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

Abstract

Multivariate multigroup data are collected in many fields of science, where the so‐called groups pertain to, for instance, experimental groups or countries the participants are nested in. To summarize the main information in such data, principal component analysis (PCA) is highly popular. PCA reduces the variables to a few components that are linear combinations of the original variables. Researchers usually assume those components to be the same across the groups and aim to apply a simultaneous component analysis. To investigate whether this assumption is reasonable, one often analyzes the groups separately and computes a similarity index between the group‐specific component loadings of the variables. In many cases, however, most variables have highly similar loadings across the groups, but a few variables, which we will call “outlying variables,” behave differently, indicating that a simultaneous analysis is not warranted. In such cases, the outlying variables should be removed before proceeding with the simultaneous analysis. To do so, the variables are ranked according to their relative outlyingness. Although some procedures have been proposed that yield such an outlyingness ranking, they might not be optimal, because they all rely on the same choice of similarity coefficient without evaluating other alternatives. In this paper, we give an overview of other options and report extensive simulations in which we investigate how this choice affects the correctness of the outlyingness ranking. We also illustrate the added value of the outlying variable approach by means of sensometric data on different bread samples.

ONVAR: A simultaneous component analysis approach for disentangling outlying and non-outlying variables

“…What to do if outlying variables are detected in the data and one is not just interested in modeling the data in the blocks correctly, but also wants to further correlate the component scores with external variables, for instance, to establish construct validity [38,5]? This brings us back to approaches like LBCM that suggest to remove the outlying variables from the data until the components extracted from the remaining variables are the same across the blocks, before proceeding with further analyses [16,17,18]. This strategy is defendable if we have a relatively large number of non-outlying variables and a few outlying ones only.…”

“…As holds for separate PCAs, this analysis sheds no direct light on which variables (i.e., attributes) are used differently in the two clusters, and to what extent. To answer this question, Gvaladze et al [17] did a follow up analysis that resulted in a ranking of the variables from most to least outlying across these two clusters of panelists. Albeit interesting, the latter analysis suffers from two limitations: First, it does not draw a clear line between outlying and non-outlying variables.…”

“…To address these issues, we applied the ONVar model to the bread data. The first set of analyses builds on the analyses of De Roover et al [8] and Gvaladze et al [17] as it uses the same partitioning of the data blocks. Thus, these analyses determine the variables that behave differently in the two clusters of panelists and shed light on their loading patterns.…”

“…If most of the correlations are very different across the blocks, it is recommended to switch to running separate PCAs. However, if the differences are concentrated around a few variables only, it may be better to flag these variables, which we call outlying, in line with [16,17,18] and continue with a simultaneous analysis approach on the subset of variables that behave (virtually) the same across the blocks and are therefore called non-outlying variables. Note that our definition of outlying variables differs from the one given by Hahs-Vaughn [19]; this author uses the term outlying variables to denote variables that hardly correlate with any of the other variables.…”