Department of Methodology of the Behavioral Sciences, University of Barcelona, Barcelona, Spain.
Behav Res Methods. 2013 Sep;45(3):873-9. doi: 10.3758/s13428-012-0306-x.
This study analyzes the robustness of the linear mixed model (LMM) with the Kenward-Roger (KR) procedure to violations of normality and sphericity when used in split-plot designs with small sample sizes. Specifically, it explores the independent effect of skewness and kurtosis on KR robustness for the values of skewness and kurtosis coefficients that are most frequently found in psychological and educational research data. To this end, a Monte Carlo simulation study was designed, considering a split-plot design with three levels of the between-subjects grouping factor and four levels of the within-subjects factor. Robustness is assessed in terms of the probability of type I error. The results showed that (1) the robustness of the KR procedure does not differ as a function of the violation or satisfaction of the sphericity assumption when small samples are used; (2) the LMM with KR can be a good option for analyzing total sample sizes of 45 or larger when their distributions are normal, slightly or moderately skewed, and with different degrees of kurtosis violation; (3) the effect of skewness on the robustness of the LMM with KR is greater than the corresponding effect of kurtosis for common values; and (4) when data are not normal and the total sample size is 30, the procedure is not robust. Alternative analyses should be performed when the total sample size is 30.
本研究分析了在具有小样本量的分割区组设计中，使用肯沃德-罗杰（KR）程序的线性混合模型（LMM）对正态性和球形度违反的稳健性。具体来说，它探讨了偏度和峰度对 KR 稳健性的独立影响，考虑了在心理和教育研究数据中最常发现的偏度和峰度系数的值。为此，设计了一项蒙特卡罗模拟研究，考虑了具有三个受试者分组因素水平和四个受试者内因素水平的分割区组设计。稳健性是根据第一类错误的概率来评估的。结果表明：（1）当使用小样本时，KR 程序的稳健性不会因违反或满足球形度假设而有所不同；（2）对于正态分布、轻微或中度偏态以及具有不同程度的峰度违反的分布，KR 的 LMM 可以成为分析总样本量为 45 或更大的样本的一个不错的选择；（3）偏度对 KR 的 LMM 稳健性的影响大于常见值的峰度对应的影响；（4）当数据不是正态分布且总样本量为 30 时，该程序不稳健。当总样本量为 30 时，应进行替代分析。
