An Order-Theoretical Extension of the Guttman Scale to Less Simple Orders

Abstract

A perfect Guttman scale is rarely found in real data. Pairwise dominance relations between items to be scaled, however, often meet the conditions for less simple orders, such as strict partial orders, interval orders, and semiorders. Examples are thus provided for an extension of the Guttman scale to less simple orders in the framework of ordinal theory, or more specifically, the theory of representations with thresholds. The study is methodologically based on ordering theory. Three illustrative constructions of less simple orders demonstrate that they much more strongly account for real data than do Guttman scales, and that some uniqueness in scale values and thresholds is found in semiorders and interval orders.