Yuan Ke-Hai, Jiang Ge, Cheng Ying
Department of Psychology, University of Notre Dame, Indiana, USA.
Br J Math Stat Psychol. 2017 Nov;70(3):525-564. doi: 10.1111/bmsp.12098. Epub 2017 May 26.
Data in psychology are often collected using Likert-type scales, and it has been shown that factor analysis of Likert-type data is better performed on the polychoric correlation matrix than on the product-moment covariance matrix, especially when the distributions of the observed variables are skewed. In theory, factor analysis of the polychoric correlation matrix is best conducted using generalized least squares with an asymptotically correct weight matrix (AGLS). However, simulation studies showed that both least squares (LS) and diagonally weighted least squares (DWLS) perform better than AGLS, and thus LS or DWLS is routinely used in practice. In either LS or DWLS, the associations among the polychoric correlation coefficients are completely ignored. To mend such a gap between statistical theory and empirical work, this paper proposes new methods, called ridge GLS, for factor analysis of ordinal data. Monte Carlo results show that, for a wide range of sample sizes, ridge GLS methods yield uniformly more accurate parameter estimates than existing methods (LS, DWLS, AGLS). A real-data example indicates that estimates by ridge GLS are 9-20% more efficient than those by existing methods. Rescaled and adjusted test statistics as well as sandwich-type standard errors following the ridge GLS methods also perform reasonably well.
心理学数据通常使用李克特式量表收集，并且研究表明，对李克特式数据进行因子分析时，在多序相关矩阵上进行比在积差协方差矩阵上进行效果更好，尤其是当观测变量的分布呈偏态时。理论上，多序相关矩阵的因子分析最好使用具有渐近正确权重矩阵的广义最小二乘法（AGLS）。然而，模拟研究表明，普通最小二乘法（LS）和对角加权最小二乘法（DWLS）的表现都优于AGLS，因此在实践中通常使用LS或DWLS。在LS或DWLS中，多序相关系数之间的关联被完全忽略。为了弥合统计理论与实证工作之间的这种差距，本文提出了一种新方法，称为岭广义最小二乘法（ridge GLS），用于有序数据的因子分析。蒙特卡罗结果表明，对于广泛的样本量，岭广义最小二乘法比现有方法（LS、DWLS、AGLS）能产生更一致准确的参数估计。一个实际数据示例表明，岭广义最小二乘法的估计比现有方法的估计效率高9% - 20%。遵循岭广义最小二乘法重新缩放和调整后的检验统计量以及三明治型标准误差也表现良好。
