VanderWeele bias calculation

Question

In his book "Explanation in Causal Inference Methods for Mediation and Interaction", Tyler VanderWeele gives the following formula for bias (also Cinelli & Hazlet).

Let $\beta'$ be the biased coefficient of our exposure of interest in a simple situation where our exposure $X$, has a confounder $U$.

\begin{equation} \beta' = \gamma \delta \end{equation}

$\gamma$ being the direct effect of the confounder $U$ on our outcome $Y$,

In the case where $U$ and $X$ are binary, we have

\begin{equation} \gamma = E[Y|X=x,U=1] - E[Y|X=x,U=0] \end{equation}

and $\delta$ the difference in proportion $U$ in $X$ ("the difference in the prevalence of the unmeasured confounder", p.68 VanderWeele),

\begin{equation} \delta = Pr[Y|X=1, U=1] - Pr[Y|X=0,U=1] \end{equation}

When I try to simulate this bias (see code below, it is not working).

Now it was established elsewhere that the bias could simply be estimated as

\begin{equation} \beta' = \beta + \frac{\Delta}{\gamma} \end{equation}

Which is basically the "selection path" ($\Delta$) divided by the direct path of $U$ to $Y$ ($\gamma$).

library(tidyverse)

set.seed(123)
n=10000

# around 70% #
plogis(1)

u = rnorm(n, 0, 1)
# x is a function of u + a random component
x = rnorm(n, 0, 1) + u
# dichotomise
x = ifelse(plogis(x) < 0.5, 0, 1)
u = ifelse(plogis(u) < 0.5, 0, 1)

# y equation
y = x*1.5 + u*2

df = data.frame(u,x,y)

# counterfactual outcomes
df$y0 = u
df$y1 = u + 1.5

df$y1y0 = df$y1-df$y0

Here I have the $\delta$ part, the difference of proportion of $U$ in $X$

# \delta -- P(U=1,X=1) - P(U=1,X=0)
count(df, u, x) %>% group_by(u) %>% mutate(n = n/sum(n)) %>% filter(u==1) %>% spread(x,n) %>% mutate(`1`-`0`)

I get 0.5 (50%)

We know that the direct effect $\gamma$ is 2, but $0.5 \times 2$ is not the bias that we find when we run a simple OLS regression, where we miss $U$.

lm(y ~ x)

We get a $\beta$ of 2.5 for x.

Once we correct for $u$, we get the correct $\beta$ of 1.5

lm(y ~ x + u)

Question 1

What I am missing in VanderWeele bias calculation?

Question 2

What is the relationship between the 2 formulas of bias

$\beta' = \gamma \delta$ and $\beta' = \beta + \frac{\Delta}{\gamma}$

Answer 1

Your replication of the formula is not correct. The correct formula in Cinelli and Hazlett (see equation 5) is:

$$ \beta' = \beta + \gamma\delta $$

This is an algebraic identity that holds for any OLS model. In your example we obtain:

beta.prime <- coef(lm(y ~ x))["x"]
beta       <- coef(lm(y ~ x + u))["x"]
imbalance  <- coef(lm(u ~ x))["x"]
impact     <- coef(lm(y ~ x + u))["u"]


> beta.prime
       x 
2.514337 

> beta + impact*imbalance
       x 
2.514337 

Your second formula does not seem to be true in general, you seem to be assuming that a single latent variable determines both $X$ and $Y$. Here is a counter-example to your formula:

> n <- 1e3
> u <- rnorm(n)
> x <- u + rnorm(n)
> y <- x + u + rnorm(n)
> lm(y ~ x)

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept)            x  
    0.00792      1.48475