Duncan’s New Multiple Range Test

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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 jth of J groups,
w_{ij} is the Calibrated Weight for the ith 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 and the resulting p-value is adjusted using p = 1 - (1 - p)^{1 / (g - 1)} (see Duncan, D B.; Multiple range and multiple F tests. Biometrics 11:1–42, 1955).

Circumstances when this test is applied

See also