Latent trait estimation for item bundle structures

Tests comprising item bundles, (i.e., groups of items related to common passages, paragraph, or graphs), have been treated as if they measure only a general trait, ignoring the within bundle dependencies. The consequences of not taking into account the specificity of each group of items are, in particular, the violation of the conditional or local independence assumption, and the consequent misspecification of the latent space. Factor analytic models seem to be the most suitable class of models available to deal with this problem because they provide a means to account for the specification of the intra-group structure. This investigation describes alternative ways to estimate latent traits through linear and non-linear models, and proposes two alternative models to measure a latent trait for a test having a within-bundle dependence structure, when items are pre-calibrated: the non-linear multidimensional confirmatory hierarchical model, obtained by the Schmid-Leiman (1957) transformation of a second-order common factor model, and an approximation of the inter-battery linear model, as described in McDonald's (1970) generalization of Bartlett's (1937) estimator. These two models were tested in a simulation study and the results compared with those obtained from the traditional non-linear unidimensional item response model.