Modeling a smooth course of learning and testing individual deviations from a global course

Abstract

Formative assessment supplies valuable feedback for teachers and learners, and has been facilitated by computerized implementations. While longitudinal within-student assessment or within-class comparisons are useful, a normative interpretation of an individual’s course of learning can only be given relative to a reference population. As current computerized assessment systems sample items from pools or adapt tests, monitored students might work on non-overlapping item sets, so that classic sum scores cannot be compared directly. To meet this challenge, the Smooth Growth and Linear Deviations Rasch Model (SGLDRM) is introduced, an extension of Rasch’s item response theory model for binary test data. With the help of spline functions a smooth global course of learning is included. The model is flexible enough to accommodate increases and/or decreases of the mean ability level, which might be more or less pronounced at each measurement occasion. On the individual level, a random slope and a random intercept with amenable interpretations modify the global course of learning. Two measurement occasions suffice to estimate person-specific courses. A likelihood ratio test allows identifying students whose performance differs from the mean course. The methodology is illustrated with data from an online dyscalculia assessment and training.