Geodesic Ability in Unidimensional IRT Models: A Geometric Perspective on Parametrization Invariance

Psychometric models use latent variables to quantify psychological traits, but statistical and substantive inferences based on these variables may depend on how the latent scale is parametrized. Consequently, the same fitted model can yield qualitatively different conclusions under different parametrizations. Building on Ramsay's (1996, Behaviormetrika, 23, 3-16) geometric perspective, we reinterpret this problem using differential geometry. While Ramsay defined ability via an arc length in Euclidean space, psychometric models can be seen instead as a curved statistical manifold with the Fisher-Rao metric. This allows for a parametrization-invariant definition of ability, which we term geodesic ability. We illustrate this concept for unidimensional item response theory models, including the Rasch and 3PL models, and compare it to Ramsay's arc-length approach.