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New item response theory models supported on [0, 1] bounded interval examinee proficiency are proposed. The proposed models are based on a truncated logistic function, and the rationales and analytic frameworks of the models are articulated parallel to the traditional logistic function item response theory models. The proposed models are established at the expense of some increase in the link function complexity, but greatly boost the interpretability and utility of the person and item difficulty parameters. A popular empirical example, five dichotomously scored items on the Law School Admissions Test, is analyzed to illustrate the proposed approach compared to the traditional models using Markov chain Monte Carlo estimator.