A geometrical interpretation of the horseshoe effect in multiple correspondence analysis of binary data

Abstract

When a set of binary variables is analyzed by multiple correspondence analysis, a quadratic relationship between individual scores corresponding to the two largest characteristic roots is often observed. This phenomenon is called the horseshoe effect, which is well known as an artifact in the analysis of the perfect scale in Guttman's sense, and also observed in the quasi scale as a result of random errors. In addition, although errors are unsystematic and symmetric, scores corresponding to erroneous response patterns lie only inside the horseshoe. This phenomenon, which we will call filled horseshoe, is explained by the concept of an affine projection of a hypercube that represents binary data. The image of the hypercube on the plane has the form of a zonotope, which is a convex and centrally symmetric polygon, and it is shown that images forming the horseshoe must lie along the vertices of the zonotope, if it exists, and hence, other images must reside inside it. © 2012 Springer-Verlag Berlin Heidelberg.