An Algebraic Computation of Weights for Thurstone's Analytic Rotation Method

Abstract

Thurscone (1954) proposed a single-plane analytic method of factor rotation in which each coordinate hyperplane is determined separacely. The solution requires an arbitrary weighting system, and che trial veccors used to approximate the coordinates of the normals to the desired hyperplanes are not obtained analytically; trial vectors are chosen from marker variables in the factor macrix or according to some hypothesis. However, once the weighting system has been chosen and the uial vectors selected (one trial vector for each factor to be rotated) the solution is complecely analytical. The criterion function of rotation was set up so char each attribute vector receives a weight which is determined solely by rhe size of che projeccion of that attribute on the crial vector. The weighcs are whole numbers ranging from 0 . . . 6 and are related to the projection by an arbitrary step funccion so as to emphasize the zero or near zero projections. The selection of adequate firsc approximarions to the desired transformation veccors is quite important in this mechod. Thurstone recommended raking normalized attribute vectors from the unrorated factor macrix. He stated that these trial vectors should represent attributes having both relatively high and relatively low correlations in their respective columns of the correlation matrix; however, his analytic rotational method does not include a procedure for the selection of these attributes. This must be done by E. In essence then, Thurstone's method of rotacion falls berween a complete hypothesis mechod and a method which is entirely analytical. An hypothesis mechod requires an escimate of the location of each large or significant loading and also each small or near-zero loading in the rotated matrix. On the other hand, in Thurstone's method only as many trial vectors have to be selected as there arc factors. Attempts have been made to obtain trial vectors analytically. For example, Cureton (1958) defined a trial vector from the relationship CI = hl- 2 R, where R' is the mean absolute value of che m numerically lowest correlations in column i of the correlation matrix, m is the number of factors, and hi is che square root of the communality. The firsc trial veccor selecred is the one for which Ci is algebraically highest; for each succeeding factor the selection is made on the arcribuce for which Ci is algebraically highest after elimination of all those having high loadings on any previously rocated factor. In any case, once a trial vector is selecred, che procedure is designed to find the mean principal axis for a whole set of weighted attribute vectors making up