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Statistical formula for the Tukey test
Notation
Description
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Number of levels for an effect.
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j-th observation at i-th level.
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Harmonic mean of the sample size.
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Number of observations at level i, i=1,...,k.
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Mean at level i, i=1,...,k.
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Sum of squares at level i, i=1,…k.
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Pooled standard deviation from all levels.
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Degrees of freedom for the within-groups mean square.
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Experimentwise error rate under the complete null hypothesis.
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Number of steps between means.
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The upper-ε critical point of the Studentized range distribution.
Range values
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for all values of r.
When finding the critical value R ε,r,f for any pair of means r columns apart, the value S ε,k,f is always used where k is the total number of columns. The lookup table for ε is queried to find the value in the appropriate row for f and the appropriate column for k, regardless of the value of r.
The confidence intervals of the mean difference are calculated using Q i,j instead of Q h.
For all pairs of means
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For any two means i and j being tested
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Constructing homogeneous subsets
Homogenous subsets are only available in diagnostics (see Diagnostics information: Tukey test).
The outermost pair of means have a significant range if:
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If so, test whether:
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Once a whole set is found to be nonsignificant testing stops.
Each time a range proves nonsignificant, the means involved are included in a single group (homogeneous subset). This mean that the columns within a nonsignificant range should be combined into a single column and the test reapplied with the collapsed sets of columns.
Multiple comparisons test for all possible pairs
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for all i < j.
See also
Tukey test