Identifiability analysis of the fixed-effects one-parameter logistic positive exponent model

In addition to the usual slope and location parameters included in a regular two-parameter logistic model (2PL), the logistic positive exponent (LPE) model incorporates an item parameter that leads to asymmetric item characteristic curves, which have recently been shown to be useful in some contexts. Although this model has been used in some empirical studies, an identifiability analysis (i.e., checking the (un)identified status of a model and searching for identifiablity restrictions to make an unidentified model identified) has not yet been established. In this paper, we formalize the unidentified status of a large class of fixed-effects item response theory models that includes the LPE model and related versions of it. In addition, we conduct an identifiability analysis of a particular version of the LPE model that is based on the fixed-effects one-parameter logistic model (1PL), which we call the 1PL-LPE model. The main result indicates that the 1PL-LPE model is not identifiable. Ways to make the 1PL-LPE useful in practice and how different strategies for identifiability analyses may affect other versions of the model are also discussed.