I have two continuous variables
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
b are positively correlated with
c. Then there are 3 possibilities:
bhave a positive interaction (e.g.
c = a + b + 2ab - 1,
c = 2a + 3b + 2 a / b + 3)
bhave a negative interaction (e.g.
c = a + b - 2ab - 1,
c = 2a + 3b - 2 a / b + 3)
bdo 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 + dwhere
dis independent of
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?