Duncan’s New Multiple Range Test

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The test statistic is:

[math]\displaystyle{ 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})}} }[/math]


[math]\displaystyle{ \bar x_1 }[/math] and [math]\displaystyle{ \bar x_2 }[/math] are the means of the two groups being compared and [math]\displaystyle{ \bar x_j }[/math] is the mean of the [math]\displaystyle{ j }[/math] of [math]\displaystyle{ J }[/math] 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 [math]\displaystyle{ x_{ij} }[/math],
[math]\displaystyle{ n_j }[/math] is the number of observations in the [math]\displaystyle{ j }[/math]th of [math]\displaystyle{ J }[/math] groups,
[math]\displaystyle{ w_{ij} }[/math] is the Calibrated Weight for the [math]\displaystyle{ i }[/math]th observation in the [math]\displaystyle{ j }[/math] group,
[math]\displaystyle{ e_j }[/math] is the Effective Sample Size for the [math]\displaystyle{ j }[/math] group,
[math]\displaystyle{ v = (J - 1)(\sum^J_{j=1} e_j - 1) }[/math] for Repeated Measures and [math]\displaystyle{ v = \sum^J_{j=1} e_j - J }[/math] otherwise.
[math]\displaystyle{ t }[/math] is evaluated using a Tukey’s Studentized Range distribution with [math]\displaystyle{ v }[/math] degrees of freedom for [math]\displaystyle{ g }[/math] groups, where [math]\displaystyle{ g }[/math] is determined by a step down analysis and the resulting p-value is adjusted using [math]\displaystyle{ p = 1 - (1 - p)^{1 / (g - 1)} }[/math] (see Duncan, D B.; Multiple range and multiple F tests. Biometrics 11:1–42, 1955).

Circumstances when this test is applied

See also