A Nonmetric Variety of Linear Factor Analysis

The numbers in each column of an n × m matrix of multivariate data are interpreted as giving the measured values of all n of the objects studied on one of m different variables. Except for random error, the rank order of the numbers in such a column is assumed to be determined by a linear rule of combination of latent quantities characterizing each row object with respect to a small number of underlying factors. An approximation to the linear structure assumed to underlie the ordinal properties of the data is obtained by iterative adjustment to minimize an index of over-all departure from monotonicity. The method is “nonmetric” in that the obtained structure in invariant under monotone transformations of the data within each column. Except in certain degenerate cases, the structure is nevertheless determined essentially up to an affine transformation. Tests show ( a ) that, when the assumed monotone relationships are strictly linear, the recovered structure tends closely to approximate that obtained by standard (metric) factor analysis but ( b ) that, when these relationships are severely nonlinear, the nonmetric method avoids the inherent tendency of the metric method to yield additional, spurious factors. From the practical standpoint, however, the usefulness of the nonmetric method is limited by its greater computational cost, vulnerability to degeneracy, and sensitivity to error variance.