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.

虽然分层线性模型经常用于社会科学研究，但在经验数据中可能会违反个体和集群水平的正态分布残差假设。以前的研究分别关注较低或较高水平的非正态性的影响。但是，同时违反所有级别的正态性假设可能会以无法预料的方式对参数估计产生偏差。本文旨在提高人们对当聚类数量从小到大时跨级别的复合非正态残差相关缺点的认识。探讨了在个体和集群水平上违反正态性假设的影响。进行模拟研究以通过操纵数据的正态性来评估模型参数估计的相对偏差和均方根。结果表明，与固定效应相比，非正态残差对随机效应的影响更大，尤其是当簇数和簇大小较小时。此外，对于简单的随机效应结构，当跨级别的复合残差显示适度的非正态性时，建议使用受限最大似然估计来改进参数估计，同时具有少量簇和大簇大小。