Uniformly implementable small sample integrated likelihood ratio test for one-way and two-way ANOVA under heteroscedasticity and normality

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.