Item response theory (IRT) models for categorical response data are widely used in the analysis of educational data, computerized adaptive testing, and psychological surveys. However, most IRT models rely on both the assumption that categories are st
rictly ordered and the assumption that this ordering is known a priori. These assumptions are impractical in many real-world scenarios, such as multiple-choice exams where the levels of incorrectness for the distractor categories are often unknown. While a number of results exist on IRT models for unordered categorical data, they tend to have restrictive modeling assumptions that lead to poor data fitting performance in practice. Furthermore, existing unordered categorical models have parameters that are difficult to interpret. In this work, we propose a novel methodology for unordered categorical IRT that we call SPRITE (short for stochastic polytomous response item model) that: (i) analyzes both ordered and unordered categories, (ii) offers interpretable outputs, and (iii) provides improved data fitting compared to existing models. We compare SPRITE to existing item response models and demonstrate its efficacy on both synthetic and real-world educational datasets.
In describing the motion of atoms and clusters, we face with choosing quantum mechanics or classical mechanics under different conditions. In principle, there exist two criteria for this choice, but they do contradict in some cases though they are in
agreement for other cases. Actually, this problem is closely related with the effective centre-of-mass method, the underlying application of quantum mechanics. It is shown that quantum mechanics must be selected for particles motion when the de Broglie wave length of the mass centre is larger than the particle size, and in such case the effective centre-of-mass can be used in Quantum Mechanics. In order to test this conclusion, an easy-manufactured experiment is suggested.
