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## Comments

Hi,

To understand this, it's probably best to start with a hypothetical 2-way interaction.

Say that you have two factors,

`A`

and`B`

, and that there is a full`A × B`

cross-over interaction in your data. In that case, there is neither a main effect of`A`

nor`B`

, but there is an`A × B`

interaction. So JASP might tell you that the BF10 for the`A + B`

model is (say) 0.1, reflecting that there are no main effects. But the BF10 for the`A + B + A × B`

model might be (say) 0.5, reflecting that there is an interaction. Specifically, these results would tell you that the data is 5 times more likely under a model with the interaction than under a model without the interaction.So if you're only interested in the interaction, then the relevant comparison is between the model with the interaction and the model without the interaction (but with all main effects). The null model is, in this case, of no concern.

In your case, you're interested in a 3-way interaction, but the logic is the same. The relevant comparison is between the model with the 3-way interaction and the model without the 3-way interaction (but with all 2-way interactions and all main effects). So this:

… is a sensible conclusion.

Does that make sense? In recent versions of JASP, you can use the 'Inclusion BF based on matched models' option, which is based on this logic, and should tell you that the 3-way interaction is reasonably well supported by your data.

You can also take a look at a post that I wrote some time ago about this question:

Cheers!

Sebastiaan

Hi Sebastiaan,

Thanks! I've read the post you suggested and it helps me understand better Baws factor and Bayes factor. I think your answer made perfect sense. But I'm still a bit confused about the meaning of the three-way interaction. Compared to the model which only lacks the three-way interaction, I can conclude that the model with the three-way interaction is 4 times better. But the BF10 of this model is way smaller than the pure null model.

It makes me doubt the real function of this three-way interaction. Because it seems the most powerful statement would be the dependent variable does not depend on anything. In other words, my question is: when should I look into the model and when should I look into the effect? If the evidence for a particular effect is abundant but the evidence for the model with this effect is small compared to the null model, do they seemingly contradict each other?

Thanks again for your clear answer!

Shuang

No they don't

Say that you have a model with three factors and all interactions:

`A + B + C + A×B + A×C + B×C + A×B×C`

. So there are three main effects, three two-way interactions, and one three-way interaction. And now say that the data does not contain any evidence for the main effects nor for the two way interactions, but does contain evidence for the three-way interaction, then there are 6 effects that drive the BF down relative to the null model, and only 1 effect that drives the BF up relative to the null model.That's why you can get a model with a three-way interaction that does poorly compared to the null, even though the three-way interaction is supported by the data. That's all there is to it. Does that make sense?

Incidentally: If you would compare a model with

onlythe three-way interaction (but no main effects nor two-way interactions) to the null model, you would find that the three-way model does best. But JASP doesn't allow you to construct such models: if you have an interaction in a model, then all the main effects and lower-order interactions must be in the model as well. This is the principle of marginality, which for some reason (unclear to me) statisticians consider a good thing.Cheers,

Sebastiaan

Thanks, this is super clear! You really help me out