Factors and Principal Components in the Near Spherical Case

A sufficient condition in terms of the unique variances of a common factor model is given for the results of factor analysis to come close to those of principal component analysis. Principal components and factor scores as well as their proxies approach each other when the unique variances (or rather the eigenvalues of the unique covariance matrix) do not differ very much (the near spherical case), more precisely, when the differences of the unique variances tend to zero relative to the smallest squared singular value of the loading matrix. A similar statement can be made with respect to the loading matrices.