The Foundations of Ordinal Factor Analysis: Affine and Spherical Geometries Induced by a Collection of Weak Orders
Richard Beals, David H. Krantz · Journal of Mathematical Psychology · 1993
The model for ordinal factor analysis represents individuals as vectors and represents their rankings on tests as projections onto suitable directed axes. This paper constructs such a vectorial model (analytic affine and spherical geometries) from a synthetic geometry whose undefined concepts are abstract orders on an arbitrary set. The first part of the construction uses only the equivalence relations obtained from the set of orderings. A series of seven axioms permits construction of linear and dual (spherical) subspaces satisfying the ordinary dimensional rules. The one-dimensional linear objects constructed in this way are called lines. The second half of the construction is based on Axiom 8, which asserts that all the orderings are consistent in the betweenness relation induced on any given line. This permits the introduction of order topology on each line. Four topological axioms are introduced, including connectedness of lines, which implies constructibility of certain intersections. A theory of affine congruence can then be developed, leading to the introduction of real-valued coordinates.