tribution Rasch models will be considered as the third way of introducing an interaction into the main effects model.
Interaction between items and methods. The basic model (Model 5) is a rather restrictive assumption because the impact of different contexts or situations, of different response formats or representation modes, or whatever the methods of a test application may be on the response behavior may be different for different items. The interaction model is obtained by substituting the two single-indexed parameters cr and ¡1. by a double-indexed parameter cr., which parameterizes the difficulty of item i with method j:
In case of a complete multimethod design, that is, if all items are administered using every method, the interaction model is equivalent to the ordinary Rasch model with as many items as there are item-by-method combinations. From this point of view, the original Rasch model is more general than the extension of the Rasch model by a third, latent additive method factor as is the case in the basic model (Model 5). In fact, the LLTM (Model 6) itself is an ingenious extension of the Rasch model because it contains item components that the Rasch model does not take into consideration.
The advantage of treating the main effects model (Model 5) as a specification of the LLTM is that a large number of intermediate stages between the main effects model (Model 5) and the interaction model (Model 7) can be specified, and a very flexible way of dealing with incomplete multimethod test designs is given. Whereas in the latter case simply canceling the lines of the design matrix Q (see earlier) that relate to a missing item-method combination solves this situation, the specification of mixed interaction and conditional main effects requires a reorganization of the Q matrix. The following matrix (see Table 18.3) illustrates the assumption that the third method does interact with the items so that there is only a conditional main effect between Methods 1 and 2. An example for such a semi-interaction between item content and item method would be an achievement test in physics, where four different item contents are crossed by three kinds of response formats, that is, open verbal response, multiple choice, and numerical calculation. On one side, the difficulties of the four content areas, say electrical circuitry, optical lenses, gravity, and density, are combined additively with the (low) difficulty of giving a multiple-choice
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