Additive and multiplicative effects in a fixed 2 × 2 design using ANOVA can be difficult to differentiate: Demonstration and mathematical reasons

Several researchers have expressed doubts as to whether ANOVA is fit to detect interaction in a fixed 2 × 2 design. To study this, values of a dependent variable are created with linear construction formulae, with and without a product term. Our simulation demonstrates that when the dependent variable results from a product term, ANOVA often fails to detect a significant interaction effect, while at the same time two strong additive main effects are indicated. As a consequence, multiplicative models are rejected in favor of an additive model. Mathematically it is shown that the ANOVA results can indicate strong main effects as the result of a solely multiplicative contribution of both factors. It is recommended to abandon the fixed ANOVA model for the study of interaction effects and use a regression model with assessed level values of each independent variable instead.