Likelihood-based statistics for validating continuous response models

The theory for the estimation and testing of item response theory (IRT) models for items with discrete responses is by now very thoroughly developed. In contrast, the estimation and testing theory for IRT models for items with continuous responses has hardly received any attention. This is mainly due to the fact that the continuous response format is seldom used. An exception may be the so-called analogous-scale item format where a respondent marks the position on a line to express his or her opinion about a topic. Recently, continuous responses have attracted interest as covariates accompanying discrete responses. One may think of the response time needed to answer an item in a computerized adaptive testing situation. In the present report, the theory of estimating and testing a model for continuous responses, the model proposed by Mellenbergh in 1994, is developed in a marginal maximum likelihood framework. It is shown that the fit to the model can be evaluated using Lagrange multiplier tests. Simulation studies show that these tests have excellent properties in terms of control of Type I error rate and power.