A computer program for ordering-theoretic data analysis

Abstract

Ordering theory, a measuremen t model which ex­ tends scalogram analysis to include nonlinear item or task hierarchies in unidimensional as well as multidimensional configurations, allows for both the determination of an item or task hierarchy and the testing of a hypothesized hierarchy among a set of items or tasks. The prerequi­ site relation employed to generate the hierarchy is derived from formal logic: Item i is prerequisite to item j if and only if the response pattern (0,1), where °rep­ resents a failure on item i and 1 represents a success on item j, occurs infrequently. That is, the response pattern (0,1) is disconfirmatory of the hierarchical relation, item i is prerequisite to item j. To overcome the deterministic limitation shared with scalogram analysis concerning an inability of the model to consider random error occurrences in the item or task response data, ordering-theoretic analysis in­ corporates preset tolerance levels to establish the num­ ber of disconfirmatory response patterns that will be accepted in defining a prerequisite relation between two items. A prerequisite relation, whether hypothesized a priori or generated by the analysis, is accepted only if the frequency of disconfirmatory response patterns is less than or equal to the frequency of such response patterns established by the tolerance level. Detailed discussions of ordering-theoretic data analysis have been provided elsewhere. Airasian and Bart (1973, Note 1) articulated the general nature of order­ ing theory. Bart and Krus (1973) described an ordering­ theoretic technique by which item or task hierarchies could be determined. Krus and Bart (1974) discussed an application of the technique to multidimensional item scaling. Moreover, the technique has been applied to diverse data, including Piagetian task responses (Bart & Airasian, 1974), propositional logic game item responses (Airasian, Bart, & Greaney, 1975), and various attitudinal data (Bart, 1972a, b). The program discussed in this paper searches out all of the prerequisite relations between all item pairs as described above, employing a user preset tolerance level(s) for each testing. Input. The job deck consists of the following cards: title cards, a problem parameters card, tolerance level card(s), output options card, a data format card, a