ANOVA under normally distributed response and heteroscedastic variances is commonly encountered in biological, behavioral, educational and agricultural sciences where the commonly used F-test is not valid. Many alternatives suggested in the literature exhibit unsatisfactory performance with respect to the type-I errors notably under a large number of small size groups. This has a direct bearing on their power performance. Anticipating that a major cause may be the existence of a large number of unknown unequal group variances as nuisance parameters, the present work attempts to provide a uniformly implementable simple solution that addresses this problem through the use of likelihood integration with respect to the nuisance parameters. The second-order accurate asymptotic \(\chi ^2\) distribution of the test is established. Simple ad hoc corrective adjustments suggested for enhancing the small sample distributional performance make the test usable even for small group sizes. Simulation studies demonstrate that the test exhibits uniformly well-concentrated sizes at the desired level and the best power, particularly under very small size groups, highly scattered group variances and/or a large number of groups under one-way and two-way ANOVA where precisely a better option is needed. Being closely competent to other peers in all other cases, it offers an universally implementable and trustworthy option in this scenario. The method is straightway extendable to multi-factor setup and has direct connection to ANOVA under log-normally distributed data. Results are illustrated with real data.

正态分布响应和异方差方差下的 ANOVA 在常用的F检验无效的生物、行为、教育和农业科学中经常遇到。文献中提出的许多替代方案在 I 类错误方面表现不佳，尤其是在大量小规模组的情况下。这直接关系到它们的功率性能。预计一个主要原因可能是存在大量未知的不等组方差作为干扰参数，本工作试图提供一个统一可实现的简单解决方案，通过使用关于干扰参数的似然积分来解决这个问题。二阶精确渐近\(\chi ^2\)测试分布确定。为增强小样本分布性能而建议的简单的特别校正调整使测试即使对于小规模的群体也可用。模拟研究表明，该测试在所需水平和最佳功效下表现出均匀集中的尺寸，特别是在非常小的尺寸组、高度分散的组方差和/或单向和双向 ANOVA 下的大量组下，其中正是需要更好的选择。在所有其他情况下与其他同行密切相关，它在这种情况下提供了一个普遍可实施且值得信赖的选择。该方法可直接扩展到多因素设置，并在对数正态分布数据下直接连接到方差分析。结果用真实数据说明。