Mixture Simultaneous Factor Analysis for Capturing Differences in Latent Variables Between Higher Level Units of Multilevel Data

Abstract

Given multivariate data, many research questions pertain to the covariance structure: whether and how the variables (e.g., personality measures) covary. Exploratory factor analysis (EFA) is often used to look for latent variables that might explain the covariances among variables; for example, the Big Five personality structure. In the case of multilevel data, one might wonder whether or not the same covariance (factor) structure holds for each so-called data block (containing data of 1 higher level unit). For instance, is the Big Five personality structure found in each country or do cross-cultural differences exist? The well-known multigroup EFA framework falls short in answering such questions, especially for numerous groups or blocks. We introduce mixture simultaneous factor analysis (MSFA), performing a mixture model clustering of data blocks, based on their factor structure. A simulation study shows excellent results with respect to parameter recovery and an empirical example is included to illustrate the value of MSFA.Keywords: factor analysis, latent variables, mixture model clustering, multilevel data Given multivariate data, researchers often wonder whether the variables covary to some extent and in what way. For instance, in personality psychology, there has been a debate about the structure of personality measures (i.e., the Big Five vs. Big Three debate; De Raad et al., 2010). Similarly, emotion psychologists have discussed intensely whether and how emotions as well as norms for experiencing emotions can be meaningfully organized in a lowdimensional space (e.g., Ekman, 1999;Fontaine, Scherer, Roesch, & Ellsworth, 2007;Russell & Barrett, 1999;Stearns, 1994). Factor analysis (Lawley & Maxwell, 1962) is an important tool in these debates as it explains the covariance structure of the variables by means of a few latent variables, called factors. When the researchers have a priori assumptions on the number and nature of the underlying latent variables, confirmatory factor analysis (CFA) is often used, whereas exploratory factor analysis (EFA) is applied when one has no such assumptions.Research questions about the covariance structure get further ramifications when the data have a multilevel structure; for instance, when personality measures are available for inhabitants from different countries. We refer to data organized according to the higher level units (e.g., the countries) as data blocks. For multilevel data, one can wonder whether or not the same structure holds for each data block. For example, is the Big Five personality structure found in each country or not (De Raad et al., 2010) Setiadi, & Markam, 2002;MacKinnon & Keating, 1989;Rodriguez & Church, 2003).When looking for differences and similarities in covariance structures, using EFA is very advantageous because it leaves more room for finding differences than CFA does. For instance, in the emotion norm example (Eid & Diener, 2001), one might very well expect two latent variables to show up in each country corresponding to approved and disap...

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