Newman-Keuls (S-N-K)

The test statistic is:

$t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{\sum _{j=1}^J\sum _{i=1}^{n_j}{w}_{ij}(x_{ij}-{\overline{x}}_j{)}^2}{v}(\frac{1}{e_1}+\frac{1}{e_1})}}$

where:

${\overline{x}}_1$ and ${\overline{x}}_2$ are the means of the two groups being compared and ${\overline{x}}_j$ 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,

$v=(J-1)(\sum _{j=1}^J{e}_j-1)$ for Repeated Measures and $v=\sum _{j=1}^J{e}_j-J$ 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).

Circumstances when this test is applied

• Computing either:
  • Column Comparisons
  • A Planned Tests Of Statistical Significance for a row or column of a table.
• This test has been selected in the Statistical Assumptions setting of Multiple comparison correction > Column comparisons

See also

• Multiple Comparisons (Post Hoc Testing)
• Planned ANOVA-Type Tests
• ANOVA-Type Tests - Comparing Three or More Groups
• How to Specify Comparisons for ANOVA-Based Tests