# Linear regression for main effect not interaction effect

I have a linear equation:

lm(Connectivity ~ (Complex-Attention + Memory)*MDD, data = D)


From this association I obtain a significant main effect but no interaction effect. Is it okay for me to report on the main effect? This would be interpreted as both (healthy controls and depressed) patients connectivity strength is significantly associated with complex attention, when memory is held constant.

Is that correct?

• Welcome. Did you intend to wrap the additive terms in parentheses? It will multiply MDD with Memory and Complex-Attention? Is that what you wanted to achieve? Also, if you want to interpret the coefficient on Complex-Attention, then you're holding memory at 0. The interpretation is quite different in the presence of the interaction terms. – Thomas Bilach Jan 25 at 22:49
• @ThomasBilach Thank you. I wanted to see if there was an interaction effect between MDD and Memory and also with MDD and Complex attention. Do you think this isn't the right approach? – J.Doe Jan 26 at 1:29
• It is the correct approach. – Thomas Bilach Jan 26 at 5:53

I often see the code written more explicitly with two multiplicative terms—only. To be clear, the following three models produce identical output:

# 1. Hard on the eyes, but it works
lm(Connectivity ~ (Complex-Attention + Memory) * MDD, data = D)

# 2. Explicit, but requires too many keystrokes
lm(Connectivity ~ Complex-Attention + Memory + Complex-Attention * MDD + Memory * MDD, data = D)

# 3. Less explicit, but allows software to do most of the work for you
lm(Connectivity ~ Complex-Attention * MDD + Memory * MDD, data = D)


I am partial to the third equation. R will return the constituent terms for you—for free.

What is even more important is your interpretation. The other variable (e.g., Memory) is held fixed—at 0. If you dropped the multiplicative term(s) entirely (i.e., no interaction), then the coefficient associated with your measure of "attention" is interpreted appropriately. However, in the presence of the interaction term(s), your coefficient of interest is now the effect of Complex-Attention on Connectivity conditional on Memory $$= 0$$. If your "memory" measure never equals 0, then this interpretation might be a bit quirky. Consider "centering" Memory at its mean to improve the interpretive value of the model.

A meaningful main effect may exist in the presence of an interaction—even when the coefficient on the interaction term is itself not significant.

Yes, it is totally fine to have and report a significant main effect. The lack of significant interaction effect just means there is no effect modification going on here; it doesn't take away from the fact that you have a significant association between your outcome and complex attention.

• Thank you @Izzy do you think that I can interpret the main effect between complex attention and outcome as an effect of memory? or would that be wrong to say? – J.Doe Jan 26 at 1:31
• @J.Doe This is incorrect. Which main effect is of interest? – Thomas Bilach Jan 26 at 6:06