Smooth Tests for Normality in ANOVA

The normality assumption for random errors is fundamental in the analysis of variance (ANOVA) models, yet it is seldom subjected to formal testing in practice. In this paper, we develop Neyman’s smooth tests for assessing normality in a broad class of ANOVA models. The proposed test statistics are constructed via the Gaussian probability integral transformation of ANOVA residuals and are shown to follow an asymptotic Chi-square distribution under the null hypothesis, with degrees of freedom determined by the dimension of the smooth model. We further propose a data-driven selection of the model dimension based on a modified Schwarz’s criterion. Monte Carlo simulations demonstrate that the tests maintain the nominal size and achieve high power against a wide range of alternatives. Our framework thus provides a systematic and effective tool for formally validating the normality assumption in ANOVA models.