Computational Statistics and Data Analysis Two Types of Single-peaked Data: Correspondence Analysis as an Alternative to Principal Component Analysis

It has been argued that principal component analysis (PCA) is not appropriate for analyzing data conforming to single-peaked response models, also referred to as unfolding models. An overview of these findings is given, which relates them to the distinction between two types of unfolding models; namely, models that are either a quadratic function of the person-to-item distances or an exponential function of these distances. This distinction is easy to recognize empirically because the inter-item correlation matrix for the two types of data typically shows different patterns. Furthermore, for both types of unfolding models, correspondence analysis (CA), which is a rival method for dimensionality reduction, outperforms PCA in terms of representation of both person and item locations, especially for the exponential unfolding model. Finally, it is shown that undoubled CA outperforms doubled CA for both types of unfolding models. It is argued that performing CA on the raw data matrix is an unconventional, but meaningful approach to scaling items and persons on an underlying unfolding scale. A real data example on personality assessment is given, which shows that for this type of data (undoubled) CA is to be preferred over PCA.