Cumulant-Based Robustness of ANOVA-like Tests in Multiple Additive Models

Multiple additive models provide a robust framework for analysing families of additive models associated with the treatments of a base design. In this context, ANOVA-like procedures can be used to assess the effects of factors through corresponding components of principal estimable functions. However, the validity of the resulting F-tests is typically justified under approximate homoscedasticity and distributional assumptions that may be violated in practice. This paper studies the robustness of ANOVA-like inference in multiple additive models under non-normal error distributions. Using cumulants of order two, three and four, we characterise departures from normality in terms of variance, skewness and excess kurtosis. A simulation study evaluates the empirical type I error and power of the proposed ANOVA-like tests under normal, heavy-tailed and skewed error distributions, including Gumbel-type components. The results identify conditions under which the F approximation remains reliable and cases where non-normality leads to distorted inference. The proposed cumulant-based perspective provides both a diagnostic and methodological framework for robust inference in multiple additive models.