Defining asymmetry in item response theory.

Asymmetric item response theory (asymIRT) has emerged as an important extension of classical IRT, motivated by empirical evidence and theoretical arguments that symmetric item response functions (IRFs) often inadequately describe real response processes. Despite rapid model development, there remains ambiguity regarding what constitutes asymmetry, how different models relate to one another, and how asymmetry should be quantified. This paper provides a unified framework for defining, interpreting, and measuring asymmetry in IRT models. Refining Samejima's notion of point symmetry, we propose general definitions of IRF symmetry based on properties of the first derivative of the IRF. These definitions clarify the status of various models, including the 3PL, unipolar models, and recently proposed asymmetric functions. We further introduce quantile-based measures of skewness as convenient indices of the magnitude and direction of item asymmetry and demonstrate how these measures behave across several asymmetric models. Through analytic results and numerical illustrations, we show that asymmetry has meaningful consequences for latent trait estimation, particularly in how items penalize or reward responses at different trait levels. This work positions asymmetry as a fundamental item characteristic, alongside difficulty and discrimination, and provides practical tools for comparing asymmetric IRT models and understanding their substantive implications.