Mixture models are a flexible tool to uncover latent groups for which separate models hold. The Rasch model can be used to measure latent traits by modeling the probability of a subject solving an item through the subjects ability and the items difficulty. A crucial assumption of the Rasch model is measurement invariance: each item measures the latent trait in the same way for all subjects. Measurement invariance is violated if, e.g., an item is of different difficulty for different (groups of) subjects. Mixtures of Rasch models can be used to check if one Rasch model with a single set of item difficulties holds for all subjects and thus measurement invariance is not violated. However, estimation of the item difficulties in a Rasch mixture model is not independent of the specification of the score distribution, which is based on the abilities. The latent groups detected with such a Rasch mixture model are not solely based on the item difficulties but also or even only the scores and thus subject abilities. If the aim is to detect violations of measurement invariance, only latent groups based on item difficulties are of interest because different ability groups do not infringe on measurement invariance.
This thesis aims at making three different yet connected contributions: The methodological, psychometric contribution is a new specification of the Rasch mixture model. It ensures that latent classes uncovered by a Rasch mixture model are based solely on the item difficulties and thus increases the models suitability as a tool to detect violations of measurement invariance. The computational contribution is open-source software in form of the R package psychomix for estimation of various flavors of the Rasch mixture model with or without concomitant variables and several options for the score distribution including the newly suggested specification. The statistical contribution connects and compares mixture models to model-based recursive partitioning. This is another method to detect subgroups in the data for which a stable set of model parameters holds and has also been applied to Rasch models to detect violations of measurement invariance. Here, mixture models and model-based recursive partitioning are presented in a unifying framework and the relative (dis-)advantages are illustrated in a simulation study.