Language Reference |
rescales qualitative data to be a least squares fit to qualitative data
When qualit is at the nominal level of measurement,
the optimal scaling transformation result is a least squares
fit to quanti, given the restriction that the
category structure of qualit must be preserved.
If element of qualit is in category
, then
element
of the optimum scaling transformation result is
the mean of all those elements of quanti that correspond
to elements of qualit that are in category
.
For example, consider these statements:
quanti={5 4 6 7 4 6 2 4 8 6}; qualit={6 6 2 12 4 10 4 10 8 6}; os=opscal(1,quanti,qualit);The resulting vector
OS 1 row 10 cols (numeric) 5 5 6 7 3 5 3 : 5 8 5The optimal scaling transformation result is said to preserve the nominal measurement level of qualit because wherever there was a qualit category
When qualit is at the ordinal level of measurement,
the optimal scaling transformation result is a least squares
fit to quanti, given the restriction that the
ordinal structure of qualit must be preserved.
This is done by determining blocks of elements of
qualit so that if element of qualit is
in block
, then element
of the result is the mean
of all those quanti elements corresponding to
block
elements of qualit so that the means are
(weakly) in the same order as the elements of qualit.
For example, consider these statements:
quanti={5 4 6 7 4 6 2 4 8 6}; qualit={6 6 2 12 4 10 4 10 8 6}; os=opscal(2,quanti,qualit);The resulting vector
OS 1 row 10 cols (numeric) 5 5 4 7 4 6 4 : 6 6 5This transformation preserves the ordinal measurement level of qualit because the elements of qualit and the result are (weakly) in the same order. It is least squares because the result elements are the means of appropriate elements of quanti. By comparing this result to the nominal one, you see that categories whose means are incorrectly ordered have been merged together to form correctly ordered blocks. This is known as Kruskal's (1964) least squares monotonic transformation. Consider the following statements:
quanti={5 3 6 7 5 7 8 6 7 8}; os=opscal(2,quanti);These statements imply that
qualit={ 1 2 3 4 5 6 7 8 9 10} ;This means that the resulting vector has the following values:
OS 1 row 10 cols (numeric) 4 4 6 6 6 7 7 : 7 7 8
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