2021-09-17T07:04:55Zhttps://nagoya.repo.nii.ac.jp/oaioai:nagoya.repo.nii.ac.jp:000023302021-03-01T21:22:36ZPrincipal Components and Canonical Variates of Several Sets of Variables<原著>複数の変数集合の主成分と正準変量村上, 隆6359MURAKAMI, Takashi6360Many of psychological correlational studies using questionnaires or tests generally consist of two steps. In the first step, some composite variates as linear combinations of a single set of variables are constructed by means of factor analytical procedures. Each composite is interpreted as a realization of an unobservable psychological construct. In the second step, correlations between composites from different variable sets are obtained. From these correlations, one wishes to establish the relationships between several psychological constructs under consideration. However, the factor analytical methods applied to a single variable set produce internally consistent composites, but they do not account for the relationships of the composites with variables belonging to other variable sets at all. The situation which we are confronted with is characterized by the bandwidth-fidelity dilemma in Cronbach's sense. Factor analysis of a single variable set overemphasizes the fidelity. The canonical correlation analysis, on the other hand, produces the composites with the highest correlations between sets, but they are often very unreliable, and are inappropriate measures of constructs. The canonical method overemphasizes the bandwidth. Because main objective of correlational studies may be finding interesting relationships between reliable measures, it is desirable to develop procedures through which one can balance the fidelity of the composites with the bandwidth of them. In this paper two methods which analyze several sets of variables simultaneously are proposed. They can be considered to be a natural extension of the principal component analysis and the canonical correlation analysis, respectively. Both of them define the first-order composites as linear combinations of variables of each set, and the second order composites as linear combinations of all the first order composites. Let Z_k be a N by n_k row-wise standardized data matrix for k-th set, then the matrix of first-order composites for k-th set is defined as F_k=V'_kZ_k, (1) and that of second order composites is defined as G=Σ__kW'_kF_k, (2) where V_k and W_k are n_k by q_k and q_k by Q weight matrices, respectively. Both methods share the optimization criterion which is the maximization of tr GG'/N under the constraint, F_kF'_k/N=I. (3) An additional constriant differs in two methods. That is, Σ__kW'_kV'_kV_kW_k=I (4) is imposed for the first one, hierarchical principal component analysis, and, Σ__kW'_kW_k=I (5) for the second, generalized canonical correlation analysis. Alternating algorithms for two methods are formulated. It is expected that the property of the first order composites is changed through altering the number of second order composites. More concretely, the most internally consistent solution can be obtained when Q is set to be equal to Σqk in hierarchical component analysis, which is equal to that of the separate principal component analysis for each set, and the most highly correlated solution is given by generalized canonical analysis when Q=max qk. Generally speaking, as Q is set to be large, correlations between composites of different sets are expected to be increased, and the fidelity of first order composites is improved as Q is smaller. Therefore, through changing the Q, one can balance the fidelity and the bandwidth of first order composites. Application to the real data revealed that the balancing mechanism of these methods works as was expected.国立情報学研究所で電子化したコンテンツを使用している。departmental bulletin paper名古屋大学教育学部1987-12-24application/pdf名古屋大學教育學部紀要. 教育心理学科34235253http://hdl.handle.net/2237/375303874796https://nagoya.repo.nii.ac.jp/record/2330/files/KJ00000137232.pdfeng