The latter type of assumption is not necessary because a multidimensional model can also be formalized without a decomposition of the difficulty parameters. The resulting model is a multidimensional Rasch model with as many dimensions as there are methods and a difficulty parameter for each item x method combination:

This model allows for both kinds of interactions— person x method and item x method. Data analyses according to this model can be done using the computer programs MULTIRA and ConQuest.

Rasch model and latent class analysis: "Some interaction" between items and persons. From a formal point of view, the third kind of interaction, that is, between items and persons, could be treated quite similar to the interaction between methods and persons. A multidimensional model could be defined in which the traits are specific for each item:

Such a multidimensional approach to modeling the interaction of persons and items would again lead to the same family of models as discussed in the last section, with just items and methods exchanged. A "complete" interaction between items and persons in the sense that each item addresses its own latent trait certainly contradicts the basic idea of Rasch's theory of measurement to separate the influence of items and persons.

With the approach of mixture distribution Rasch models, a moderate way of introducing interactions between items and persons is provided. The mixed Rasch model is a combination of latent class analysis and the Rasch model. As described, LCA is used to identify groups of subjects, called classes, who respond to the test items in a qualitatively similar way. This is formalized by class-specific solving probabilities for each item. But LCA does not take into account the quantitative differences between persons within each class. This extension is made by the mixed Rasch model (Rost 1990, 1991), which states that the Rasch model holds for each and every class, however, with its own set of parameters for each class. It assumes that both the item parameters and the person parameters are class specific, which means that the model assigns different item parameters to the same item and different person parameters to the same person depending on the latent class.

The ^-parameters are the unconditional probabilities of belonging to class c and are sometimes called the class size parameters or the mixing proportions. 0vc symbolizes the class-specific person parameter, and <7 stands for the class-specific item parameter.

There are many different ways of generalizing the mixed Rasch approach to multimethod data. A straightforward way would be to separate the effects of items and methods within each class, namely, to assume the main effects model (Model 5) for each latent class c:

A more general model structure would be obtained when a latent additive decomposition of the item-method parameters is introduced, exP(0vc-5>mA)

which would be the mixture distribution LLTM (Rost, 2001). The basic idea of this model is the assumption of different latent classes in which the LLTM holds. Obviously the mixture distribution

LLTM (Model 14) is a restricted case of the mixed Rasch model (Model 12).

Moreover, the possibility of defining so-called hybrid models (which are mixture models in which a different kind of model holds in each latent class) inflates the family of multimethod Rasch models to an intractable framework of thousands of models, and this situation would not really help us understand the psychometric issues of multimethod data.

Instead of searching for the model that is most general but not applicable to any data set (because no appropriate software would be available), we will restrict ourselves to the simple case of the mixed Rasch model. This model takes an interaction between items and methods into account and restricts the interaction with persons to a limited (small) number of latent classes:

c vijc

The crc parameters are combined item-method parameters and specific for each class c. The unidimensional d parameters refer to all tasks, so that (similar to interaction Model 7) there is no difference between items and methods in the formalization of the model. Both may contribute to the identification of subgroups of persons. Therefore, it is not a model that is specific for the multimethod situation. Rather, it provides us with an elegant heuristic tool to identify interactions between items and methods and the latent trait.

All models described in this section have been defined for dichotomous data. All these models can also be defined for ordinal response variables, for example, for rating scales as a response format or partial credit scoring in the case of achievement tests.

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