Principal Components Analysis With Nonlinear Optimal Scaling Transformations for Ordinal and Nominal Data

Abstract

This chapter focuses on the analysis of ordinal and nominal multivariate data, using a special variety of principal components analysis that includes nonlinear optimal scaling transformation of the variables. Since the early 1930s, classical statistical methods have been adapted in various ways to suit the particular characteristics of social and behavioral science research. Research in these areas often results in data that are nonnumerical, with measurements recorded on scales having an uncertain unit of measurement. Data would typically consist of qualitative or categorical variables that describe the persons in a limited number of categories. The zero point of these scales is uncertain, the relationships among the different categories is often unknown, and although frequently it can be assumed that the categories are ordered, their mutual distances might still be unknown. The uncertainty in the unit of measurement is not just a matter of measurement error because its variability may have a systematic component. For example, in the data set that will be used throughout this chapter as an illustration, concerning feelings of national identity and involving 25,000 respondents in 23 different countries all over the world (International Social Survey Programme [ISSP], 1995), there are variables indicating how close the respondents feel toward their neighborhood, town, and country, measured on a 5-point scale with labels ranging from not close at all to very close. This response format is typical for a lot of behavioral research and definitely is not numerical (even though the categories are ordered and can be coded numerically).