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Statistical formula for the column means test
The column means test is performed separately for each relevant pair of columns within a row that contains mean values and so the formula is presented in terms of one row and one pair of columns.
Notation
Description
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Weighted count of cases contributing to the mean in column i.
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Sum of squared weights for column i.
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Weighted count of cases contributing to the mean for the overlap.
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Sum of squared weights for the overlap.
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Weighted sum of the values in column i.
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Weighted sum of the squared values in column i.
The mean in each column i is
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If the effective base is being used, the effective base in each column i is
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Otherwise
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The test is not performed if:
w i <= 0
The effective base is being used and q i <= 0
The mean values in the two columns being tested are identical
The sample variance in column i is
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If we set
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Then the pooled estimate of the population variance is
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The t value is:
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With no overlap, Z and e o are both zero. With overlap, Z is 1.0, except in the case of grids, where it is:
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where:
X12 is the weighted sum, for respondents in both columns, of the value in column 1 multiplied by the value in column 2
all X and Y terms in Z refer to respondents who are in both columns.
The degrees of freedom, DF, are
DF = e1 + e2 - e0 - 2
See also
Column means test