Department of Methods and Statistics, Faculty of Social Sciences, Utrecht University Utrecht, Netherlands ; Optentia Research Program, Faculty of Humanities, North-West University Vanderbijlpark, South Africa.
Front Psychol. 2013 Oct 23;4:770. doi: 10.3389/fpsyg.2013.00770. eCollection 2013.
Measurement invariance (MI) is a pre-requisite for comparing latent variable scores across groups. The current paper introduces the concept of approximate MI building on the work of Muthén and Asparouhov and their application of Bayesian Structural Equation Modeling (BSEM) in the software Mplus. They showed that with BSEM exact zeros constraints can be replaced with approximate zeros to allow for minimal steps away from strict MI, still yielding a well-fitting model. This new opportunity enables researchers to make explicit trade-offs between the degree of MI on the one hand, and the degree of model fit on the other. Throughout the paper we discuss the topic of approximate MI, followed by an empirical illustration where the test for MI fails, but where allowing for approximate MI results in a well-fitting model. Using simulated data, we investigate in which situations approximate MI can be applied and when it leads to unbiased results. Both our empirical illustration and the simulation study show approximate MI outperforms full or partial MI In detecting/recovering the true latent mean difference when there are (many) small differences in the intercepts and factor loadings across groups. In the discussion we provide a step-by-step guide in which situation what type of MI is preferred. Our paper provides a first step in the new research area of (partial) approximate MI and shows that it can be a good alternative when strict MI leads to a badly fitting model and when partial MI cannot be applied.
测量不变性（MI）是跨组比较潜在变量得分的前提。本文基于 Muthén 和 Asparouhov 的工作以及他们在 Mplus 软件中应用贝叶斯结构方程建模（BSEM）的工作，介绍了近似 MI 的概念。他们表明，通过 BSEM，可以用近似零来替代严格的 MI 的精确零约束，从而在不严格符合 MI 的情况下仍然可以得到拟合良好的模型。这种新的机会使研究人员能够在 MI 的程度和模型拟合的程度之间做出明确的权衡。在整篇文章中，我们讨论了近似 MI 的主题，然后通过一个实证示例来说明，在这个示例中，MI 的检验失败了，但是允许近似 MI 会得到一个拟合良好的模型。使用模拟数据，我们研究了在哪些情况下可以应用近似 MI，以及在什么时候它会导致无偏的结果。我们的实证示例和模拟研究都表明，在存在（许多）组间截距和因子载荷存在小差异的情况下，近似 MI 在检测/恢复真实潜在均值差异方面优于完全或部分 MI。在讨论中，我们提供了一个逐步指南，说明了在什么情况下应该选择哪种类型的 MI。本文为（部分）近似 MI 的新研究领域迈出了第一步，并表明在严格 MI 导致模型拟合不良且无法应用部分 MI 的情况下，它可以是一个很好的替代方案。
