A variety of techniques exist for addressing the fundamental question: What does a test measure? The answer to that question often depends on how one asks the question of the data. At one extreme we see correlations of a test with other editions of that test (e.g., parallel form reliability estimates). At the other extreme we see exploratory factor analyses of item-level data. In this paper we argue that item-level and test-score dimensionality are not necessarily identical; define mathematical relationships among item-level factors, test-score factors, and parcel-level factors; and advocate that the choice of dimensionality technique should be based on the purpose of the dimensionality analysis. Finally, we illustrate how different approaches yield different answers with SAT-Verbal data.
