Orthonormal Polynomial Principal Component Analysis as a Transformation of Multiple Correspondence

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  • 多重対応分析の変換としての正規直交多項式主成分分析
  • 多重対応分析の変換としての正規直交多項式主成分分析 : Likert型項目の探索的分析のための新たな手続き
  • タジュウ タイオウ ブンセキ ノ ヘンカン ト シテ ノ セイキ チョッコウ タコウシキ シュセイブン ブンセキ : Likertガタ コウモク ノ タンサクテキ ブンセキ ノ タメ ノ アラタ ナ テツズキ
  • Analysis: A New Procedure for Exploratory Analysis of Likert-Type Items
  • —Likert 型項目の探索的分析のための新たな手続き—

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Abstract

<p> We proposed a method for transforming solutions provided by multiple correspondence analysis (MCA) to the form of principal component analysis (PCA) to justify the exploratory factor analysis of Likert-type items, and to extend it. We began by reformulating MCA as the maximization of the sum of variances of quantified variables, defined as the sum of quantified scores for each categorical variable. Next, we obtained a PCA formulation that yielded the same quantified scores as did the MCA through orthonormalizations of quantified scores by singular value decomposition of each block of a matrix of quantification weights. Owing to entire indeterminacies under orthonormal transformations of quantified scores for each variable, we proposed a way of providing metrics to ordered categories by orthonormal polynomials. We also proposed a method for computing a component pattern matrix after rotating a matrix of weights for PCA. The method can be viewed as Harris-Kaiser's independent cluster rotation. Finally, we demonstrated the application of the proposed procedures and interpreted the output using a real data set consisting of university student responses to Likert-type items asking experiences of positive and negative emotions in academic situations.</p>

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