| An empirical study was carried out to examine the structure of two-dimensional dichotomous test data with underlying compensatory and noncompensatory ability interactions. Two multidimensional Item Response Theory (IRT) models, one compensatory and one noncompensatory, were used to simulate four data sets with matched ability correlations ranging from.00 to.80. The real test data were derived from four administrations of the Mathematics Problem-Solving subtest of the Iowa Tests of Basic Skills (ITBS). To facilitate comparisons with the simulated test data, items from the ITBS subtests were further subsetted into compensatory and noncompensatory item subtests as identified in a previous study. Principal components analysis, factor analysis, and NOHARM, a nonlinear factor analytic program were employed and the results were examined and compared.;The principal components analysis confirmed each of the eight simulated data sets to be two-dimensional while the real data displayed two major dimensions with possibly one or two minor dimensions. Comparing eigenvalues in the compensatory and noncompensatory data revealed first eigenvalues from the compensatory data sets to be consistently larger than those from matched noncompensatory data. The ratio of the first eigenvalue to the second was also consistently larger in the compensatory data with trends in the real and the simulated data being very similar. Additionally, the magnitude of first eigenvalues increased as the correlation between the ability dimensions increased in the simulated data. A factor analysis of the compensatory and noncompensatory data showed that for each of the factor solutions investigated, the percentage of variance accounted for by a fixed number of factors was larger for solutions in the compensatory data than for matched factor solutions in the noncompensatory data. Again there was a high degree of similarity in the compensatory/noncompensatory contrasts for the simulated and the real test data. The program NOHARM provided accurate estimates in the simulated compensatory data and the simulated noncompensatory data. |
