The Rasch Model for Multimethod Data

A straightforward way to define a Rasch model for multimethod data, that is, a data cube consisting of persons X items X methods, would be to follow the principle of latent additivity and to define the probability of the response of person v on item i with method j as an additive logistic function of a person parameter 0v, an item parameter <x, and a method parameter fj... Here the method parameter /i. stands for the specific and item-independent contribution of method j (that is, its main effect) to the probability of solving an item:

VIJ > d where dy[. denotes 1 plus the exponential function of the numerator, in this case dvi. = 1 + exp(0v - cr + fi). This generalized Rasch model was described as early as 1970 by Micko (1970; in German) and has been discussed by Linacre (1989) as the multifacets model. Depending on the conceptualization of "methods," this model can be applied to different situations. For example, Rost and Spada (1983) and Spiel (1994) used it in the context of measurement of change as a model of global learning. Here, the method parameter represents the effect of learning over time, and each "method" j stands for a different point in time. Another example is revealed if the researcher also deems the raters to be "methods" to judge certain person characteristics. Then the model described can be used to identify the amount of rater bias, that is, the tendency of raters to rate generally higher or lower than other raters. Because the method parameter is independent of the item parameter, the application of this model is limited to the identification of main effects of biased ratings and cannot identify interactions between raters and items.

Applications of the model discussed in Europe, in particular by Gerhard Fischer and his students (Fischer, 1995b), used computer programs of the linear logistic test model (LLTM; Fischer, 1973). The LLTM is a general model structure for defining any kind of latent additive (but unidimensional) Rasch model. For this purpose, the item parameters are considered as a linear function of some basic parameters r¡h:

where the design matrix Q covers the weights qjh of the item component h in item i. By means of an appropriate specification of the design matrix Q, the Rasch model for three-dimensional data structures can be defined as a LLTM model. Table 18.2 shows the Q matrix for four items and three methods. The Q matrix is built by taking account of the combination of method j and item i. For example, because the first item is an item-l-type item that has been administered according to Method 1, the first component (defining all item-l-type items) and the fifth component (defining all Method 1 items) were marked in the row of the "real" item 1. An empty cell in a Q matrix indicates a weight of 0.

Although there are certain advantages to specifying a model within the framework of a general model structure (e.g., for dealing with incomplete test designs and for model control and hypothesis testing), it may be more convenient to have a computer program for that particular model. Such a program is FACETS (Linacre, 1989), which provides many useful fit diagnostics for the multifacets Rasch model. A prototypical application of FACETS is given by Eckes (2004).