Abstract

To handle the nonnormal data issue, Browne proposed an unbiased distribution free (DF) estimator (${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$) and an asymptotically distribution free estimator (${\widehat{\mathbf{\Gamma }}}_{\mathbf{ADF}}$) of the covariance matrix of sample variances/covariances $\mathbf{\Gamma }$ to calculate robust test statistics and robust standard errors. However, ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ is ignored in methodological and substantive research, and has not been extended to models with mean structures. To improve robust standard errors and the model fit statistic for nonnormal data with mean structures (e.g., growth curve models), we propose an unbiased distribution free estimator with mean structures considered. In growth curve models, we apply ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ to four robust statistics that have relatively simple forms and denote them as $T_{SB}^U,T_{\mathit{MVA}}^U,T_{\mathit{MVA}2}^U,$ and $T_{\mathit{COR}1}^U.$ We compare their performance with 7 robust test statistics that employ ${\widehat{\mathbf{\Gamma }}}_{\mathbf{ADF}}.$ We find that with the same model fit statistic, ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ generally leads to smaller Anderson-Darling distances from the theoretical distribution than ${\widehat{\mathbf{\Gamma }}}_{\mathbf{ADF}},$ except $T_{\mathit{MVA}2}^U$ in some skewed cases. Additionally, the p-values from $T_{\mathit{MVA}2}^U$ are distributed closest to the theoretical distribution $\mathit{Uniform}(0,1)$ among the 12 examined statistics. In terms of Type I error rates, $T_{\mathit{MVA}2}^U$ and $T_{\mathit{MVA}2}$ are the most stable statistics. Additionally, ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ provides smaller relative biases of the robust SE estimates than ${\widehat{\mathbf{\Gamma }}}_{\mathbf{ADF}}.$ Hence, we suggest using ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}}$ in both model fit statistics and robust SE calculation. Among the model fit statistics using ${\widehat{\mathbf{\Gamma }}}_{\mathbf{DF}}^{\mathbf{U}},$ we suggest $T_{\mathit{MVA}2}^U.$

摘要

为了处理非正态数据问题，Browne 提出了一种无偏分布自由 ( DF ) 估计器 (${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$) 和一个渐近分布自由估计器 (${\widehat{\mathbf{\Gamma }}}_{\mathbf{自动进纸器}}$) 样本方差/协方差的协方差矩阵$\mathbf{\Gamma }$计算稳健的检验统计量和稳健的标准误差。然而，${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$在方法论和实质性研究中被忽略，并且尚未扩展到具有平均结构的模型。为了改进具有均值结构（例如，增长曲线模型）的非正态数据的稳健标准误差和模型拟合统计量，我们提出了一个考虑了均值结构的无偏分布自由估计器。在增长曲线模型中，我们应用${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$到四个具有相对简单形式的稳健统计数据并将它们表示为${吨}_{\mathrm{小号}乙}^{ü},{吨}_{\mathit{MVA}}^{ü},{吨}_{\mathit{MVA}\mathrm{2个}}^{ü},$和${吨}_{\mathit{心率}\mathrm{1个}}^{ü}.$我们将它们的性能与 7 个稳健的测试统计数据进行比较，这些统计数据采用${\widehat{\mathbf{\Gamma }}}_{\mathbf{自动进纸器}}.$我们发现使用相同的模型拟合统计，${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$通常导致与理论分布的 Anderson-Darling 距离小于${\widehat{\mathbf{\Gamma }}}_{\mathbf{自动进纸器}},$除了${吨}_{\mathit{MVA}\mathrm{2个}}^{ü}$在一些偏斜的情况下。此外，p值来自${吨}_{\mathit{MVA}\mathrm{2个}}^{ü}$分布最接近理论分布$\mathit{制服}(0,\mathrm{1个})$在 12 个检查的统计数据中。在 I 类错误率方面，${吨}_{\mathit{MVA}\mathrm{2个}}^{ü}$和${吨}_{\mathit{MVA}\mathrm{2个}}$是最稳定的统计数据。此外，${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$提供稳健的SE估计的相对偏差小于${\widehat{\mathbf{\Gamma }}}_{\mathbf{自动进纸器}}.$因此，我们建议使用${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}}$在模型拟合统计和稳健的SE计算中。在模型拟合统计中使用${\widehat{\mathbf{\Gamma }}}_{\mathbf{测向仪}}^{\mathbf{ü}},$我们建议${吨}_{\mathit{MVA}\mathrm{2个}}^{ü}.$