Factorial ANOMC Tests: A Comparative Analysis of Type I Error and Power under Multifactor Experimental Designs

Abstract This study introduces a factorial extension of the Analysis of Means with Covariate (ANOMC), building on the classical ANOM framework by incorporating auxiliary covariate information. The proposed factorial ANOMC approach is designed for experiments involving two fixed factors and a continuous covariate, where traditional ANOM or factorial ANOM may lose efficiency. Six variants of the factorial ANOMC test are developed using regression and ratio-type estimators, and their performance is evaluated against the standard factorial ANOM test, which does not utilise covariate information. A comprehensive Monte Carlo simulation study is conducted to assess these tests under diverse conditions, including normal and non-normal error distributions, varying correlation structures, different sample sizes, multiple levels of each factor, and both homogeneous and heterogeneous variances. Performance is examined through empirical Type I error rates and statistical power. The findings show that while factorial ANOM maintains stable Type I error rates under ideal settings, several factorial ANOMC variants (i.e., $${ANOMC-MR1}_{F},$$ $${ANOMC-MR2}_{F},$$ $${ANOMC-MR4}_{F}$$ and $${ANOMC-Reg}_{F}$$ ) achieve improved detection capability, especially when regression estimators are used. Some ratio-based versions also perform well under specific correlation and distribution structures. Power increases noticeably for factorial ANOMC when sample sizes or factor-level combinations grow. Overall, the factorial ANOMC framework provides a more adaptable and informative alternative for multifactor experiments involving covariates.