Extending Liu’s ordering theory for cognitive diagnosis and remedial instruction

Since the Ordering Theory (OT) does not consider the item nonindependence, and the Item Relational Structure Theory (IRST) does not satisfy the item ordering consistence property, the first author of this paper proposed his improved theory, Liu’s Ordering Theory (LOT), which has considered both the item ordering consistence property and item non-independence. However, all of above-mentioned item ordering theories can only be used for dichotomous items, not for polytomous items. In this paper, an extensional Liu’s Ordering Theory was proposed, it can be used for not only dichotomous items, but also polytomous items, since any dichotomous item can be viewed as a special case of a polytomous item, and this new method and its extensional item ordering consistence property were proposed by the first author of this paper. We called this new theory as Liu’s Polytomous Ordering Theory, denoted as LPOT, and this new theory is more reasonable and useful than before.