主因子分析における因子数に関する新基準の提案

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Here we propose a new computational procedure for the number of factors in Principal Factor Analysis (PFA). The basic notion of our computational procedure is based upon that Thurstone-type (1947) iterative PFA achieves a <minimization of the off-diagonal residuals>___- in a given correlation matrix on condition that the main-diagonal residuals are zeroes (Harman & Jones; 1966). The method is computationally a <modification>___- of Thurstone-type iterative PFA. We assume ths standard score of a test j, z_j, and its variance as fol1ows: [numerical formula] (1) and [numerical formula] (2) Here, F_j, a_<ji>, U_j, and a_j are the i-th common factor, the factor loading of test j on the i-th common factor, unique factor of test j, and the unique factor loading respectively. And we assumse [numerical formula] (3) If l<__-r<__-m, we have from (2) and (3) [numerical formula] (4) And if m<__-s<__-n, we may have the following possibility in test j under s [numerical formula] (5) (cf. Huttman; 1940). According to (5), we start with the estimation of <total variance>___- of at least one test. We perform our computation assuming that a given number of factors under which the total variance of at least one test is defined or that it is unity. In every iteration, we set at unity the largest estimated test variance in the diagonal of a given correlation matrix, and perform computation under a given number of factors. Then, after all estimated test variances are stabilized, we compare each estimated test variance to its corresponding calculated test variance. When the former is equal to the latter, that is, the minimization of the sum of the squares of off-diagonal residuals (i. e., the minimization of the minres___- criterion of Harman and Jones (1966)) is achieved, we have at least one test whose total variance is defined perfectly under the number of factors. (5) indicates that the number of factors should not___- be adopted as the number of common factors m in the matrix, and that the greatest number of factors among all the numbers of factors under which each estimated test variance is greater than its corresponding calculated test variance should be the number of common factors m. By using two illustrative hypothetical examples, we show that our method seems always to give the same number of factors as the assumed one. We apply the same rationale to empirical matrices and show that it is possible to classify empirical matrices into several types. And then we propose criteria concerning the number of factors each of which has its correspondence to a specific type by our classification. Finally, we discuss the estimation of communalities in our method, and we suggest the use of Thurstone-type iterative PFA under the number of common factors m.
日本教育心理学会の論文
1967-09-30

主因子分析における因子数に関する新基準の提案

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