Correction for overlap in principal component loadings

Abstract

Previous investigators have proposed procedures to correct for overlap in correlations when an item or variable is correlated with a total score in which it is included. The present investigation extends the research on item-total correlation correction into the area of principal component analysis. Since the interpretation of component loadings in principal component analysis involves the correlation of an item or variable with a weighted linear combinations of which it is a part, a similar problem of overlap or inflation of loadings (correlations) is present. The study involved two phases: (1) derivation of two correction formulas for overlap in principal component loadings; and (2) analysis of the effects of correction for overlap on interpretation of components under different matrix conditions. The first phase involved the adaptation of previously developed item-total correlation correction formulas for the weighting of both individual items and variance of original components. One correction method simply scales each of the loadings on the component by the square-root of coefficient alpha for that component. The other corrects for the unreliability of items or variables by substituting a rational equivalent for the original item or variable. The effects of the number of variables in the matrix and the magnitude of the correlations were investigated in Phase 2. The two correction methods were applied to four types of matrices in an attempt to show the effect that overlap may have on the psychological interpretation of principal component analyses The types of matrices used were as follows: (1) small number of variables - low intercorrelations; (2) small number of variables - moderate intercorrelations; (3) large number of variables - low intercorrelations; and (4) large number of variables - moderate intercorrelations. Application of the two formulas developed indicated that reduction of component loadings, as a result of correction for overlap, is considerably greater when the number of variables or items entering into the analysis is relatively small. For the examples used, correlation magnitude appeared to have minimal influence in the correction procedures. However, interpretation of component structures following correction for overlap showed considerable change for both large and small matrices.