{ij}}$,
${\displaystyle n_{j}}$ is the number of observations in the ${\displaystyle j}$th of ${\displaystyle J}$ groups,
${\displaystyle w_{ij}}$ is the Calibrated Weight for the ${\displaystyle i}$th observation in the ${\displaystyle j}$ group,
${\displaystyle e_{j}}$ is the Effective Sample Size for the ${\displaystyle j}$ group,
${\displaystyle v=(J-1)(\sum _{j=1}^{J}e_{j}-1)}$ for Repeated Measures and ${\displaystyle v=\sum _{j=1}^{J}e_{j}-J}$ otherwise.
${\displaystyle t}$ is evaluated using a Tukey’s Studentized Range distribution with ${\displaystyle v}$ degrees of freedom for ${\displaystyle g}$ groups, where ${\displaystyle g}$ 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).