How to derive the size of a simple bias in a mediation setting? Consider the following DAG which shows the direct and indirect effect of $U$ on $Y$.
The total effect of $U \rightarrow Y$ is simply $(2\times4) + 3 = 11$.
I am looking for the derivation of the biased effect of $A \rightarrow Y$.
When I run simulation I get a biased effect of about 5.5
How can I retrieve mathematically this 5.5?

# R simulation

n = 100000
u = rnorm(n, 0, 1)
a = u*2

y = 3*u + 4*a

lm(y ~ a) # biased effect
lm(y ~ u) # total effect
lm(y ~ a + u) # correct effects


2 ways to compute the bias effect, which is what I observe from the simulation, I worked out but that do not make sense to me why it works. Is there a general formula to compute expected bias?

*

*Divide the total effect of $U$ on $Y$, by the path $U \rightarrow A$, $\frac{11}{2}=5.5$


*Let's label the paths: $U \rightarrow A = a$, $A \rightarrow Y = b$, and $U \rightarrow Y = c$, then we can compute the bias with: a + ($\frac{c}{b}$), $4 + \frac{3}{2} = 5.5$

The OLS coefficient is the ratio of covariance to variance. That is, let
$$
y = \beta a + \nu.
$$
Then $\beta = \frac{Cov (y,a)}{Var(a)} = \frac{Cov(11u,2u)}{Var(2u)} = \frac{22}{4} \frac{Var(u)}{Var(u)} = 5.5$.
(We can skip the intercept as both variables have mean 0)
More details for example here: https://jrnold.github.io/intro-methods-notes/regression-anatomy.html
