A novel generalized item relational structure theory based on Liu’s normalization and consistency criteria

In this study, based on weighted mean, we can extend any item ordering theory from dichotomous scoring to polytomous scoring. However, before this study, there is no validity criterion to detect any item ordering theory for polytomous scoring whether is valid or not; this paper defines finite correlation coefficient and item difficulty for polytomous scoring corresponding to dichotomous scoring; based on these new definitions, we propose the generalized criteria of completeness, normalization and consistency for polytomous scoring corresponding to the original ones for dichotomous scoring, respectively. Two well-known item ordering theories: Takeya’s IRS and Liu et al.‘s LIRS, can be extended from dichotomous scoring to polytomous scoring, denoted as GIRS and GLIRS. And then, several important properties of them and counter examples are provided. This paper points out that not only does IRS not satisfy the three above-mentioned original criteria, but also its generalization, GIRS, does not satisfy the generalized criteria of them, and only the new theory, GLIRS, can satisfy both of the generalized and the original criteria of completeness, normalization and strict consistency.