Three-Way Anova: What does a significant three way interaction tell you conceptually?

Here's a made up example. Let's say that I have two factors, energy drink (Gatorade vs water) and gender (male and female). The outcome variable is mile time. A significant two way interaction between energy drink and gender would suggest that the effect of energy drink on mile time differs between males and females. I have no problem understanding this conceptually.

However, let's say that I add a third factor, age (40+ vs 0-40). I understand that a significant three way interaction would tell you that the interaction of energy drink and gender differs across the levels of age. However, I'm not sure what the utility of interpreting the three way interaction term is. What is it telling me conceptually/ What's the point?


Less is more. Stay pure. Stay poor.
Well you shouldn't even explore interactions unless you have a theoretical justification. Just searching for interactions for interaction's sake risks finding spurious associations. So given you have a justification for a plausible 3-way interaction it is showing that result. Given you have three binary variables, it is showing you the same thing as the 2-way interaction but stratified by the other variable now - so you have two figures to compare. You should think about the variable you stratify by and if you can manipulate it like the exposure vs sex. This is given you want to posit manipulable differences. If your third variable really is dichotomized age, you shouldn't discretize it, but examine it as a spline in a model.
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Fortran must die
The higher the level of interaction the more complex it becomes and in social science the more doubtful it really exists. I have to disagree with hlsmith, maybe the first time ever :p, that you should not explore interaction without theory, however. First, there are lots of fields like mine where there is not a lot of empirical theory. Second, if interaction does exist (regardless of why it exists) you have to be very careful about interpreting the impact of X on Y (main effects). Because with interaction simple effects become critical rather than normal regression slopes.


TS Contributor
I come from a different background (i.e., industrial statistics) from others on this forum. There are a number of experimental principles that have been developed that apply broadly in industry. I don't know if they hold true for other fields. One of these principles is called The Sparsity of Effects principle. It states that the numbers of relatively important effects in a factorial experiment are small. and that 3-way (and higher) interactions are very rare. There are a few exceptions to the latter point, which are thermodynamic, chemical, biological and nuclear systems, all of which are extremely complex systems. In electro-mechanical systems, not only are 3-way interactions exceedingly rare, but when one is found, they are also very weak and of no practical value.

One major difference that I have observed between industrial statistics and social statistics, is a focus on practical value. A factor might be significant (p = 0.001), but if it doesn't have an effect large enough to be of practical value, it is of no practical interest.


Fortran must die
Miner I think the view in social science is pretty much the same as what you suggest. Interaction beyond 3 way is rarely considered I suspect-_ don't think I have ever seen such an analysis . It becomes, among other things, very difficult to interpret substantively.