ABSTRACT
In Chapter 6 we considered the Spearman single-factor model as a simple model for a homogeneous test—a test whose items measure just one attribute in common. We saw that the Spearman model is a good approximation to the behavior of quantitative items such as Likert items, and may be an acceptable approximation to the behavior of binary items. The model, with its special cases, was used to answer the primary question whether the test is indeed unidimensional, and to assess the error of measurement of the attribute by the test score. Chapter 12 extended this work to give item response models that should more precisely fit a homogeneous set of binary items. Then Chapter 13 applied these to assess the error of measurement of the attribute by sufficient statistics from the response patterns. In these nonlinear models, the error variance is a function of the latent trait/common factor. Chapter 9 introduced the multiple-factor model, in which the item responses are multidimensional, depending on two or more common factor attributes. So instead of asking if the items form a homogeneous set, with just one attribute constituting a common dimension of variation of the respondents, we could ask how many attributes are measured by the items, that is, on how many dimensions do the respondents vary. In the context of careful test design, we could also ask whether the items group into independent clusters, each making a separate homogeneous subtest, or, in a partly exploratory study, if some of them are complex, measuring two or more common attributes simultaneously. The hierarchical model was used as a way to see if a multidimensional test is nevertheless sufficiently dominated by a general factor to be treated as unidimensional. Chapter 10 used the multiple factor model as a way to measure construct validity.
