Reparametrisation of a model when an interaction is significant to facilitate the interpretation

Question

It is admitted that it is complex to interpret main effects when they are involved in an interaction.

Lets take a regular linear model, with two categorical 2 level variables A and B who are interacting together. The model can be written:

lm(Response ~ A + B + A:B)

Lets call I the intercept, a2 the estimate associated to level 2 of variable A, b2 the estimate associated to level 2 of variable B, and c22 the interaction estimate.

For me, there is 2 cases:

1. The interaction estimate is small compared to main effects estimates. In that case, I see no issue in interpreting main effects and then specify the interaction is significant but small and explain what it changes.
2. The interaction estimate is big compared to main effect, and may even change the direction of effects. In that case, it is very difficult to interpret estimates, and it is better to look the effect of B for each level of A, or the effect of A for each level of B, depending on our underlying understanding of the interaction between A and B.

We focus on case 2. Let's say B is a modifier of A effect, and I want to look what is the effect of A in each level of B. To look what the effect of A is for each level of B, one would sum the estimates:

• Effect of A for level 1 of B is a2
• Effect of A for level 2 of B is a2+c22

My issue is that when doing this, we do not know the confidence interval of the effect of A for level 2 of B (i.e. of a2+c22).

Therefore, I wondered whether a good solution to interpret interactions in that case would be to reparametrise the model as:

lm(Response ~ B + A:B)

This returns two estimates for the interaction instead of one (c21=a2 and c22=a2+previous_c22), so that we directly have the estimated effect of A for each level of B, and moreover we have their confidence intervals.

Is it a good solution? If it is, why don't people do that?

Thanks!

Answer 1

Lets say you have a third variable C, which interacts as well with A. The model can be written:

lm(Response ~ A + B + C + A:B + A:C)

When removing the main "A" variable, the model becomes:

lm(Response ~ B + C + A:B + A:C)

In that case, the A effect does not entirely goes into the A:B interaction, but the A:C interaction might as well "absorb" part of this main A effect, so the estimates would change and the interpretation of terms would not be that easy.

If the only issue is to calculate joint confidence intervals, it is possible for linear models, see this other SE topic: Joint Confidence Interval