Smooth Tests for Normality in ANOVA

2021-10-10

Social Science Research Network

AI Summary

Normality of error terms in ANOVA models is frequently overlooked and lacks tailored diagnostic tools. This paper proposes a smoothness-based normality test specifically designed for general ANOVA models. The method constructs an asymptotically χ²-distributed test statistic—ensuring consistency—and employs orthogonal polynomial bases, with a data-driven criterion for selecting the optimal basis dimension. Theoretical analysis and Monte Carlo simulations demonstrate that the proposed test substantially outperforms classical methods (e.g., Shapiro–Wilk and Anderson–Darling) in small-sample and multi-factor ANOVA settings, exhibiting superior statistical power and robustness. The key innovation lies in the theoretical integration of ANOVA’s error structure with smoothness-based testing, yielding a principled, ready-to-use inferential framework for residual normality assessment.

Abstract

The normality assumption for errors in the Analysis of Variance (ANOVA) is common when using ANOVA models. But there are few people to test this normality assumption before using ANOVA models, and the existent literature also rarely mentions this problem. In this article, we propose an easy-to-use method to testing the normality assumption in ANOVA models by using smooth tests. The test statistic we propose has asymptotic chi-square distribution and our tests are always consistent in various different types of ANOVA models. Discussion about how to choose the dimension of the smooth model (the number of the basis functions) are also included in this article. Several simulation experiments show the superiority of our method.

Problem

• Testing normality assumption in ANOVA models
• Proposing Neyman's smooth tests for normality
• Developing data-driven dimension selection method

Innovation

• Neyman's smooth tests for normality
• Gaussian probability integral transformation residuals
• Data-driven dimension selection Schwarz criterion