Effects of Compounded Nonnormality of Residuals in Hierarchical Linear Modeling

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.