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In this article we derive two simple tests for two-way ANOVA under unequal variances requiring some constraints, which cannot be solved by classical regression formulas. We do so by taking the generalized p-value approach and provide explicit formulas to handle the constraints. The first test is an extension of fiducial one-way ANOVA test, which tends to assure the intended size of the test, but tends to be conservative, and the second test reduces that drawback, but may slightly exceed the intended size of the test. The second test is also a generalized test that is numerically equivalent to the parametric bootstrap (PB) test, which contains some unintended glitches in formulas, which we will rectify. Moreover, our approach does not require good estimators of parameters as PB approach does. Hence, one can take our approach dealing with such distributions as Weibull and Gamma that are used in analysis of lifetime data in engineering and analysis of Survival data in studies of public health. By taking similar approach, researchers are also encouraged to derive generalized tests in other ANOVA applications, such as higher-way ANOVA, ANCOVA and MANOVA under heteroscedasticity, especially in Mixed Effects Model applications such as repeated measures (RM) ANOVA, where the PB approach run into difficulties.

Heteroscedastic two-way ANOVA under constraints