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I would like to regress basal metabolic rate (BMR) on height, weight, age and gender. How do I do this taking into consideration the interaction between the variable? Wouldn't I have to consider interactions upto fouth order?

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Theoretically yes, you could go up to fourth-order interaction, but practically we rarely go beyond second- (or perhaps third-) order interactions, I think. They are very hard to assign any meaningful interpretation (i.e. medical meaning).

To test whether interactions are needed, you can simply use a Wald-type test or LM-test, as models with/without interaction represent nested models.

(For large samples they might be always significant, in such scenarios it might also be useful to have a look at how $\chi^2$ changes when the interaction is added.)

And of course, it is always important to consider -- in addition to the statistical points -- how meaningful these interactions are from the clinical point of view.

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Fourth order interactions are going to be very hard to interpret. If you have theoretical reasons to suspect them, then, in your case, I'd stratify by gender and then look at three way interactions in each stratum. Those will be hard enough to interpret.

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