Department of Biostatistics, Van Yüzüncü Yıl University, Van, Turkey.
Department of Statistics, Ankara University, Ankara, Turkey.
Comput Math Methods Med. 2019 Jul 18;2019:2173638. doi: 10.1155/2019/2173638. eCollection 2019.
In this study, we investigate how Wilks' lambda, Pillai's trace, Hotelling's trace, and Roy's largest root test statistics can be affected when the normal and homogeneous variance assumptions of the MANOVA method are violated. In other words, in these cases, the robustness of the tests is examined. For this purpose, a simulation study is conducted in different scenarios. In different variable numbers and different sample sizes, considering the group variances are homogeneous ( = = ⋯ = ) and heterogeneous (increasing) ( < < ⋯< ), random numbers are generated from Gamma(4-4-4; 0.5), Gamma(4-9-36; 0.5), Student's (2), and Normal(0; 1) distributions. Furthermore, the number of observations in the groups being balanced and unbalanced is also taken into account. After 10000 repetitions, type-I error values are calculated for each test for = 0.05. In the Gamma distribution, Pillai's trace test statistic gives more robust results in the case of homogeneous and heterogeneous variances for 2 variables, and in the case of 3 variables, Roy's largest root test statistic gives more robust results in balanced samples and Pillai's trace test statistic in unbalanced samples. In Student's distribution, Pillai's trace test statistic gives more robust results in the case of homogeneous variance and Wilks' lambda test statistic in the case of heterogeneous variance. In the normal distribution, in the case of homogeneous variance for 2 variables, Roy's largest root test statistic gives relatively more robust results and Wilks' lambda test statistic for 3 variables. Also in the case of heterogeneous variance for 2 and 3 variables, Roy's largest root test statistic gives robust results in the normal distribution. The test statistics used with MANOVA are affected by the violation of homogeneity of covariance matrices and normality assumptions particularly from unbalanced number of observations.
在这项研究中，我们研究了当多变量方差分析（MANOVA）方法的正态和同方差假设被违反时，Wilks 的 lambda、Pillai 的迹、Hotelling 的迹和 Roy 的最大特征根检验统计量会如何受到影响。换句话说，在这些情况下，检验的稳健性将被检验。为此，在不同的场景中进行了模拟研究。在不同的变量数量和不同的样本大小下，考虑到组方差是同质的（ = = ⋯ = ）和异质的（递增）（ < < ⋯< ），从 Gamma(4-4-4; 0.5)、Gamma(4-9-36; 0.5)、Student's (2) 和正态(0; 1)分布中生成随机数。此外，还考虑了组中观测值的平衡和不平衡数量。在 10000 次重复后，为每个检验计算了 = 0.05 的Ⅰ型错误值。在 Gamma 分布中，在 2 个变量的同质和异质方差情况下，Pillai 的迹检验统计量给出了更稳健的结果，在 3 个变量的情况下，Roy 的最大特征根检验统计量在平衡样本中给出了更稳健的结果，而在不平衡样本中则给出了 Pillai 的迹检验统计量。在 Student's 分布中，在同质方差的情况下，Pillai 的迹检验统计量给出了更稳健的结果，而在异质方差的情况下，Wilks 的 lambda 检验统计量给出了更稳健的结果。在正态分布中，在 2 个变量的同质方差情况下，Roy 的最大特征根检验统计量给出了相对更稳健的结果，而在 3 个变量的情况下，Wilks 的 lambda 检验统计量给出了更稳健的结果。此外，在 2 个和 3 个变量的异质方差情况下，Roy 的最大特征根检验统计量在正态分布中给出了稳健的结果。用于 MANOVA 的检验统计量受到协方差矩阵同质性和正态性假设违反的影响，特别是受到不平衡观测数量的影响。
