Department of Statistics, University of Washington, Box 354320, Seattle, WA, 98195, USA.
Psychometrika. 2017 Jun;82(2):295-307. doi: 10.1007/s11336-017-9564-y. Epub 2017 Mar 13.
This paper considers the reflection unidentifiability problem in confirmatory factor analysis (CFA) and the associated implications for Bayesian estimation. We note a direct analogy between the multimodality in CFA models that is due to all possible column sign changes in the matrix of loadings and the multimodality in finite mixture models that is due to all possible relabelings of the mixture components. Drawing on this analogy, we derive and present a simple approach for dealing with reflection in variance in Bayesian factor analysis. We recommend fitting Bayesian factor analysis models without rotational constraints on the loadings-allowing Markov chain Monte Carlo algorithms to explore the full posterior distribution-and then using a relabeling algorithm to pick a factor solution that corresponds to one mode. We demonstrate our approach on the case of a bifactor model; however, the relabeling algorithm is straightforward to generalize for handling multimodalities due to sign invariance in the likelihood in other factor analysis models.
本文考虑了验证性因素分析（CFA）中的反射不可识别问题，以及其对贝叶斯估计的相关影响。我们注意到，在 CFA 模型中，由于载荷矩阵中所有可能的列符号变化而导致的多模态性，与由于混合成分的所有可能重新标记而导致的有限混合模型中的多模态性之间存在直接的类比。借鉴这一类比，我们推导出并提出了一种简单的方法来处理贝叶斯因子分析中方差的反射问题。我们建议在不限制载荷旋转的情况下拟合贝叶斯因子分析模型——允许马尔可夫链蒙特卡罗算法探索完整的后验分布——然后使用重新标记算法选择一个与一个模式相对应的因子解。我们在双因子模型的情况下演示了我们的方法；然而，对于由于似然中不变性导致的其他因子分析模型中的多模态性，重新标记算法很容易推广。
