Difference between controlling for other variables in additive models vs. interaction model

Question

In a regression model with multiple predictors, the word "control" is used to refer to the inclusion of other variables than the one relevant to a specific question (e.g., the effect of alcohol consumption on ability to focus, while controlling for fatigue). Partial regression coefficients are defined as the isolated association between a single X and Y when none of the other predictors are changing. But If they are not changing are they considered to be set at a specific value? (i.e., 0)

It is my understanding that in an interaction model, when looking at the simple slopes, we are looking at the effect of the predictor when the moderator is at 0 (which itself can mean different thing depending on whether we centered the variable or not, and at what value it's centered).

model1 <- lm ( outcome ~ predictor * moderator, data = data)

But my confusion is regarding additive models.

model2 <- lm (outcome ~ predictor + control, data = data)

This is also used to "control" for variables. And conceptually it refers to the idea of "accounting for the part that is explained by the other variable", but what does it mean really? Is it also looking at the effect of the predictor when the control variable is at 0? But that cannot be the case, because if the predictor variable varied based on he level of the control variable it would have been a moderation. Is it just part of the residuals that are now explained by the control variable? But sometimes the control variable affects the predictor (like with mediators). But what confuses me the most is that if the control variable is a mediator, it reduces the effect of the predictor. How can this happen? How can the control variable affect the value of the predictor?

I apologize in advance if this is basic regression knowledge, I find that most of the time textbooks and information online are either very conceptual or too advanced with mathematics I can't understand.

Answer 1

You are correct that the nature of "controlling for" other variables differs depending on whether the control is via an additive term or an interaction.

With only an additive term the association of your main predictor with outcome doesn't depend directly on the value of the control variable, as you note. But the added control variable(s) in linear regression can remove the omitted-variable bias that otherwise affects the estimate for the predictor variable if a control variable is associated both with the predictor and with outcome. You can think about that as "part of the residuals ... are now explained by the control variable" if you wish. I prefer the compelling visual example in this answer showing how including a control variable ("group" in that case) can flip the sign for a continuous predictor. In models like logistic or Cox survival regressions, you can have bias even if an omitted outcome-associated control variable isn't correlated with the predictor of interest, so including such variables is even more important.

... sometimes the control variable affects the predictor (like with mediators).

With a mediator, the control (mediator) variable doesn't affect the predictor. Causality is in the opposite direction. With a mediator, at least part of the apparent association of a predictor with outcome is because it affects the value of the mediator, which in turn is associated with outcome.