A Note on Equivalent and Nonequivalent Parametrizations of the Two-Parameter Logistic Item Response Model

The two-parameter logistic (2PL) item response model is typically estimated using an unbounded distribution for the trait theta. In this article, alternative specifications of the 2PL models are investigated that consider a bounded or a positively valued theta distribution. It is highlighted that these 2PL specifications correspond to the partial membership mastery model and the Ramsay quotient model, respectively. A simulation study revealed that model selection regarding alternative ranges of the theta distribution can be successfully applied. Different 2PL specifications were additionally compared for six publicly available datasets.