Binomial Regression with Monotone Splines: A Psychometric Application

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Abstract
A binomial regression function p(x, θ) models the probability of rj successes in nj trials as a function of the values of an observed covariate xj and/or a latent variable θj (j = 1, …, J). This article explores the use of monotone regression splines to define p, and applies them to the representation of test items as functions of examinee ability. Some illustrative data suggest that the flexibility of monotone splines permits the detection of item characteristics not observable using logistic-based or log-linear approaches. A simulation study indicates that estimates of both item-characteristic curves and ability are reasonably precise for numbers of items and examinees typical of large university lectures. Given a set of such binomial regression functions, it can be useful to study the principal components of functional variation. The extension of multivariate principal-components analysis to permit the analysis of many item-characteristic curves is described.

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