The development and investigation of a method for removing the effects of the distribution of mean values from the relationships among binary variables

Abstract

As is well known, difficulties arise in attempts to analyze the relationships among binary data through methods which assume the characteristics of normally distributed, continuous data. From the factorial point of view, the problem is that factorization of phi coefficients for variables with disparate distributions can be expected to yield a confounding of results due to common trait variance, with results which are a function of variance associated with the distribution of mean values of the variables. This problem has been discussed most often under the rubric of "difficulty factors"; in this paper it was referred to as the "disparate distributions problem." Two methods designed to overcome the disparate distributions problem were developed and investigated. One method (which was designated RSBD) called for the definition of a hypothetical simplex matrix as a corollary to an observed binary matrix and the regression of the simplex on the observed matrix. The other method (RMF) involved the definition of reference vectors which would represent the distribution of mean values in the space of the observed matrix, and the removal of variance which could be accounted for by these vectors. Each of these methods was designed to yield a residual correlation matrix which would be free from the confounding effects of the distribution of mean values of the variables. The procedures of RSBD and RMF were illustrated through analyses of small hypothetical binary data matrices. In addition, an RSBD was carried out on a set of data composed of the scores of 300 subjects for 28 items from the Guilford-Zimmerman Temperament Survey, and for 2 randomly generated variables. This analysis was followed by a principal components analysis (PCA) of the residual matrix that was obtained, and a PCA of the matrix of phi coefficients for the 30 items. The results of these PCAs were compared in an attempt to determine the kinds of changes which might be found in the factor structure of a matrix before and after an RSBD. The results of the applications of RSBD and RMF were in accordance with expectations. That is, both of these methods accomplished the removal of the effects of the disparate distributions from the relationships among the binary variables under study. However, computational difficulties which make these methods impracticable for analyses of large sets of data were encountered. In addition, conceptual difficulties associated with RMF were brought to light. The attempt to develop a method for dealing with the disparate distributions problem was only partially successful. Neither of the methods which were developed provided the practical solution which was sought. However, a substantial amount of information regarding the characteristics of the problem and the obstacles to overcoming it was obtained. This information and the results of RSBD and RMF for small sets of data led to the conclusion that these methods may offer potentially effective and practicable solutions to the disparate distributions problem.