順序性をもつデータ構造と因子分析

―完全尺度・準尺度に従う0・1変数相関行列の性質―

抄録

A Γ-shaped scatter diagram of factor scores has been reported previously in common with the data relating to cerebral and neural diseases; language tests of aphasia, finger function tests of hemiplegia, and symptoms in mild disturbance of consciousness. The factors were inter pretable as situated successively on severity continuum of the diseases. Scale analysis suggested a data structure based on Guttman perfect scale. Then Guttman perfect scale and quasi-scale models were investigated in respect to attributes of correlation matrices of dichotomous scale variables. Eigenvalues and eigenvectors were obtained in case of the perfect scale that subjects had a uniform distribution on the continuum. According to the number of eigenvalues greater than 1, the number of factors is expected as √t-½in the perfect scale, where t is the number of variables. The score of the 2-nd factor before factor rotation is represented by a parabola of the 1-st factor. The oscillation law of eigenvectors has a characteristic manifestation on the factor loadings after rotation, and on the Γ-shaped distribution of factor scores. Successive factors have high loadings on successive groups of variables located in the order determined by Guttman scale. In quasi-scale models, these properties undergo certain changes, so that the Γ-shape becomes ambiguous depending on the size of deviation from the perfect scale.