Psychological research often builds on between-group comparisons of (measurements of) latent variables; for instance, to evaluate cross-cultural differences in neuroticism or mindfulness. A critical assumption in such comparative research is that the same latent variable(s) are measured in exactly the same way across all groups (i.e., measurement invariance). Otherwise, one would be comparing apples and oranges. Nowadays, measurement invariance is often tested across a large number of groups by means of multigroup factor analysis. When the assumption is untenable, one may compare group-specific measurement models to pinpoint sources of noninvariance, but the number of pairwise comparisons exponentially increases with the number of groups. This makes it hard to unravel invariances from noninvariances and for which groups they apply, and it elevates the chances of falsely detecting noninvariance. An intuitive solution is clustering the groups into a few clusters based on the measurement model parameters. Therefore, we present mixture multigroup factor analysis (MMG-FA) which clusters the groups according to a specific level of measurement invariance. Specifically, in this article, clusters of groups with metric invariance (i.e., equal factor loadings) are obtained by making the loadings cluster-specific, whereas other parameters (i.e., intercepts, factor (co)variances, residual variances) are still allowed to differ between groups within a cluster. MMG-FA was found to perform well in an extensive simulation study, but a larger sample size within groups is required for recovering more subtle loading differences. Its empirical value is illustrated for data on the social value of emotions and data on emotional acculturation.

中文翻译：

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混合多组因子分析，用于解开多个组的因子加载非不变性。

心理学研究通常建立在潜在变量（测量）的组间比较之上；例如，评估神经质或正念的跨文化差异。这种比较研究中的一个关键假设是相同的潜在变量在所有组中以完全相同的方式测量（即测量不变性）。否则，就会比较苹果和橙子。如今，测量不变性通常通过多组因子分析在大量组中进行测试。当假设不成立时，可以比较特定组的测量模型以查明非不变性的来源，但成对比较的数量随着组的数量呈指数增长。这使得很难从非不变性中解开不变性以及它们适用于哪些组，它增加了错误检测非不变性的机会。一个直观的解决方案是根据测量模型参数将组聚类成几个集群。因此，我们提出了混合多组因子分析（MMG-FA），它根据特定的测量不变性水平对组进行聚类。具体来说，在本文中，具有度量不变性（即，等因子载荷）的组簇是通过使载荷集群特定来获得的，而其他参数（即，截距、因子（协）方差、残差方差）仍然允许集群内的组之间存在差异。MMG-FA 在广泛的模拟研究中表现良好，但需要更大的组内样本量来恢复更细微的负载差异。