Penalized Non-Linear Principal Components Analysis for Ordinal Variables with an Application to International Classification of Functioning Core Sets

Ordinal data occur frequently in the social sciences. When applying principal components analysis (PCA), however, those data are often treated as numeric implying linear relationships between the variables at hand, or 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 is maximized. We propose a penalized version of non-linear PCA for ordinal variables that is an intermediate between standard PCAon 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 as well as better performance on validation data than unpenalized non-linear PCA and/or standard linear PCA. In particular, an application of penalized non-linear PCA to ordinal data as given with the International Classification of Functioning, Disability and Health (ICF) is provided.