Principal components analysis and canonical correlations analysis for three-mode data and their applications to questionnaire data

传统的利用因子分析法构建心理个体差异量表的程序存在两个明显的缺陷：(1)它基于项目之间的内部一致性，忽略了与外部信息的关系。结果表明，（子）量表的分数虽然具有足够高的信度，但有时缺乏建设性效度。(2)措辞详尽的题目往往被从子量表的定义中删除为“剩余题”，因为它们与措辞简单的问题的相关性相对较低，尽管它们可以通过反映预期构念的一些微妙方面来提高效度。这两个命题表明，人们应该探索不同的优化准则，这些准则或多或少地为从标准主成分分析（PCA）中获得的项目响应的线性复合产生不同的权重。我们提出了两种新开发的心理测量技术，并证明了它们克服上述缺陷的可能性。第一个是改进的tucker2（三模式主成分分析的简化版本），它表明，当内部一致项的子集与外部变量的相关性高于其他项时，它产生的权重集与普通PCA显著不同。此外，还表明可以通过改变二阶分量的数量来平衡由相关性差异引起的“带宽保真困境”。第二（组）方法，正交和斜直接procrustes技术，表明人们可以在任何条件下获得（子）尺度，其模式近似于理论上指定的目标矩阵，而如果目标与给定数据的相关结构不一致，则解释方差的数量将非常小。斜向法优化的是模式（标准回归权重矩阵）而不是结构（相关性矩阵），效果优于正交法。少

The traditional procedure of scale construction for measuring psychological individual differences by the use of factor analytical method has two distinctive drawbacks:(1)It is based on internal consistency between items and ignores the relationships with external information. As the results, scores of (sub)scales sometimes lack of constructive validity although they have sufficiently high reliability.(2)Items with elaborated wordings tend to be removed from the definitions of subscales as 'remainder items' due to relatively low correlations with questions with simple wordings although they could contribute to increasing validity by reflecting some subtle aspects of intended constructs.These two propositions suggest that one should explore different optimization criteria which generate more or less different weights for linear composites of item responses from ones obtained from standard principal components analysis (PCA). We proposed two newly developed psychometric techniques, and d … More emonstrated their possibility to overcome drawbacks mentioned above. The first one, modified tucker2 (a simplified version of three-mode principal components analysis), showed that it generates remarkably different sets of weights from ordinary PCA when a subset of internally consistent items have higher correlations with external variables than others. In addition, it was also shown that one can balance the 'bandwidth-fidelity dilemma' caused by the differences of correlations by changing the number of the second-order components. The second (set of) methods, orthogonal and oblique direct procrustes techniques, demonstrated that one may obtain (sub)scales whose pattern approximate the theoretically specified target matrix in any conditions while the amount of explained variances would be very small if the target is not congruent to the correlational structure of the given data. Oblique method optimizing the pattern (the matrix of standard regression weights) rather than structure (the matrix of correlations) worked better than orthogonal one. Less