Optimized item relational structure theory: Using the differentiations of multinomial functions as an example

For considering the ordering relation of any two items, the first method, called Ordering Theory (OT), was proposed by Airasian and Bart. Since two items with ordering relation must be not independent, Takeya proposed his improved method called Item Relational Structure Theory (IRS); however, the thresholds of both with a fixed constant are lack of objectivity. In this paper, we consider the family of Q-matrix based IRS with different thresholds among all of the possible values, based on the ideal item structure theory proposed by the first author of this paper, an improved Q-matrix based IRS theory with the best threshold among all ordering coefficients called Optimized IRS is proposed, it is trivial that the IRS may be a special case of this new method, and the new one is always better than the traditional IRS and the Q-matrix based IRS. A real data experiment of the differentiations of multinomial functions by using Liu's validity index is conducted. The performances of the family Q-matrix based IRS with different thresholds from 0 to 1 are compared. Experimental results show that the Optimized IRS outperforms the traditional IRS and other new IRSs with different thresholds.