A Geometrical Approach to Item Response Theory

How critical is the concept of the latent trait to modern test theory ? The appeal to some unobservable characteristic modulating response probability can lead to some confusion and misunderstanding among users of psychometric technology. This paper looks at a geometric formulation of item response theory that avoids the need to appeal to unobservables. It draws on concepts in differential geometry to represent the trait being measured as a differentiable manifold within the space of possible joint item response probabilities given conditional independence. The result is a manifest and in principle observable representation of the trait that is invariant under one-to-one transformations of trait scores. These concepts are illustrated by analyses of an actual test.