I recently saw a paper with a four-way interaction. That already is difficult to interpret (maybe if you have 1 or more categorical variables but definitely near impossible to interpret if all continuous) and you’re almost certainly underpowered for it.
But beyond the four-way, the researcher modeled only one main effect, two 2-way interactions, and two 3-way interactions. So the authors dropped a bunch of lower-order effects in their (already difficult to interpret) four-way interaction.
I know this is a problem. In terms of consequences, it will inflate the likelihood of your interaction effect appearing as a false positive when you omit the lower-order effects.
But I still struggle a bit as to why mathematically this is a problem? Why is it a bad idea or mathematically a problem to not simultaneously model the lower-order effects of an interaction? Can someone break down or explain to me why it is essential to model your lower-order effects in an interaction? Why is it necessary? Any other details or input would be appreciated.