As far as I can tell this question has never been asked before. There are several questions that touch on related issues, but as far as I can see none of them have provided a definitive answer to this question. Furthermore, in at least one case the most upvoted and second most upvoted answer have implicitly disagreed with each other.
Textbooks often caution against interpreting main effects in Two-Way ANOVA when the interaction effect is significant. Here's a (mild) example of this, from pp562-563 of Hatcher's "Step-by-step Basic Statistics Using SAS: Student Guide":
When you perform a two-way ANOVA, it is possible that you will find
that (a) the interaction term is statistically significant, and (b)
one or both of the main effects are also statistically significant.
When you prepare a report summarizing the results, you will certainly
discuss nature of your significant interaction. But is it also
acceptable to discuss and interpret the main effects that were
significant?There is some disagreement between statisticians in answering this
question. Some statisticians argue that, if the interaction is
significant, you should not interpret the main effects at all, even if
they are significant. Others take a less extreme approach. They say
that it is acceptable to interpret significant main effects, as long
as long as the primary interpretation of the results focuses on the
interaction (if it is significant).
Do I assume correctly that the hard-line stance ("you should not interpret the main effects at all") is incorrect, but that the interpretation must nonetheless change if the interaction effect is significant? If so, how does the interpretation change?
Best Answer
There is less to this issue than it seems. The real answer isn't that you cannot interpret the main effects at all, but rather that it is very difficult to interpret them correctly. The reason for the warning not to interpret the main effects is because people will inevitably interpret them incorrectly.
If there isn't an interaction term included in the model, the main effects have a straightforward meaning: is there variation amongst the levels of the factor in question? If there is an interaction in the model, the main effects don't mean that. In fact their meaning is hard to convey and it depends on how the model was fit and how it was tested. In the abstract, I cannot tell you exactly what they mean in any given model. However, interpreting them as you would if there weren't an interaction would be incorrect. What is important for this issue is not whether or not the interaction is significant, but whether or not the interaction was included in the model in the first place.
If the interaction is sufficiently non-significant for your purposes, and you want to test and interpret the main effects, the simplest thing to do would be to drop the interaction and re-fit / re-test the model. Note that this procedure, if not a-priori, comes with all the usual caveats about fishing and threats to the validity of the hypothesis tests.