Solved – Can we skip the lower order terms in interactions?

This question is about three-way interaction and the possibility of applying without second lower terms with keeping the main variables in the equation not like the other questions. In fact the other answers suggest there is possibility of applying . I am not here to find the best solution because I know it and I already included in my question, but to know whether is it possible regardless if it is preferable or not. thank you and please open my question for discussion

The widely known regression equation for assessing the three-way interaction is

$$ Y= B_1 X+B_2 Z+B_3 W +B_4XZ+B_5XW+B_6ZW+B_7XZW+B_0 $$

All lower order terms is included in the regression equation for the B7 coefficient to
represent the effect of the three-way interaction on Y.

Is there possible way to skip the lower order terms and include only the higher term? as in:

$$ Y= B_1 X+B_2 Z+B_3 W +B_4XZW+B_0 $$

And how many observations do I need to perform such equation if X & Z are continuous variables and W is dummy variable ?

I will be thankful if anyone can provide me with any suggestions

In short, yes.

A little longer answer: you need to consider how you got to this particular collection of explanatory variables and their combinations.

If you used any kind of model selection/ likelihood ratio test/… then the final p-values are conditional on what you did before. see for example Efron's paper.

If in your specific data the 3-way interaction is meaningful from before you saw/collected the data then your model is representing what you want and can be used .

Additional thought:

If your data is discrete 3-way interaction is not just one thing XZW, it can be also (1-X)Z(1-W) and the results will be very different.

If your data is continuous it is even more messy.

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