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Statistical formula for the net difference test
The following table shows the formulae used for conducting the net difference test in UNICOM Intelligence Professional.
Formula for proportions
For any row, and any of the four columns being tested (i=1,2,3, and 4):
Notation
Description
W i
Sum of the weights (weighted base) for column i.
Q i
Sum of the squared weights for column i.
E i = (W i * W i) / Q i
Effective base for column i.
P i
Proportion in column i
For a table with overlap or a grid table, and any pair of columns from the four being tested (i and j=1,2,3, and 4):
Notation
Description
W ij
Sum of the weights (weighted base) for respondents in both columns.
Q ij
Sum of the squared weights for respondents in both columns.
E ij = (W ij * W ij) / Q ij
Effective base for respondents in both columns.
P ij
Proportion for respondents belonging in the row being tested for both columns.
The formula is:
where
numer = (P 3 - P 4) - (P 1 - P 2)
and for a non-grid, non-overlap table
For a table with overlap or a grid table
where
The degrees of freedom are:
where, for a non-grid, non-overlap table
and
For a table with overlap or a grid table
and
Formula for means
For any row, and any of the four columns being tested (i=1,2,3, and 4):
Notation
Description
W i
Sum of the weighted base for column i.
Q i
Sum of the squared weights for column i.
E i = (W i * W i) / Q i
Effective base for column i.
X i
sum of values for column i
Y i
sum of squared values for column i
M i
mean for column i=X i/W i
The values can be either numeric values or factor values.
For a table with overlap or a grid table, and any pair of columns from the four being tested (i and j=1,2,3, and 4):
Notation
Description
W ij
Sum of the weighted base for respondents in both columns.
Q ij
Sum of the squared weights for respondents in both columns.
E ij = (W ij * W ij) / Q ij
Effective base for respondents in both columns.
The intermediate term SX is:
The tstat is
where
numer = (M 3 - M 4) - (M 1 - M 2)
and for a grid, non-overlap table,
For a table with overlap or a grid table
where
For a non-grid table with overlap, R ij reduces to 1.
Grid tables
For a grid table, it is not possible to display the net difference if the mean is a numeric mean rather than a factor mean. In this case, an error is returned.
For a grid table with factor means:
Notation
Description
X i*
The weighted sum of factors for column i for all respondents belonging in the mean for column i and in the base of column j.
X *j
The weighted sum of factors for column j for all respondents belonging in the mean for column j and in the base of column i.
Y i*
The weighted sum of squared factors for column i for all respondents belonging in the mean for column i and in the base of column j
Y *j
The weighted sum of squared factors for column j for all respondents belonging in the mean for column j and in the base of column i
Y ij
The weighted sum of (factor for column i) * (factor for column j) for all respondents belonging in the mean for both columns.
Using the preceding terms:
where:
Degrees of freedom
The degrees of freedom are:
where, for a non-grid, non-overlap table:
and
For a table with overlap or a grid table:
and
For more on the theory of overlapping samples, see Kish, L (1965), Survey Sampling, New York: John Wiley and Sons. ISBN 0-471-48900-X.