因子数推定のための線形回帰基準について

柏木 繁男; 石塚 智一

doi:10.4992/jjpsy.44.167

A LINEAR REGRESSION CRITERION FOR ESTIMATE OF THE NUMBER OF FACTORS

SHIGEO KASHIWAGI, TOMOICHI ISHIZUKA

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JOURNAL FREE ACCESS

1973 Volume 44 Issue 4 Pages 167-178

DOI https://doi.org/10.4992/jjpsy.44.167

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Published: 1973 Received: May 08, 1973 Available on J-STAGE: July 16, 2010 Accepted: - Advance online publication: - Revised: -
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Date of correction: July 16, 2010 Reason for correction: - Correction: ABSTRACT Details: Wrong : A new criterion which is based on the principle of linear regression analysis is proposed concerning the estimation of the number of factors in multiple factor analysis.
In the first section, immediately after a short historical review on the number of factors is presented, the emphasis is placed on the necessity of a further criterion which will assure greater invariance with regard to the change of the number of variables.
In the second section, as indicated in the title of this paper, the following linear regression formula is proposed. That is _lY=Xβ+E (4) where _lY is the criterion vector which consists of the deviated off-diagonal elements of the reproduced correlation matrix with the number of factors l, X the predicted vector which consists of the deviated off-diagonal elements of the original correlation matrix β, the linear regression coefficient, and E the error vector. In (4), if the standard deviation of _lY is almost the same as that of X, the least square estimate of the linear regression coefficient β may be β≈r_{X_lY}≈1 (6) where r_{X_lY} is the correlation coefficient between X and _lY. If the criterion (6) is satisfied with the increase of the number of factors l, l is expected to be a powerful estimate of the true number of factors. Here we assume that the change of the value of r_{X_lY} is monotone with regard to the increase of the number of factors.
The proposal above is applied to two hypothetical and twelve actual correlation matrices and the functional relationships between the change of the value of r_{X_lY} and the increase of the number of factors and is investigated in the third section. The most important parts of results are presented in Table 1, Fig. 1-a, Fig. 1-b, and Fig. 1-c. And it is suggested that the value of r_{X_lY}. 900 may be a useful index for the estimate of the number of common factors.
In the last section, some related topics such as the sign reflection to assure the maximization of the value of r_{X_lY} in the neighborhood of the true number of factors are discussed.
Right : A new criterion which is based on the principle of linear regression analysis is proposed concerning the estimation of the number of factors in multiple factor analysis. Following a short historical review on the number of factors, the emphasis is placed on the necessity of a new type of criterion, and a linear regression criterion is proposed based on the regression of the reproduced off-diagonal elements with an assigned number of factors on the corresponding elements in the original correlation matrix. In the last section some related important topics which are necessary for further sophistication of our proposal are discussed.

Abstract

A new criterion which is based on the principle of linear regression analysis is proposed concerning the estimation of the number of factors in multiple factor analysis. Following a short historical review on the number of factors, the emphasis is placed on the necessity of a new type of criterion, and a linear regression criterion is proposed based on the regression of the reproduced off-diagonal elements with an assigned number of factors on the corresponding elements in the original correlation matrix. In the last section some related important topics which are necessary for further sophistication of our proposal are discussed.

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