Interpretation of significant interaction

Question

I look for an intuitive understanding of interaction effect in ANOVA or regression. Let's keep things simple as the following.

Suppose we have a standard 2 x 2 factorial design, where each factor variable has two levels, e.g. Factor A has 0 and 1, so does factor B. Let's denote the groups as:

             B
          0      1
      ---------------
    0 |  G1  |  G2  |
 A    ---------------
    1 |  G3  |  G4  |
      ---------------

If we include the interaction term in a two-way ANOVA or linear regression for some dependent variable Y, and it turns out that the interaction between A and B is significant . Does this mean that:

1. The difference in Y between G1 and G3 and that between G2 and G4 must be different?

2. At the same time, the difference in Y between G1 and G2 and that between G3 and G4 must also be different?

In other words, are the two conditions above necessary and sufficient for the interaction to be significant?

A related conceptual question: if I'm interested in showing that combining A and B (i.e. G4) enhances Y compared to having only either A (i.e. G3) or B (i.e. G2), do I have to show that the interaction between A and B is significant? If not, what are my options here?

Answer 1

The first part of your question is so good it can be answered with "yes", with respect to your first two conditions. Your related question is also yes but it's a bit hard to tell what you mean. I think you're also asserting that when A = 0 that means no A but there are lots of other interpretations of that.

There are other options but all of the correct ones result in a significant interaction and require a test of your 1 or 2.

Something you might want to consider now is the symmetry of your little matrix if main effects are removed.