2022-05-27T03:55:11Zhttps://nagoya.repo.nii.ac.jp/oaioai:nagoya.repo.nii.ac.jp:000021232021-03-01T21:27:01ZDESCRIPTION OF FACTOR CHANGE AND QUASI THREE-MODE FACTOR ANALYSIS因子変化の記述と準 3 相因子分析村上, 隆5534MURAKAMI, Takashi5535A simplified three-mode factor analysis model is investigated as a procedure for factor analysis of data matrices obtained on several occasions and for assessment of factor change. The model, termed quasi three-mode factor analysis, approximates the raw data x_<ijk>, which designates i^<th> individual's score on j^<th> scale on k^<th> occasion, by the following equations. [numerical formula] where x_<ijk>'s are assumed to be standardized to have zero mean and unit variance for each scale on each occasion. That is [numerical formula] The only difference of this model from Tucker's original one is that occasion factor coefficients are removed in Eq. (1), but this minor change makes this model applicable to any data with n_k≧2. Let X_k be N× n_j matrix of x_<ijk> for k^<th> occasion, A be the N×M matrix, B be the n_j×P matrix, and G_k be the M×P matrix, which is called the core matrix, then equation (1) is written in matrix form as X_k≅AG_kB' k=1,..., n_k (3) Matrix A is obtained by the factor analysis of N×n_jn_k data matrix X, X=(X_1...X_<n_k>) as X≅AL' (4) where A is factor score matrix which is subjected to constraints 1/NA'A=I In the analysis, factor scores are taken to be the same on each occasion. The loading matrix L, however, contains n_k loading matrices for separate occasions. That is L=(L_1...L_<n_k>) This factoring method of three-mode data is called method I. In method II, on the other hand, we begin with n_kN×n_j data matrix X^^^<∿>. <X'>^^^<∿>=(X_1'...<X'>_<n_k>) (<X'>^^^<∿> is transposed matrix of X^^^<∿>.) and B is obtained as factor loading matrix in the following factor analysis of X^^^<∿>. X^^^<∿>≅FB' (5) where B is, in this model, a single factor loading matrix common for each occasion and F is a following super matrix which involves n_k factor score matrices. F=(F_1...F_<n_k>) F_k's have zero mean for each factor, and F is orthogonal and has unit variance for each factor in the following sense. 1/NF'F.=I Two methods described above are usual factor analytic treatment of three-mode data. In quasi three-mode model, matrix A and B are considered to be the criterion for description of factor change by core matrices. Using L_k in method I and B in method II, core matirx G_k is obtained by G_k=<L'>_kB(B'B)^<-1> k=1,..., n_k (6) in the least squares sense. G_k may be submitted to orthogonal simple structure transformation, if necessary. These procedures do not yield the overall least squares solutions for Eq. (1), but make the quasi three-mode model interpretable as the summarized form of method I and method II. The core matrix G_k has a useful interpretation that it is the covariance matrix of factor scores in F_k with factor scores in A. That is G_k=1/N<F'>_kA (7) This shows that G_k gives the relations between factors on k^<th> occasion and the criterion factors, thus assessing the factor change through n_k occasions. Moreover, the core matrix can distinguish factor loadings change from factor scores change. As the matter of facts, G_k's are shown to exibit the characteristic patterns corresponding to the four types of factor changes suggested by Baltes & Nesselroade (1973).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1979-12-22application/pdf名古屋大學教育學部紀要. 教育心理学科26116http://hdl.handle.net/2237/354603874796https://nagoya.repo.nii.ac.jp/record/2123/files/KJ00002361550.pdfjpn