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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 has the following values:
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 , there is now a result category label . The transformation is least squares because the result element is the mean of appropriate elements of quanti. This is Young's (1981) discrete-nominal transformation.
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 has the following values:
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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