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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?

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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.

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  • $\begingroup$ 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? $\endgroup$
    – David B
    Jan 4, 2023 at 1:43
  • $\begingroup$ 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 $\operatorname{Corr}(A_1B_1, C_1D_1) \neq \operatorname{Corr}(A_2B_2, C_2D_2)$. $\endgroup$
    – Zhanxiong
    Jan 4, 2023 at 1:52

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