Iteration of Partially Specified Target Matrices: Application to the Bi-Factor Case

The current study proposes a new bi-factor rotation method, Schmid-Leiman with iterative target rotation (SLi), based on the iteration of partially specified target matrices and an initial target constructed from a Schmid-Leiman (SL) orthogonalization. SLi was expected to ameliorate some of the limitations of the previously presented SL bi-factor rotations, SL and SL with target rotation (SLt), when the factor structure either includes cross-loadings, near-zero loadings, or both. A Monte Carlo simulation was carried out to test the performance of SLi, SL, SLt, and the two analytic bi-factor rotations, bi-quartimin and bi-geomin. The results revealed that SLi accurately recovered the bi-factor structures across the majority of the conditions, and generally outperformed the other rotation methods. SLi provided the biggest improvements over SL and SLt when the bi-factor structures contained cross-loadings and pure indicators of the general factor. Additionally, SLi was superior to bi-quartimin and bi-geomin, which performed inconsistently across the types of factor structures evaluated. No method produced a good recovery of the bi-factor structures when small samples (N = 200) were combined with low factor loadings (0.30–0.50) in the specific factors. Thus, it is recommended that larger samples of at least 500 observations be obtained.

本研究提出一种新型双因子旋转方法——带迭代目标旋转的施密德-莱曼法（Schmid-Leiman with iterative target rotation, SLi），该方法基于部分指定目标矩阵的迭代，以及由施密德-莱曼正交化（Schmid-Leiman, SL）得到的初始目标。当因子结构包含交叉载荷、近零载荷，或二者兼具时，SLi可改善此前提出的两类SL双因子旋转方法（即SL与带目标旋转的施密德-莱曼法SLt）所存在的部分局限。本研究通过蒙特卡洛模拟（Monte Carlo simulation）检验了SLi、SL、SLt，以及两种解析型双因子旋转方法——双四次极小值旋转（bi-quartimin）与双几何极小值旋转（bi-geomin）的性能表现。结果表明，SLi在绝大多数实验条件下均可精准恢复双因子结构，整体性能优于其余旋转方法。当双因子结构包含交叉载荷与一般因子（general factor）的纯指示变量时，SLi相较SL与SLt的性能提升最为显著。此外，SLi的性能优于双四次极小值旋转与双几何极小值旋转，这两种方法在受试的各类因子结构间表现并不稳定。当样本量较小（N=200）且特殊因子（specific factors）的载荷水平处于0.30~0.50的较低区间时，所有方法均无法实现双因子结构的良好恢复。因此建议研究中采用至少包含500个观测值的更大样本。