Oblique rotation in correspondence analysis: a step forward in the search for the simplest interpretation.

Correspondence analysis (CA) is a popular method that can be used to analyse relationships between categorical variables. It is closely related to several popular multivariate analysis methods such as canonical correlation analysis and principal component analysis. Like principal component analysis, CA solutions can be rotated orthogonally as well as obliquely into a simple structure without affecting the total amount of explained inertia. However, some specific aspects of CA prevent standard rotation procedures from being applied in a straightforward fashion. In particular, the role played by weights assigned to points and dimensions and the duality of CA solutions are unique to CA. For orthogonal simple structure rotation, procedures recently have been proposed. In this paper, we construct oblique rotation methods for CA that take into account these specific difficulties. We illustrate the benefits of our oblique rotation procedure by means of two illustrative examples.

Clustering preference data in the presence of response-style bias.

Preference data, such as Likert scale data, are often obtained in questionnaire-based surveys. Clustering respondents based on survey items is useful for discovering latent structures. However, cluster analysis of preference data may be affected by response styles, that is, a respondent's systematic response tendencies irrespective of the item content. For example, some respondents may tend to select ratings at the ends of the scale, which is called an 'extreme response style'. A cluster of respondents with an extreme response style can be mistakenly identified as a content-based cluster. To address this problem, we propose a novel method of clustering respondents based on their indicated preferences for a set of items while correcting for response-style bias. We first introduce a new framework to detect, and correct for, response styles by generalizing the definition of response styles used in constrained dual scaling. We then simultaneously correct for response styles and perform a cluster analysis based on the corrected preference data. A simulation study shows that the proposed method yields better clustering accuracy than the existing methods do. We apply the method to empirical data from four different countries concerning social values.

Constrained Dual Scaling for Detecting Response Styles in Categorical Data.

Dual scaling (DS) is a multivariate exploratory method equivalent to correspondence analysis when analysing contingency tables. However, for the analysis of rating data, different proposals appear in the DS and correspondence analysis literature. It is shown here that a peculiarity of the DS method can be exploited to detect differences in response styles. Response styles occur when respondents use rating scales differently for reasons not related to the questions, often biasing results. A spline-based constrained version of DS is devised which can detect the presence of four prominent types of response styles, and is extended to allow for multiple response styles. An alternating nonnegative least squares algorithm is devised for estimating the parameters. The new method is appraised both by simulation studies and an empirical application.