Discussion: Why do we teach ANOVA for factorials?

In experimentation, the power of manipulating two (or more) factors simultaneously has been recognized for at least a century and the analysis of such experiments is considered among the most basic analyses in biostatistics. I’m sure this topic has been studied in every possible way in the literature and in practice. However, as I’ve refined how I teach this topic to graduate students at UC Davis and discuss such experiments with collaborators I’ve found the way most textbooks describe two-factor experiments to be non-intuitive and focused on mathematically precise answers to the wrong questions. What am I missing? Do you start by estimating main effects and interactions? In what context is a “main effect” actually interesting? Instead, I favor talking about two-factor experiments as estimating a collection of “simple effects” or “subgroup effects” and then comparing them to look for consistency or heterogeneity. This approach is no more difficult, rarely much less powerful, produces more descriptive results, and I think provides a more unified perspective through which to view both simple and complex experiments. I would like feedback on this “treatment effect variability”-first perspective, both in the context of factorials and blocked designs.