Causality - Can adding predictors unblock causal effects?

Question

My question pertains to the following empirical situation.

You have a bivariate model (e.g. regression), and you find no effect of $X$ on $Y$.

Then you add a covariate in your model, $Z$, and now you find an effect of $X$ on $Y$. In other words, conditional on $Z$, an effect appears between $X$ and $Y$.

I sometimes see this empirical example in research, but it does not make sense to me.

According to Pearl, two variables show an association in three scenarios

1. One causes the other
2. They have a common cause (i.e. confounding bias)
3. You (mistakenly) condition on common effects (collider bias)

Therefore, I do not see how it is possible that an "effect" is "revealed" by introducing predictors to "account for selection", other than via the collider bias.

If there is no bivariate association, how can there be "selection" to account for?

The only exemption I can think of is that $Z$ is a predictor of $Y$ but not $X$, and that $X$ is measured with a lot of noise, thereby by controlling for $Z$, we reduce the variation in $Y$, and make the estimate of $X$ more precise, potentially "revealing" an effect.

What do you think? Am I missing something?

Answer 1

I think you're missing the fact that confounding bias (nothing to do with colliders) can be of arbitrary magnitude, including equal to the treatment effect. So if the treatment effect is $\tau$ and the bias due to confounding is $-\tau$, then the unadjusted effect estimate should be close to 0 and the adjusted effect estimate should be $\tau$. Here's an example:

set.seed(1234)
n <- 1e4

x <- rnorm(n)
a <- .5 * x + rnorm(n, 0, sqrt(.75))
y <- a - 2 * x + rnorm(n)

lm(y ~ a) |> coef()
#> (Intercept)           a 
#> -0.01176063  0.00993723
lm(y ~ a + x) |> coef()
#>  (Intercept)            a            x 
#>  0.004169566  0.998852233 -2.014340925

We can see that without adjusting for the confounder, there appears to be no association between treatment a and outcome y, and after adjusting for confounder x we see that the true nonzero association is revealed. Again, nothing to do with colliders, this can happen purely with confounding.

Another example is to do with suppression in mediation. I give an example of this here. When you have two compensatory pathways from the treatment to the outcome, you will see no effect of the treatment, but when you adjust for one of the pathways, the direct effect of the treatment is revealed. This happens in systems that maintain homeostasis.