Handling ordinal variables in three-way analysis of quantification matrices for variables of mixed measurement levels

For the analysis of variables of mixed measurement levels a class of methods can be used that is based on three-way analysis of quantification matrices for nominal or quantitative variables. This class of methods incorporates some well-known techniques but also offers a series of interesting new alternatives for the analysis of nominal or quantitative variables. Ordinal variables have received hardly any attention in this class of methods, and are usually treated as if they are quantitative variables. In the present paper this gap is filled by constructing quantification matrices for ordinal variables via optimal scaling of the ordinal variables, thus yielding optimal quantification matrices for these variables. Algorithms for this optimal scaling procedure are developed, and the optimal scaling procedures are compared to optimal scaling of ordinal variables in the context of principal components analysis and multidimensional scaling