# How to use the likelihood ratio test (LRT) to test whether a three-way interaction is significant?

Our hypothesis is that there is a 3-way interaction between A, B, and C.

I have defined a model as follows:

Y=A+B+C+AB+AC+BC+ABC+error

I aim to use the likelihood ratio test (LRT) to determine if the three-way interaction effect is statistically significant. To do this, I constructed a nested model without the three-way interaction term:

Y=A+B+C+AB+AC+BC+error

Then, I plan to employ the LRT to test the significance of the 3-way interaction.

However, one of my co-authors disagrees with my approach. He suggests that the nested model should be simplified to:

Y=A+B+C+error

From this, we would use the LRT to test the significance of all interaction effects, including the three-way interaction.

I'm currently uncertain about which approach is appropriate. Could someone provide clarity on this matter?

Your approach tests the coefficient on the three-way interaction term with a null hypothesis of $$\beta_{ABC}=0$$ and an alternative hypothesis that the null is false.
Your collaborator's approach tests the coefficients on all of the interaction terms with a null hypothesis of $$\beta_{AB} = \beta_{BC} = \beta_{AC} = \beta_{ABC}=0$$ and an alternative hypothesis that the null is false (at least one of those $$\beta$$ values is nonzero).
Both tests are reasonable, depending on the research question, so that is what you need to sort out with your collaborator. If all you want to know is if the three-way interaction is nonzero, then you would use your test. If you wanted to know that and tested as your collaborator wants, you would run the risk of rejecting because one of the other $$\beta$$ values is nonzero, even if the coefficient on the three-way interaction term is zero.