# Correlation between two interaction terms

I'm working on a project involving regression models with multiple interaction terms. An example regression would be:

$$Y$$ ~ $$A$$ + $$B$$ + $$C$$ + $$D$$ + $$AB$$ + $$CD$$

where $$AB$$ = $$A*B$$ and $$CD$$ = $$C*D$$

All variables are continuous, normally distributed, and symmetric, with a mean of 0 and sd of 1.

I know the correlations between every term and $$Y$$, the correlations between $$A$$,$$B$$,$$C$$, and $$D$$, as well as the correlations between the two interaction terms ($$AB$$ and $$CD$$) and $$A$$,$$B$$,$$C$$, and $$D$$.

My intuition is that using this information I can figure out the correlation between $$AB$$ and $$CD$$ without first simulating them, but I'm not sure how to do it. Does anyone have any ideas?

I don't think you would need the conditions related to the response $$Y$$, because what you are interested is completely determined by the joint distribution of the $$4$$-dimensional random vector $$(A, B, C, D)$$. In fact, \begin{align} \operatorname{Corr}(AB, CD) = \frac{\operatorname{Cov}(AB, CD)}{\sqrt{\operatorname{Var}(AB)\operatorname{Var}(CD)}}, \end{align} where \begin{align} & \operatorname{Var}(AB) = E[A^2B^2] - (E[AB])^2, \\ & \operatorname{Var}(CD) = E[C^2D^2] - (E[CD])^2, \\ & \operatorname{Cov}(AB, CD) = E[ABCD] - E[AB]E[CD]. \end{align}

With the conditions you stated, $$E[AB], E[CD]$$ can be determined. However, higher-order moments $$E[A^2B^2], E[C^2D^2], E[ABCD]$$ cannot be determined.

For example, consider $$(A_1, B_1, C_1, D_1) \sim N_4(0, I_{(4)})$$ and $$(A_2, B_2) \sim N_2(0, I_{(2)})$$, $$C_2 = (2\xi - 1)A_2, D_2 = (2\xi - 1)B_2$$, where $$\xi \sim \text{Bin}(1, 1/2)$$ is independent of $$(A_2, B_2)$$. It is then easy to verify that

1. $$A_i \sim N(0, 1), B_i \sim N(0, 1), C_i \sim N(0, 1), D_i \sim (0, 1), i = 1, 2$$.
2. All pairwise correlations of $$(A_i, B_i, C_i, D_i)$$ are $$0$$, $$i = 1, 2$$.
3. All pairwise correlations of $$(A_iB_i, A_i, B_i, C_i, D_i)$$ and $$(C_iD_i, A_i, B_i, C_i, D_i)$$ are $$0$$, $$i = 1, 2$$.

Therefore both $$(A_1, B_1, C_1, D_1)$$ and $$(A_2, B_2, C_2, D_2)$$ give exactly the same information in your question. However, \begin{align} & E[A_1B_1C_1D_1] = E[A_1]E[B_1]E[C_1]E[D_1] = 0, \\ & E[A_2B_2C_2D_2] = E[(2\xi - 1)^2A_2^2B_2^2] = 1. \end{align}

This shows $$\operatorname{Corr}(A_1B_1, C_1D_1) = 0$$, while $$\operatorname{Corr}(A_2B_2, C_2D_2) = 1$$, i.e., the correlation between interaction terms cannot be uniquely determined given the conditions presented.

• Thanks for your answer! A couple follow-up questions. First, I'm not following your subscript notation, sorry! What is the difference between A1 and A2 here? Second, I have tried to do some reading on this - I found this paper, which states on pg. 315 that given my assumptions (I meant to include multivariate normality previously, sorry!) that E[ABCD], Var(AB), and Var(CD) are known (equations 7 & 11). Does the additional assumption help to resolve things? Jan 4, 2023 at 1:43
• If you would like to assume multivariate normality, then $E[ABCD]$ (hence the correlation you are interested in) is of course uniquely determined. What I was trying to demonstrate is that, without the assumption of jointly normal, you can construct two random vectors $(A_1, B_1, C_1, D_1)$ and $(A_2, B_2, C_2, D_2)$ such that each of them satisfies the conditions you listed, while $\operatorname{Corr}(A_1B_1, C_1D_1) \neq \operatorname{Corr}(A_2B_2, C_2D_2)$. Jan 4, 2023 at 1:52