The Multiple Group Method of Factor Analysis and Rotation to a Simple Structure Hypothesis
THE MULTIPLE group method of factor anI alysis developed by Thurstone (9) is one of the most rapid computational techniques available. It is much more rapid than the centroid method also developed by Thurstone. The success of the multiple group method in reproducing the correlation matrix depends somewhat on the se lection of the variables to be included in e a c h group. In describing the multiple group method, Fruchter (3) suggests that a preliminary cluster analysis of the correlation matrix may provide the basis for the grouping of variables. Gutt man (4), on the other hand, points out that a fac tor analysis study like any other scientific in vestigation should begin with certain initial hy potheses and proceed to test these hypotheses. The implication seems to be that in setting up a factor study one should have a hypothesis as to the primary variables in the particular domain under investigation and should include measures of these variables in one's test battery. This point of view seems to be in line with that of Thurstone (10), although his rotational proced ures have not in general presupposed any knowl edge of the primary factors in the system to be investigated. In any case, suppose one has a hypothesis as to the variables in the correlation matrix which have appreciable loadings on each of the primary factors. One can then construct a hypothe sis matrix consisting of "0" 's and "1" 's, such that if a test is assumed to have no loadings for a given factor, a zero will appear in the corres ponding position, and if it is assumed to have a loading a "1" appears. One may then use this hypothesis matrix as a basis for grouping the variables to be factored by the multiple gr o up method. In a recent article Horst (5) has shown how some of the computations outlined by Thurstone for the multiple group method can be simplified. This simplified procedure gives the same results as Thurstone's method and likewise provides no unique solution in the sense that the results depend on the particular order inwhichthe I groups of variables are arranged. If there are n groups then the number of different solutions is ni A method for obtaining a unique solution which does not depend on the order of the groups has been outlined by Adcock (1). However, this method does not allow for correlations among the primaries since it involves an orthogonal so lution. A method has been developed by Tucker (11) for rotating any arbitrary factor loading matrix to a simple structure hypothesis. A simplified approximate solution has been given by Horst (6) for the case of a centroid factor loading matrix. If a simple structure hypothesis is available the multiple group method and Tucker's rotational method may be combined to effect a great econ omy of labor in computing the simple structure factor loading matrix. This solution is unique in that it does not depend upon the order of groups. The procedure for this method will be described with the aid of a numerical example. The vari ables used are from a study by Schaie (7) which was concerned with investigating the dimension ality of rigidity. The details of this study will be reported in a separate paper by Schaie (8). Table I gives the intercorrelations for ten var iables for a sample of 216 cases. The highest correlation in each column has been used as an estimate of the test communality. A prelimin ary inspection of this table together with certain a priori considerations suggested that the vari ables be grouped as follows: (1, 2, 3), (4, 5, 6), and (7, 8, 9,10). As pointed out by Guttman (4), it is entirely possible for a variable to be includ ed in more than one group. For example, vari able 7 because of its relatively high correlation of . 404 with variable 3, might have been includ ed in both the first and third groups. Table II gives the hypothesis matrix according to which the variables were grouped. Table HI shows the the factor loadings obtained by the computation al procedure outlined in (5).