How to detect which variables are causing differences in component structure among different groups

Abstract

When comparing the component structures of a multitude of variables across different groups, the conclusion often is that the component structures are very similar in general and differ in a few variables only. Detecting such Boutlying variables^is substantively interesting. Conversely, it can help to determine what is common across the groups. This article proposes and evaluates two formal detection heuristics to determine which variables are outlying, in a systematic and objective way. The heuristics are based on clusterwise simultaneous component analysis, which was recently presented as a useful tool for capturing the similarities and differences in component structures across groups. The heuristics are evaluated in a simulation study and illustrated using crosscultural data on values.

Detecting outlying variables in multigroup data: A comparison of different loading similarity coefficients

“…Given this outlyingness ranking, the heuristics implement a different stopping rule to decide how many variables are outlying. Overall, on the basis of the simulation results reported by De Roover et al 26 and Gvaladze et al, 27 the methods look promising, however.…”

“…A second aim of this paper is to further investigate the outlyingness ranking procedure. The results of De Roover et al 26 and Gvaladze et al 27 are incomplete because they only evaluated the Tucker's congruence index, whereas many other indices have been proposed in the literature: the Rv coefficient 28,32,33 and its modified variant, 34 the angle between the subspaces formed by the components, 35,36 and the root‐mean‐square difference (RMSD) between the loadings 37,38 . On the one hand, these measures differ in which loadings are involved in the computation of the index (all loadings, loadings on a specific construct, and loadings of a specific variable).…”

“…Applying such a target linear transformation before investigating the loading patterns' similarity is a common practice in psychological research 30,37,54,55 . In this paper, we follow De Roover et al 26 and use yet another option. Specifically, we first apply a VARIMAX transformation 56 to the SCA‐P loadings, followed by an oblique Procrustes transformation (i.e., target transformation) of the group‐specific PCA loadings towards the VARIMAX‐rotated SCA‐P ones.…”

“…Within psychometrics, De Roover et al 26 and Gvaladze et al 27 recently developed and evaluated multiple heuristics to do just that. All these heuristics imply that one runs separate PCA analyses on the different groups, performs some kind of oblique Procrustes transformation (also known as rotation), and finally assesses the similarity of the resulting loadings by means of Tucker's congruence index 28–31 .…”

“…In Section 3, we recapitulate the PCA‐based approaches for analyzing column‐coupled data. Section 4 discusses how the outlyingness ranking is obtained in the above‐mentioned heuristics, 26,27 followed by a brief discussion of the associated limitations. Section 5 recapitulates the different similarity measures that can be plugged into the heuristics as an alternative to Tucker's congruence.…”