Some background (due to Sewall Wright’s method of path analysis) Given a generating model:
\[\begin{equation} y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 \end{equation}\] where \(x_3 = x_1 x_2\); that is, it is an interaction variable.
The total effect of \(x_1\) on \(y\) is \(\beta_1 + \frac{\mathrm{COV}(x_1, x_2)}{\mathrm{VAR}(x_1)} \beta_2 + \frac{\mathrm{COV}(x_1, x_3)}{\mathrm{VAR}(x_1)} \beta_3\).
If \(x_3\) (the interaction) is missing, its component on the total efffect is added to the coefficient of \(x_1\).
This is a skeleton post
Standardized variables (Wright’s rules) n <- 10^5 # z is the common cause of g1 and g2 z <- rnorm(n) # effects of z on g1 and g2 b1 <- 0.7 b2 <- 0.7 r12 <- b1*b2 g1 <- b1*z + sqrt(1-b1^2)*rnorm(n) g2 <- b2*z + sqrt(1-b2^2)*rnorm(n) var(g1) # E(VAR(g1)) = 1 ## [1] 1.001849 var(g2) # E(VAR(g2)) = 1 ## [1] 1.006102 cor(g1, g2) # E(COR(g1,g2)) = b1*b2 ## [1] 0.