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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

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    $\begingroup$ I am only interested in the higher order term, so such long equation is just noisy for me $\endgroup$
    – Funn Me
    Commented Aug 28, 2014 at 17:23
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    $\begingroup$ This is generally a bad idea. Try reading the linked thread for more information. If there is something more you need to know or still don't quite understand after reading through that, come back here and edit your Q to explain what you learned & what you are still confused about; we can re-open this to address the needed issues. $\endgroup$
    – gung - Reinstate Monica
    Commented Aug 28, 2014 at 17:53
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    $\begingroup$ If you just want to know if it is possible to do, despite being a bad idea, you don't really need to ask; just try it. (As mentioned below, yes it can be done.) $\endgroup$
    – gung - Reinstate Monica
    Commented Aug 28, 2014 at 18:22
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    $\begingroup$ Please don't prevent others from further discussion. There is no point of closing the question and you can see it is different. I hope you understand $\endgroup$
    – Funn Me
    Commented Aug 28, 2014 at 18:27
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    $\begingroup$ Since you know it can be done, please give me a source or open my question $\endgroup$
    – Funn Me
    Commented Aug 28, 2014 at 18:39

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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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  • $\begingroup$ thanks a lot for your answer. I am using fixed effects in panel data, but sorry I didn't understand your last sentence. Do you have any book or article that support or used this approach because all the books I have read follow the traditional way (first equation). thanks again @pes $\endgroup$
    – Funn Me
    Commented Aug 28, 2014 at 17:30
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    $\begingroup$ i just mean that XZW is a very particular interaction, why not use Xlog(Z)tan(W) or product of other transformations. but for discrete variables X,Z,W say in (0,1), there are exactly 8 combinations and interaction XZW means exactly one of them (1,1,1), another form, say (1-X)Z(1-W), will point to another combination (0,1,0) $\endgroup$
    – pes
    Commented Aug 28, 2014 at 18:23

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