Three approaches to determining the dimensionality of binary items

Abstract

A monte carlo investigation of three approaches to assessing the dimensionality of binary items used a population model that allowed sampling of items and examinees and provided for variation and control of important parameters. The model was realistic of performance of binary items in current tests of cognitive abilities. Three indices were investigated : one based on the property of local independence of unidimensional tests (the independence index), one based on patterns of second factor loadings derived from simplex theory (the pattern index), and one that reflects the shape of the curve of successive eigenvalues (the ratio of differences index). The last index was used for matrices of phi coefficients, tetrachoric correlations, and variances-covariances. The local independence index reported here was the most accurate dimensionality index. The pattern index was accurate under many combinations of parameters, but decreased substantially at the highest level of factor correlations and the widest dispersion of item difficulties. None of the eigenvalue indices produced satisfactory accuracy, except under the most favorable combinations of parameters. Nonetheless, the eigenvalues of variance-covariance matrices provided a more accurate basis for dimensionality decisions than tetrachoric correlations, which have been the statistic of choice of many investigators. Recommendations for use are also given. Index terms: binary items, dimensionality, factor analysis, phi correlations, tetrachoric correlations.