The Statistical Assessment of Latent Trait Dimensionality in Psychological Testing

Assuming a nonparametric item response theory model, a large sample procedure for testing the unidimensionality of the latent ability space is proposed. Under the assumption of unidimensionality, the asymptotic distribution of the test statistic is derived, thereby establishing an asymptotically valid statistical test of unidimensionality. A rigorous mathematical definition of dimensionality is proposed as an alternative to the classical item response theory definition of dimensionality. This new definition, while item response theory based, is more analogous to the factor analytic notion of dimensionality and as such is more congruent with the conception of dimensionality held by applicators. The statistical test procedure is shown to have asymptotic power 1 for multidimensional in our sense tests. Monte Carlo studies, using a specially written Fortran program, indicate good agreement with the nominal level of significance when unidimensionality holds and good power performance for examinee sample sizes and psychological test lengths often encountered in practice. This adherence to the nominal level of significance and good power performance is also the case, as desired, when the items are multiple determined, thereby producing a classical dimensionality much greater than 1. Finally, an analogous test procedure is proposed for testing H d less than or d sub 0 vs. A d sub O for fixed d sub 0 greater than or 2, d denoting dimensionality.