Distance Functions and Scoring Functions in IRT Models

IRT models require scoring functions, i.e., the assignment of scores to response categories or scale points. These can play a role in measuring response styles or biases, as well as conventional IRT measures such as expected scoring functions or item information functions. There are potentially useful connections between scoring functions and functions for measuring the distance between an empirical probability distribution and a hypothetical reference distribution. Every IRT scoring function can be interpreted as corresponding to a distance between the empirical pdf, f, and a referent pdf, g, by assigning a set of destination bins to match the scoring function maxima and then constructing g by minimizing distance between f and g via a distance metric. Distance functions for measuring the difference between g and f can be normed and also "inverted" to measure the similarity between f and a response style operationalized by g. These interconnected functions provide a novel way to measure response style or response bias.