Man Kaiwen, Schumacker Randall, Morell Monica, Wang Yurou
University of Alabama, Tuscaloosa, AL, USA.
U.S. Food and Drug Administration, Silver Spring, MD, USA.
Educ Psychol Meas. 2022 Apr;82(2):330-355. doi: 10.1177/00131644211010234. Epub 2021 May 10.
While hierarchical linear modeling is often used in social science research, the assumption of normally distributed residuals at the individual and cluster levels can be violated in empirical data. Previous studies have focused on the effects of nonnormality at either lower or higher level(s) separately. However, the violation of the normality assumption simultaneously across all levels could bias parameter estimates in unforeseen ways. This article aims to raise awareness of the drawbacks associated with compounded nonnormality residuals across levels when the number of clusters range from small to large. The effects of the breach of the normality assumption at both individual and cluster levels were explored. A simulation study was conducted to evaluate the relative bias and the root mean square of the model parameter estimates by manipulating the normality of the data. The results indicate that nonnormal residuals have a larger impact on the random effects than fixed effects, especially when the number of clusters and cluster size are small. In addition, for a simple random-effects structure, the use of restricted maximum likelihood estimation is recommended to improve parameter estimates when compounded residuals across levels show moderate nonnormality, with a combination of small number of clusters and a large cluster size.
虽然分层线性模型常用于社会科学研究，但在实证数据中，个体和聚类层面残差呈正态分布的假设可能会被违背。以往的研究分别聚焦于较低或较高层面非正态性的影响。然而，所有层面同时违背正态性假设可能会以不可预见的方式使参数估计产生偏差。本文旨在提高人们对聚类数量从小到多变化时各层面复合非正态残差相关弊端的认识。探讨了个体和聚类层面违背正态性假设的影响。通过操纵数据的正态性进行了一项模拟研究，以评估模型参数估计的相对偏差和均方根。结果表明，非正态残差对随机效应的影响大于固定效应，尤其是在聚类数量和聚类规模较小时。此外，对于简单随机效应结构，当各层面的复合残差呈现中度非正态性，且聚类数量少而聚类规模大时，建议使用限制最大似然估计来改进参数估计。
