Psychometric Studies on Quantification and Simple Structure Analysis of Multivariate Categorical Data

本课题的目的是研究多重对应分析（MCA）中溶液的旋转。本文首先提出了一种正交旋转法，用待解释指标的维数矩阵来给出一个变量的简单结构。在这里，使用正交准则来定义简单性。根据作为变量的实体是类别还是项目，该方法分为类别选项和项目选项。在前一种选择中，类别得分作为指标，在后一种选择中使用歧视措施。如果将MCA表述为指标矩阵的低秩近似方法，则该解可以斜旋转。因此，我们接下来提出了一种使用Promax准则的斜旋转方法。这种方法也分为两种选择。在类别选项中，类别数据向量和对象分数向量之间的余弦用作要解释的索引，而项目子空间和对象分数向量之间的余弦用于项目选项。我们发现上述正交旋转和斜旋转方法在实际数据中的应用是有用的。此外，我们研究了MCA溶液的拟合优度（GOF）度量，并提出使用正确分类率（CCR）作为度量。CCR定义为根据解将物体分类到正确类别的情况所占的比例。仿真分析表明，CCR相对于其他基于特征值的GOF测度具有优越性。

The purpose of this project was to study the rotation of the solution in multiple correspondence analysis (MCA). We first proposed an orthogonal rotation method for giving simple structure to a variables by dimensions matrix of indices to be interpreted. Here, the Orthomax criterion is used for defining simplicity. The method is classified into a category option and an item option, according whether the entities regarded as variables are categories or items. In the former option, category scores are used as indices, while discrimination measures are used in the latter option.The solution can be rotated obliquely, if MCA is formulated as a method for the low-rank approximation of indicator matrices. Thus, we next proposed an oblique rotation method using the Promax criterion. This method is also classified into the two options. In the category option, the cosines between category data vectors and object score vectors are used as the indices to be interpreted, whereas the cosines between the sub-spaces for items and object score vectors are used in the item option. We found the usefulness of the above orthogonal and oblique rotation methods in their applications to real data.Additionally, we studied goodness-of-fit (GOF) measures of the MCA solution and proposed to use a correct classification rate (CCR) as the measure. CCR is defined as the proportion of the cases where objects are classified into correct categories according to the solution. Simulation analysis showed the superiority of CCR to other eigenvalue-based GOF measures.