Each of the four general classes of models described above—classical, generalizability, item response, and latent class—can be extended to incorporate more than one attribute of the student. Doing so allows for connections to a richer substantive theory and educationally more complex interpretations. In multidimensional IRM, observations are hypothesized to correspond to multiple constructs (Reckase, 1972; Sympson, 1978). For instance, performance on mathematics word problems might be attributable to proficiency in both mathematics and reading. In the IEY example above, the progress of students on four progress variables in the domain of science was mapped and monitored (see Box 4–2, above). Note that in this example, one might have analyzed the results separately for each of the progress variables and obtained four independent IRM estimations of the student and item parameters, sometimes referred to as a consecutive approach (Adams, Wilson, and Wang, 1997).
There are both measurement and educational reasons for using a multidimensional model. In measurement terms, if one is interested, for example, in finding the correlation among the latent constructs, a multidimensional model allows one to make an unbiased estimate of this correlation, whereas the consecutive approach produces smaller correlations than it should. Educationally dense longitudinal data such as those needed for the IEY maps can be difficult to obtain and manage: individual students may miss out on specific tasks, and teachers may not use tasks or entire activities in their instruction. In such a situation, multidimensional models can be used to bolster sparse results by using information from one dimension to estimate performance on another. This is a valuable use and one on which the BEAR assessment system designers decided to capitalize. This profile allows differential performance and interpretation on each of the single dimensions of IEY Science, at both the individual and group levels. A diagram illustrating a two-dimensional IRM is shown in Figure 4–9. The curved line indicates that the two dimensions may be correlated. Note that for clarity, extra facets have not been included in this diagram, but that can be routinely done. Multidimensional factor analysis can be represented by the same diagram. Among true-score models, multivariate G-theory allows multiple attributes. Latent class models may also be extended to include multiple attributes, both ordered and unordered. Figures analogous to Figure 4–9 could easily be generated to depict these extended models.
The measurement models considered thus far have all been models of status, that is, methods for taking single snapshots of student achievement in