Vertical Scaling with Moderated Nonlinear Factor Analysis

Vertical scales are intended to establish a common metric for scores on test forms targeting different levels of development in a specified domain. They are often constructed using common item, nonequivalent group designs that implicitly rely on the linking items being effectively free from differential item functioning (DIF) or the DIF being symmetric to produce unbiased linking constants. Moderated Nonlinear Factor Analysis (MNLFA) is a measurement model that can be used to understand both the presence of DIF among vertical scale common items and the extent to which the presence of DIF may bias grade-to-grade score distributions. Simulation and real data applications show how models that do and do not account for DIF in vertical scale common items can produce very different answers to the fundamental question of how much students grow from one grade to the next, but that when DIF is not present, MNLFA provides effectively identical growth estimates to traditional concurrent and characteristic curve approaches to vertical linking.