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I have two continuous variables a and b and I would like to test whether they non-linearly interact to affect a dependent variable c. Without loss of generality, let's assume that a and b are positively correlated with c. Then there are 3 possibilities:

  1. a and b have a positive interaction (e.g. c = a + b + 2ab - 1, c = 2a + 3b + 2 a / b + 3)

  2. a and b have a negative interaction (e.g. c = a + b - 2ab - 1, c = 2a + 3b - 2 a / b + 3)

  3. a and b do not have a consistent interaction or have no interaction (e.g. c = a + 2b + Normal(0, 1), c = a + b + ab(a - b), c = a + b + d where d is independent of a and b, c = 2a + b + 5).

I would like both a two-tailed test for interaction that gives a low p value in cases 1 and 2 but not 3, and one-tailed tests for positive interaction (low p value for case 1 only) and negative interaction (low p value for case 2 only). How should this test be formulated?

Note that this is more complicated than testing whether two continuous variables are independent or non-linearly related, although ideas from those posts might be helpful in solving this problem.

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