Investigation of justifiability of Likert scaling using nonmetric principal components analysis and multiple correspondence analysis

统计学家和心理学研究的初学者经常质疑Likert标度法的可验证性，即将对问卷项目的分类反应总结为定量变量，因为分类反应应被视为顺序标度而不是像Likert方法那样被视为区间标度。虽然多重对应分析（MCA）被推荐作为Likert标度的替代方案，但它总是产生与正交或倾斜旋转的主成分分析（PCA）不同的输出，并且输出包括难以解释的特殊非线性分量。通过使用非度量PCA及其广义版本管理人格量表所获得的几个数据集的分析表明，只要多重对应分析的基本原理是可以接受的，因为所获得的负载矩阵和得分矩阵通常与经典PCA所获得的相似，将分类反应视为区间尺度，则将分类反应视为区间尺度的方式是合理的。虽然包括“马蹄铁”的非线性成分被认为是由于任何指标矩阵中的虚假维度而产生的伪像，但它们有时可以被解释为肢体反应的测量，这是一种与要测量的心理结构内容无关的反应集。特别是，它被发现，第三个组件，往往有一个类别数的三次函数的形状是可用于发现主题的反应是一致的极端点的结构，但他们的分数几乎是中等，因为他们的表达是温和的。

Both Statisticians and beginners of psychological research often challenge the justifiability of Likert scaling method, summing up categorical responses to questionnaire items as quantitative variable, because categorical responses should be treated as ordinal scales instead of as interval scales as is in Likert method. While multiple correspondence analysis (MCA) is recommended as an alternative to Likert scaling, it always produces different output from that of principal components analysis (PCA) with orthogonal or oblique rotation, and the output includes special nonlinear components that are difficult to interpret. Analyses of several datasets obtained by administering personality inventories by the use of nonmetric PCA and its generalized version demonstrated that the way of treatment of categorical responses as interval scales is justified as long as the rationale of multiple correspondence analysis is acceptable since the obtained matrices of loadings and of scores are generally similar to those obtained by classical PCA which regards categorical responses as interval scales. Although nonlinear components including 'horse-shoe' are considered to be artifacts due to spurious dimensions in any indicator matrices, they could sometimes be interpreted as measures of extremity responses, a kind of response sets irrelevant to contents of psychological constructs to be purported to measure. Especially, it was found that the third component that has often the shape of a cubic function of category numbers was available for discovering subjects whose responses are consistent to extreme points of the constructs, but their scores are almost medium because their expressions are modest.