A Conjugate Gradient Algorithm for the Multidimensional Analysis of Preference Data

Abstract

In continuation of earlier work on a new individual difference model for the multidimensional analysis of preference data (Schonemann and Wang, 1972), a relatively efficient algorithm for applying the model to fallible data was developed. It is based on the Method of Conjugate Gradients and thus does not require storage for second order derivatives. Several different versions of such an algorithm were compared ill terms of robustness, accuracy, and speed of convergence. The results strongly suggest that the so-called "intervening conjugate gradient method" (which iterates for only two of the three sets of unknowns and solves for the third set algebraically at each stage) is the most effective method for most purposes. The algorithm was applied to a relatively large set of political choice data which had bee previously analyzed by a different method. The outcome of this empirical study not only confirmed the earlier results but also led, as a consequence of the stronger metric structure of the present model, to a more detailed an informative description of the data.