  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, 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 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 as
well as 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.
