A Comparison of Oblique and Orthogonal Factor Solutions

Abstract

THE CHOICE OF rotational methods is one of the major unresolved issues in the field of factor analysis. In this country, the principal discrepant viewpoints on this matter seem best represented by Guilford and Cattell. The position of Cattell (2) is essentially that oblique factors better represent the fundamental psychological entities with which we are dealing, for these entities as they occur in na ture are correlated. Guilford (3) feels less cer tainty that this alleged fact has been demonstrated but grants that our data may eventually force us to an oblique-factor model. He prefers meanwhile to deal with orthogonal factors o n the ground that the appearance of obliqueness is commonly a conse quence of inadequate test sampling or of insufficient factor extraction. A common argument for oblique rotation is that it enables us to achieve better simple structure, or a better approximation to simple structure, than is possible with orthogonal rotation. But the is sue cannot be resolved on the grounds of economy alone, for each rotational method affords simplicity on one level at the expense of simplicity on another. The one economy cannot readily be weighed against the other, since simplicity of factor pattern (or struc ture) and correlational independence of factors are not commensurate phenomena. If we look beyond such technical obstacles as the sampling of tests and persons and envision the state of affairs that we should have with optimal sampling from both popu lations, we can see that the ultimate solution to the controversy depends on establishing a meaningful fit for our data. The fundamental question is not the economy of the mathematical model, but rather its elegance and theoretical productivity. The lat ter will be best provided by the model that best en ables us to reconcile hypotheses ar is ing from a multitude of related, but independent, factor iza tions. In this paper, we shall not attempt to re solve the issue of factor r eproducibility, but we shall consider a number of subsidiary problems. To approach a solution to the problem of rota tional methods, we must consider the relationship between the alternative procedures with reference to contexts in which we already possess good infor mation regarding factor s t r u c ture. We can most justifiably assume the possession of such informa tion when we are dealing with simple physical meas urements. Thurstone's box problems provide a useful illustration. In the earlier of these (7), Thurstone assigned scores on 20 variables to 20 hy pothetical boxes, each variable being a function of one or more of the three fundamental measurements. In a later study (8), scores on 26 similar variables were obtained for a sample of 30 actual boxes. Each study culminated in three correlated factors which were clearly identifiable as corresponding to the fundamental dimensions of height, length and width. In the hypothetical case, of course, Thurstone was led to an oblique solution because he had deliberate ly allowed the fundamental measurements to corre late as they would in an ordinary empirical sample. In the box-problem situation, it is clear that or thogonal rotation in three dimensions would not yield factors which correspond quite so neatly to the three known factors. Thomson (6), h o w e v er, has thrown some light on the relationship between oblique and orthogonal factors in this realm. Using a simplified form of Thurstone's earlier box prob lem, with eight boxes and seven variables, he de monstrates that the oblique solution can be convert ed to an equivalent orthogonal solution with four fac tors. His calculations yield three simple-structure factors which correspond to Thurstone's factors and one general factor which corresponds to the second-order size factor which Thurstone's proce dures should yield. Thomson's analysis adds some weight to the argument that apparent obliqueness may result from incompleteness of f ac tor extrac tion, since his orthogonal solution could have been obtained directly by rotation infour dimensions. Not many orthogonalists will rotate infour dimen sions, of course, when tests of completeness of ex traction clearly indicate three factors. On a similar mathematical basis, Schmid and Leiman (5) have more recently proposed converting oblique solutions to hierarchical orthogonal solu tions. Their procedure yields as many orthogonal factors of a common order as there are factors of all orders combined in the oblique solution. Cor responding to the oblique factors of highest order, there will be very general factors. Lower orders