Natural Variances of Mental Tests, and the Symmetry Criterion

Abstract

WHEN two correlation matrices R₁, and R₂ have a symmetrical product R₁R₂ = R₂ R₁ they have the same latent vectors^1,2, and Prof. Cyril Burt has recently proposed³ to use this as a criterion that two batteries of mental tests contain the same mental ‘factors’. I do not myself think that one can draw any safe conclusions about identity of factors in two batteries unless some tests are common to both batteries. But if one accepts Burt's criterion, I wish to point out that when R₁R₂ is asymmetrical, the product of the covariance matrices D₁R₁D₁ and D₂R₂D₂ may conceivably be symmetrical, where D 1 and D₁ are diagonal matrices of standard deviations. If an experimenter therefore has psychological reasons for thinking that two batteries contain identical factors, but finds R₁R₂ to be asymmetric, he may be able to discover variances which would make the product D₁R₁D₁D₂R₂D₂, symmetric. In that case he would have considerable reason for assuming these to be the natural variances of these tests.