Parsimonious item response theory modeling with the cauchit link: Revisiting the rationale of the four-parameter logistic model.

Abstract

Application of the four-parameter logistic model (4PLM) in item response theory (IRT) research is contentious due to the complexities of estimating the asymptotes that correspond to upper and lower asymptote effects. We introduce the cauchit IRT model (i.e., a model that utilizes a link function based on the Cauchy distribution) as a compelling parsimonious alternative to the 4PLM. Through comprehensive simulation studies and real-data analysis, we determine that the cauchit model, distinguished by its symmetric error distribution and pronounced tails, provides a streamlined solution, because the tail-pronounced symmetric error distribution captures key features of the 4PLM with only one item parameter. The 4PLM requires large sample sizes (e.g., N > 5000), medium item difficulty, and high discrimination when both upper and lower asymptote effects are present. In contrast, we show that the cauchit model works well with drastically smaller sample sizes (e.g., N = 100). Our study further discusses the versatility of the cauchit model, underscoring its adaptability, especially in small sample research situations.