The test statistic is:

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

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).

## 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)