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The indices i, j and k are used for subjects, variables and occasions (or methods), respectively. The model can be written as; X_k≅A_oG_k\u003cB\u0027\u003e_o; k=1,..., p (1) where X_k\u0027s are N×n data matrices for p occasions; A_o and B_o are N×Q factor scores matrix and n×q factor loading matrix, respectively, and they are common to each k; G_k\u0027s are Q×q core matrices which represent the factor change across p occasions. While A^^^^_k=A_oG_k can be considered to be a factor score matrix for k^\u003cth\u003e occasion with the factor loadings in B_o, B^^^^_k=B_oG\u0027_k can be regarded as a loading matrix for k^\u003cth\u003e occasion with the factor scores in A_o. Therefore, the model may give a natural summarization of the results of ordinary factor analysis applied to 3way data. In this paper, a rationale of least squares algorithm for the model was formulated in the manner of the alternating least squares method and a practical computational procedure requiring only the multitraitmultimethod type correlation matrix is developed. This algorithm not only gave the least squares solution of quasi threemode factor analysis in the strict sense, but also prompted the interpretations of the parameters. Regarding to B_o estimated by the algorithm, it is shown that the factor loading, the element of B_o, is the correlation of the raw score with the factor score for each occasion. That is, the following equation holds. [numerical formula] (2) The elements of core matrices, on the other hand, are the covariances between the factor scores common to all occasions and those for each occasion, such that G_k=1/NA\u0027_oA_k; k=1,..., p (3) A slightly modified model in which the core matrix is transformed into the correlation matrix rather than the covariance matrix was discussed. The core matrix in this revised model can be regarded as a sort of higher order loading matrix, for its elements are the correlations of the factor scores for each occasion with the factor scores in common which might be, so to speak, the factor of factors. Kelly and Fiske\u0027s multitraitmultimethod matrix including five personality traits (Assertive, Cheerful, Serious, Unshakable poise and Broad interests) rated by three methods was reanalyzed in this model. Two factors, which represent Assertive and Broad interests, and Cheerful and Unshakable poise, respectively, had highly invariant loadings and stable factor scores over all methods, therefore their convergent validity was considered to be high. 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<原著>最小 2 乗法による準 3 相因子分析のアルゴリズム
https://doi.org/10.18999/bulfep.28.39
https://doi.org/10.18999/bulfep.28.39dc3743b62458449283fd3f80d893320d
名前 / ファイル  ライセンス  アクション 

KJ00000137189.pdf (1.7 MB)


Item type  紀要論文 / Departmental Bulletin Paper(1)  

公開日  20060106  
タイトル  
タイトル  <原著>最小 2 乗法による準 3 相因子分析のアルゴリズム  
その他のタイトル  
その他のタイトル  AN ALGORITHM OF QUASI THREEMODE FACTOR ANALYSIS BY THE LEAST SQUARES METHOD  
著者 
村上, 隆
× 村上, 隆× MURAKAMI, Takashi 

抄録  
内容記述  The quasi threemode factor analysis is a method for the factor analytic treatment of 3way data matrix which has elements x_<ijk> such that i=1,..., N, j=1,..., n, k=1,..., p. The indices i, j and k are used for subjects, variables and occasions (or methods), respectively. The model can be written as; X_k≅A_oG_k<B'>_o; k=1,..., p (1) where X_k's are N×n data matrices for p occasions; A_o and B_o are N×Q factor scores matrix and n×q factor loading matrix, respectively, and they are common to each k; G_k's are Q×q core matrices which represent the factor change across p occasions. While A^^^^_k=A_oG_k can be considered to be a factor score matrix for k^<th> occasion with the factor loadings in B_o, B^^^^_k=B_oG'_k can be regarded as a loading matrix for k^<th> occasion with the factor scores in A_o. Therefore, the model may give a natural summarization of the results of ordinary factor analysis applied to 3way data. In this paper, a rationale of least squares algorithm for the model was formulated in the manner of the alternating least squares method and a practical computational procedure requiring only the multitraitmultimethod type correlation matrix is developed. This algorithm not only gave the least squares solution of quasi threemode factor analysis in the strict sense, but also prompted the interpretations of the parameters. Regarding to B_o estimated by the algorithm, it is shown that the factor loading, the element of B_o, is the correlation of the raw score with the factor score for each occasion. That is, the following equation holds. [numerical formula] (2) The elements of core matrices, on the other hand, are the covariances between the factor scores common to all occasions and those for each occasion, such that G_k=1/NA'_oA_k; k=1,..., p (3) A slightly modified model in which the core matrix is transformed into the correlation matrix rather than the covariance matrix was discussed. The core matrix in this revised model can be regarded as a sort of higher order loading matrix, for its elements are the correlations of the factor scores for each occasion with the factor scores in common which might be, so to speak, the factor of factors. Kelly and Fiske's multitraitmultimethod matrix including five personality traits (Assertive, Cheerful, Serious, Unshakable poise and Broad interests) rated by three methods was reanalyzed in this model. Two factors, which represent Assertive and Broad interests, and Cheerful and Unshakable poise, respectively, had highly invariant loadings and stable factor scores over all methods, therefore their convergent validity was considered to be high. However, factor scores of the remaining one factor representing Serious were unstable across the methods and the convergent validity of the traits might be suspicious.  
内容記述タイプ  Abstract  
内容記述  
内容記述  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。  
内容記述タイプ  Other  
出版者  
出版者  名古屋大学教育学部  
言語  
言語  jpn  
資源タイプ  
資源  http://purl.org/coar/resource_type/c_6501  
タイプ  departmental bulletin paper  
ID登録  
ID登録  10.18999/bulfep.28.39  
ID登録タイプ  JaLC  
ISSN（print）  
収録物識別子タイプ  ISSN  
収録物識別子  03874796  
書誌情報 
名古屋大學教育學部紀要. 教育心理学科 巻 28, p. 3959, 発行日 19811225 

フォーマット  
application/pdf  
著者版フラグ  
値  publisher  
URI  
識別子  http://hdl.handle.net/2237/3594  
識別子タイプ  HDL 