| The first part of this thesis, Chapter 2 through Chapter 4, explores procedures for assessing latent test dimensionality. In Chapter 2, the structure and theoretical properties of conditional item pair covariances given a latent composite are thoroughly investigated. Such information is useful because estimates of such conditional covariances form the basic building blocks of several important psychometric procedures for assessing multi-dimensionality.Kim (1994) proposes a data-driven index of dimensionality--DETECT. Its purpose is to detect whatever dimensionality structure exists. Unfortunately, there has been little theoretical justification given for the use of DETECT there is no appropriate algorithm to execute the difficult maximization step of the procedure and the method used in DETECT to correct statistical bias needs to be improved. In Chapter 3, first, a theoretical index of dimensionality that is being estimated by DETECT, called theoretical DETECT, is proposed to provide a strong theoretical foundation for DETECT. Under certain reasonable conditions, theoretical DETECT is proven to be maximized at the correct dimensionality-based cluster partition of the test, where the number of clusters in this partition corresponds to the number of dominant dimensions present in the test and each cluster in this partition corresponds to a distinct dominant dimension. Second, a new bias-corrected version of DETECT is presented, thereby considerably improving the performance of the original DETECT proposed by Kim. Finally, a genetic algorithm is developed to quickly and accurately search for the maximum DETECT value over all possible item cluster partitions, thereby turning DETECT into a practical statistical procedure.In Chapter 4, the asymptotic normality of a statistic quite similar to DETECT is established, thereby providing support for DETECT's asymptotic behavior.The second part of this thesis, Chapter 5, is concerned with an asymptotic representation for the manifest test response probabilities. Holland (1990a) conjectures that a special quadratic form is a limiting one for all "smooth" unidimensional item response models as test length tends to infinity. This conjecture has the important and surprising implication that there can be at most two parameters per item consistently estimated for long unidimensional tests. Three counterexamples are presented to demonstrate that Holland's conjecture does not hold in general. |
