| The present study consisted of two phases that addressed the issue of dimensionality and provided an application to real data. In Phase I, the variables that were manipulated included the number of dimensions (one versus three), the correlation between dimensions (0.00, 0.35, 0.70), test length (12 versus 24), type of method (group average, centroid, Ward's and EFA) and cluster stopping criterion (pseudo F and pseudo t2 statistics). One hundred datasets were generated using RESGEN, for each combination of test length, number of dimensions, and correlation between dimensions. The number of times the method identified the true dimensionality across 100 replications was recorded. Phase II of this study explored the dimensionality of New Standards Mathematics Reference Examination using the preferred methods identified in Phase I To explore differences between the underlying framework for the test and the empirical results, further analyses were performed.; In the one-dimensional case, EFA recovered the underlying structure and all hierarchical cluster methods overestimated the underlying structure. In the three-dimensional case, the results demonstrated that EFA and Ward's method recovered the true dimensionality best. For the 0.00 and 0.35 correlated dimension condition, EFA and Ward's method recovered the true dimensionality nearly 100% of the time for both 12 and 24 items. In addition, use of either stopping criterion for Ward's method yielded the same results. However, for the 0.70 correlated dimension condition, Ward's method recovered the true dimensionality for pseudo F and pseudo t2 statistics better than EFA for the 12 item case. For the 24 item case, no differences between the two methods emerged. With regard to the application of the methods to the New Standards Mathematics Reference Examination, Ward's cluster method identified 6 clusters. Using the empirical results, it appeared that the tasks could be classified based on conceptual and procedural complexity. A description of the tasks in relation to their conceptual and procedural complexity was provided in the dissertation. |
