Selecting out main effects when locational-scale-invariance isn't an issue(?)

Question

I'm starting a new thread to ask a specific question that I'm left with after reading this old, good thread:

Do all interactions terms need their individual terms in regression model?

The gist of the problem is this: one starts with a complex model motivated by theory. One would like to simplify it -- practically to reduce multicolinearity, and "inferentially" to be able to conduct better inference on the model terms.

The standard advice -- well-articulated in the linked thread -- is that it is generally a bad idea to remove main effects of interaction effects, even when AIC improves. This is because the resultant model is no longer invariant to locational shifts in the variables. I emphasize locational.

But now say you're fitting a model where all the variables on the right hand side have real meanings in the physical world. Lets take, for example, the pH of a substance, or temperature in Kelvin. These things cannot be locationally shifted without changing their meaning. They can be multiplicatively scaled into different systems (i.e.: inches to cm) -- but multiplicative scaling should not affect inference in the ways described in the linked thread.

Furthermore, any sort of additive scaling would fundamentally change their meaning -- we'd no longer be talking about the absolute effect of a real thing, but rather the effect relative to some other thing.

In my context, I've got a large number of interactions, and I'm agnostic about whether many of the variables should lead to a level effect, or simply moderate the slopes of the responses driven by other variables. For example, I don't know if temperature has its own effect on the phenomenon of interest, or whether it simply affects response to change in pH. So I don't think that I'm making any serious errors in creating an interpretable, plausible model.

Would appreciate if anyone could confirm my logic here, or point out any flaws that might be lurking.

Answer 1

I think there's actually another, simpler reason why you should almost always include the main effects when you include an interaction. It has to do with what the model says. You mention pH and temp (K). Fine. So, your model without main effects is:

$y = b_0 + b_1pHT + e$

Now, let's take some values. Temp = 10 and 20. p_h = 2 and 4 (just for illustration). Then:

$Y(10,2) = b_0 + 20b_1$

$Y(10,4) = b_0 + 40b_1$

$Y(20,2) = b_0 + 40b_1$

$Y(40,2) = b_0 + 80b_1$

That is, you are forcing a temp of 10 and ph of 4 to give the same result as a temp of 20 and a pH of 2; along with other relationships.

Is this reasonable? You say you don't know if temperature has its own effect. That would seem to me to indicate that it is not reasonable to exclude the main effect.