This paper alarmed us that the model data fit can be lead us to the wrong way in some situations. In specifically, fake latent classes would be generated in mixture Rasch model if the data do not fulfill some assumptions. This paper would focus on one class 3PL and then 2PL models. First, an empirical data set which fitting the 3PL model was analysis using mixture Rasch model to illustrate underfit items would lead to two or more class. Secondary, in order to eliminate other factors which might not related to unfit of items but would influence the conclusion from empirical data analysis, a series simulation studies were conducted in a logical flow of question raising and problem solving. The most important one is data generated by one class 2PL model but would lead to fitting more than one class mixture Rasch model and it is due to the item discrimination parameters which is the most common violation of assumption in Rasch family models. The main contribution of this paper is that it raised and proved fake latent classes in mixture Rasch model and we should be careful when employing mixture Rasch model.
This paper only investigated one kind of violation of assumptions in Rasch model (underfit items), but the other type of violations were not investigated. Although the 2PL and 3PL is the most usually use model, the data which collected in real world is not that simple and might involving other impurities, such as multidimensionality, local dependence and so on.
Also, if the above direction is correct, how about the possible spurious parameters in other models which developed to fulfill the assumption violations except the group invariance? A further question might be asked that how are the interactions between these violation to the spurious parameters?
As the undetailed study 3, the author raised a reasonable but still interesting question--Do two classes in a mixture Rasch model always collapse into one class 2PL? And the not so stable conclusion is that the answer is not! But we can figure it out after a common sense reasoning. Certainly, this findings also implies the other factors or variables should be investigated in which influencing the spurious latent classes in mixture Rasch model, such as multidimensionality, local independence, distribution of abilities and so on.