Ordinal data occur frequently in the social sciences. When applying principal component analysis (PCA), however, those data are often treated as numeric, implying linear relationships between the variables at hand; alternatively, non-linear PCA is applied where the obtained quantifications are sometimes hard to interpret. Non-linear PCA for categorical data, also called optimal scoring/scaling, constructs new variables by assigning numerical values to categories such that the proportion of variance in those new variables that is explained by a predefined number of principal components (PCs) is maximized. We propose a penalized version of non-linear PCA for ordinal variables that is a smoothed intermediate between standard PCA on category labels and non-linear PCA as used so far. The new approach is by no means limited to monotonic effects and offers both better interpretability of the non-linear transformation of the category labels and better performance on validation data than unpenalized non-linear PCA and/or standard linear PCA. In particular, an application of penalized optimal scaling to ordinal data as given with the International Classification of Functioning, Disability and Health (ICF) is provided.

中文翻译：

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用于功能核心集国际分类的有序变量的惩罚最优缩放

序数数据经常出现在社会科学中。然而，在应用主成分分析 (PCA) 时，这些数据通常被视为数字，暗示手头变量之间存在线性关系；或者，在有时难以解释所获得的量化结果的情况下应用非线性 PCA。用于分类数据的非线性 PCA，也称为最佳评分/缩放，通过为类别分配数值来构造新变量，使得这些新变量中由预定义数量的主成分 (PC) 解释的方差比例最大化。我们提出了一种用于序数变量的非线性 PCA 的惩罚版本，它是类别标签上的标准 PCA 和目前使用的非线性 PCA 之间的平滑中间体。与未惩罚的非线性 PCA 和/或标准线性 PCA 相比，新方法绝不仅限于单调效应，并且提供了类别标签的非线性变换的更好解释性和验证数据的更好性能。特别是，提供了对国际功能、残疾和健康分类 (ICF) 给出的有序数据的惩罚最佳缩放的应用。