Two Simple Tests for Heteroscedastic Two-Way ANOVA

Heteroscedastic two-way ANOVA models are frequently encountered in experimental sciences, e.g., biology, psychology and physics among others. In the literature, classical F-tests are often blindly employed although they are often biased even for moderate heteroscedasticity. To overcome this problem, several approximate tests, e.g., generalized P-values, bootstrap, and permutation-based tests among others, have been proposed in the literature. These tests, however, are computationally intensive or do not work well in terms of size controlling and power. In this paper, we study tests of linear hypotheses in heteroscedastic two-way ANOVA via proposing a modified Bartlett (MB) test and a parametric bootstrap (PB) test. The MB test is easy to implement via using the usual χ-distribution and it is shown to be invariant under affine transformations, different choices of the contrast matrix used to define the same hypothesis and different labelling schemes of the cell means. The PB test, on the other hand, is easy to implement but it is time-consuming. Simulations and real data applications show that the MB test and the PB test are comparable and they both outperform the classical F-test in terms of size controlling and power for various sample sizes and parameter configurations.