ANOVA with interactions


I did an ANOVA (attached) on 3 factors at 3 levels, could you tell me if my conclusion is ok :

The main factors have no significant effect on Y directly, only via their interactions.

The adjusted coefficient of determination indicates that the model accounts for 49% of the data dispersion.

The analysis of the interaction chart shows that:

- for AB interaction:
Y is maximized when A is low and B is low.
Y is also maximized, but slightly less than previously, with A at the high level and B at the high level.

- for the AC interaction:
Y is maximized when A is low and C is low.
Y is also maximized with A at high level and C at high level.

The following symbolic model is kept : Y = I + AB + AC
with I = constant term
the main factors are not preserved



TS Contributor
When testing for interactions, it is generally best to test first for the highest order interaction (A*B*C). If the highest order is significant, you shouldn't test any terms that are nested within that higher term. So if ABC has a significant test, you wouldn't test A*B, A*C, B*C, A, B, or C.

In your case, ABC is not significant, so I would refit the model without ABC (in general, this will help increase power on subsequent tests). If A*B and A*C still are significant, then you wouldn't test any of the main effects of A, B, or C because the definition of interaction means that A, B, and C are all statistically useful for prediction/explanation of the dependent variable. If you don't refit the model without ABC, I would still adjust your conclusions about main effects since it's better not to test them given the significant test on a higher order term that they belong to.

In other words, a significant interaction of A*B means that the relationship between A and Y depends on the value of B (or that the relationship between B and Y depends on the value of A). This necessitates that A and B are important in our model-- removing either main effect implies that the removed term doesn't impact the outcome, which contradicts our conclusion about interaction (it also disrupts model hierarchy).

Your interpretation of adjusted R-squared is fine.

If you refit the model, without A*B*C, you could post the results again if you want. Otherwise, I hope this helped!


TS Contributor
I can't edit my post anymore, so another thing to think about.....

Since your sample size is not too large, I might recommend (for future consideration) thinking about grouping tests. In other words, do a nested F-test for ALL interactions (if you only have few, such as here). If it is significant, then test the highest term for significance, stopping if a test is significant. So if ABC is significant, then test no further. If ABC is not significant on its test, then remove from the model and test remaining interactions. Alternatively, you could immediately test the one 3-way interaction alone. If significant, no more tests, but if not significant, then test the group of 2-way interactions. If significant, you could stop unless your goal is to find which interactions are significant. Again, you should not test lower order terms nested in significant higher order terms. There really are many ways to approach it, but any method should be well planned in advance and the plan should contain the contingencies (like a road map for IF-THEN). The main idea would be to minimize the number of tests you conduct, control the experiment-wise error rate, and investigate the terms you need to look into for your research.
Thanks a lot for your detailed answer!

I put you 2 ANOVA analyzes:
- without ABC
- only with AB and AC

My project is in 2 stages:

1 / Complete plan
Y = I + A + B + C + AB + AC + BC + ABC
which allows me to select the model most fitted to the data

2 / Fractional plan
Y = I + AB + AC
I then select the Taguchi plan L9 (3) 4
A => column1
B => column 2
C => column 3
in the model I keep only AB and AC

What do you think ?



TS Contributor
Instead of a model with ONLY AB and AC, I would have also included the main effects of A, B, and C, because the interactions say all three are "important" (by definition of the interaction, including the cross product).

However, if theory or prior knowledge tells you ABC, AB, AC, or BC is an important or justified interaction, you may want to leave it in the model to be consistent with theory. I would also recommend checking the assumptions on each model before looking at p-values to determine significance. Sorry I forgot to mention that earlier.