# Why do we need to model lower-order effects in models with interactions?

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.

Thank you!

• I believe looking at the mathematics alone it isn't necessarily a problem. From a formal point of view there can be a true model that does not have all the lower order interactions connected to an existing higher order one, and such a model can be fitted without problems if it is correctly assumed. Apr 7, 2022 at 16:01
• The problem is rather that in reality it is hard to imagine that you have a higher order interaction in a situation in which the involved lower order interactions are truly zero (also this is kind of difficult to interpret), and if they are not, of course problems will ensue if you don't have a term in your fitted model that is truly there. (Keep in mind that non-significance doesn't mean an effect is truly zero, so if you omit insignificant interactions, you shouldn't feel confident that they are really not there.) Apr 7, 2022 at 16:03
• At stats.stackexchange.com/a/408855/919 I discuss some of the subtleties. In a circumstance where your explanatory variables are arbitrarily selected linear combinations of original variables (such as when you are using principal components), that post explains why--mathematically--you really ought to include all the lower-order terms. But in many (if not most) cases, theory and experience suggest which interactions might be real or of interest, not mathematics.
– whuber
Jun 21, 2022 at 21:13

But usually there is some linear dependency and then the model would have to crank the linear dependencies into the quadratic effect. This is similar to fitting a linear model without an offset. E.g., consider linear data like $$y = 2\cdot x + 100$$. Fitting this without an offset in you model would result in very bad results.