Is it appropriate to examine an interaction effect that is almost statistically significant? I am conducting a 2x2 within-subjects design. When I plot my results it appears that there will be an interaction between my variables but unfortunately none is emerging p=0.08. I find it a shame that I can't explore this further with simple main effects. Does anyone have any suggestions?
 A: Worship not p = 0.05. Explore away.
Additionally, in some contexts, relying on p = 0.05 for an interaction threshold is actually a bit flawed, as interaction tests are typically fairly low powered, and you can and should be using a somewhat higher threshold to accept statistical evidence of interaction. Sander Greenland or Miguel Hernan undoubtedly have a paper discussing the problem.
A: "Can't"? Who says you can't? There's nothing magic about p = .05. You can certainly explore the interaction.
The question is how you deal with complaints from people who say you can't do this. In addition to works by Greenland or Hernan (see @EpiGrad's response) you can look for papers by Jacob Cohen or Paul Meehl or the book "The Cult of Statistical Significance" by Ziliak or the book "Statistics as Principled Argument" by Abelson
A: You might consider switching to mixed effects modelling, which in some cases provides superior power over ANOVA.
A: I agree with the others that you certainly can explore this interaction, but, if it's not significant, you might not have much power to aid your analysis.
A: To say something similar to the other answers in slightly different words.
I would do the following:
Report that the (hopefully expected) interaction is almost or marginal significant or that there is a trend towards significance (these expressions are all common, at least in psychology). Then, state that therefore I inspect this interaction further with follow up simple main effects analyses or contrasts.
It is absolutely no problem to do so, if your main hypotheses are within in this interaction. As said before, omnibus tests of interaction do not have the highest power.
See also here.
