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 𝜃. In this article, alternative specifications of the 2PL models are investigated that consider a bounded or a positively valued 𝜃 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 𝜃 distribution can be successfully applied. Different 2PL specifications were additionally compared for six publicly available datasets.