The Relations of the Newer Multivariate Statistical Methods to Factor Analysis

Abstract. A survey of developments in multivariate analysis during the last thirty years shows that some, though not all, of the purposes for which factor analysis has been used may now be better accomplished by other procedures, e.g. by regression, multiple correlation, the study of relations between two sets of variates, or of the dimensionality of a set of variates. To determine whether two or more groups of persons differ significantly in their mean values or their covariance matrices, the most appropriate procedures consist of methods of multivariate analysis of variance. Such methods have increased rather than diminished the advantages of using an external criterion instead of making a purely internal analysis. The problem of estimating the dimensionality of a continuous multivariate population by means of a sample admits a simple and exact answer: the rank of the population is equal to that of the sample, provided the number of degrees of freedom among the individuals in the sample exceeds the number of the variates. If, however, the problem is re‐interpreted so as to imply that each observation is the sum of a real part and a random error, then the hypothesis that the real part has a rank smaller than the number of variates can be tested only when a proper estimate of the errors is available, i.e. by making suitable replications. The appropriate method is again multivariate analysis of variance rather than factor analysis. In a third form of the problem the object is, not to ascertain a true dimensionality, but to assess the error resulting from discarding (on practical grounds) a dimension that may really exist. Here the problem is somewhat indeterminate: a practical expedient may involve a special form of factor analysis, viz. the calculation of the least, not the greatest, latent roots. Thus, in many cases in which they have hitherto been used, factor analyses of the usual kinds are inferior to other procedures: nevertheless, the results of such analyses may have heuristic and suggestive value, and may uncover hypotheses which are capable of more objective testing by other methods.