Editing Guttman scale (section)

== Guttman scale in qualitative variables ==
Guttman's<ref name=":1" /> original definition of a ''scale'' allows also for the exploratory scaling analysis of qualitative variables (nominal variables, or ordinal variables that do not necessarily belong to a pre-specified common attribute). This definition of Guttman scale relies on the prior definition of a ''[[simple function]]''.

For a totally ordered set ''X'', say, 1,2,...,''m'' , and another finite set, ''Y'', with ''k'' elements ''k'' ≤ ''m'', a function from ''X'' to ''Y'' is a ''simple function'' if ''X'' can be partitioned into ''k'' intervals which are in a one-to-one correspondence with the values of ''Y'' .

A Guttman scale may then be defined for a data set of ''n'' variables, with the ''j''<sup>th</sup> variable having ''k<sub>j</sub>'' (qualitative, not necessarily ordered) categories, thus:

Definition: ''Guttman scale'' is a data set for which there exists an ordinal variable, ''X'', with a finite number ''m'' of categories, say, 1,...,''m'' with ''m''≥ max''<sub>j</sub>''(''k<sub>j</sub>'') and a permutation of subjects' profiles such that each variable in the data set is a simple function of ''X''.

Despite its seeming elegance and appeal for exploratory research, this definition has not been sufficiently studied or applied.