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

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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.