4

Collider bias occurs when there is no association between X and Y but when a third variable which is caused by both X and Y is controlled for, this "opens a path" between X and Y and leads to a biased estimate/distorted association.

enter image description here

Meanwhile, if you control for a confounder-- a third variable that causes X and Y-- it closes the path and the association between X and Y is removed.

enter image description here

An aspect of this that confuses me is that if X and Y are related to this third variable for both confounders and colliders, why does in one instance controlling for it open the path (collider) and in another instance close the path (confounder)? What is the reason that X and Y causing Z means that controlling for it opens the path, while if X and Y are caused by Z, controlling for it closes the path? This is a distinction I struggle with a bit since in both instances, they are related to Z/third variable.

JElder
  • 919
  • You might find a mathematical intuition here in the case of a system of linear models. There is a general expression for the correlation between two variables conditioning on a third, and when the third is a collider the resulting expression yields a nonzero condition correlation between the first two. A question may still remain about the intuition for that expression or for more general scenarios, but at least it can be a start. – Noah Jan 08 '24 at 18:51

2 Answers2

1

For relationships among $3$ variables, say, $X, Y,$ and $Z,$ there are only $3$ possibilities. Technically, there are $4,$ but $X\to Y\to Z$ and $Z\to Y\to X$ are similar enough to count as one case, since you can simply interchange the role of $X$ and $Z$ to flip from one to the other. Those $3$ relationships are:

  1. Chain: $X\to Y\to Z.$ Conditioning on $Y$ stops the information flow from $X$ to $Z.$ If $Y$ is not conditioned, then causal information can flow from $X$ to $Z.$
  2. Fork: $X\leftarrow Y\to Z.$ Conditioning on $Y$ stops the information flow from $X$ to $Z.$ If $Y$ is not conditioned, then causal information can flow from $X$ to $Z.$
  3. Collider: $X\to Y\leftarrow Z.$ This is the special case: if $Y$ is not conditioned on, then the path from $X$ to $Z$ is closed: information can not travel from $X$ to $Z.$ However, if you condition on $Y,$ you actually open the path, and information can travel from $X$ to $Z.$

So you can see that whether the unconditioned $Y$ allows information through it, and whether the conditioned $Y$ allows information through it, is utterly and completely dependent on the directions of the arrows! It is not in the slightest dependent on the fact that $Y$ is connected to $X$ and $Z,$ one way or another.

Now if you want to ask the question about why colliders are different from forks and chains, that would be a question for the likes of Judea Pearl himself, I suspect. No doubt playing around with some simulations could provide some intuition, and there are also examples in Pearl's The Book of Why that illustrate why this is so.

Adrian Keister
  • 5,862
  • You're also technically right that there are 3 possibilities... that is, 3 directed graphlets! See Enumeration of Automorphism Orbits of Graphlets | HackSeq | Talk. – Galen Dec 22 '23 at 07:30
  • Thank you for your response! But I suppose I was saying I am aware of those rules of conditioning on a chain, fork, and collider and that direction does matter, but I am asking WHY does direction matter. I was hoping for some sort of mechanistic explanation why direction matters so much in conditioning on a third variable, rather than a statement of that it is the case.

    I have read some of Pearl's work and some descriptions but still struggle with the intuition. I'll look into it.

    Thanks!

     – JElder Dec 22 '23 at 15:30
  • @JElder I thought that was likely the case; I only included the basic mechanism explanation to level-set. I would recommend, in particular, looking at The Book of Why, pages 115ff, and the "explain-away effect." I think that's a fairly intuitive explanation of collider bias, and why conditioning on a collider opens up sometimes-spurious correlations. – Adrian Keister Dec 22 '23 at 15:50
1

It's probably easier here to write out the structural model implied by the DAGs and look at how the corresponding functions behave when intervened on. I'll do this briefly for confounders and colliders. Admittedly, some of this is a little new to me too, so apologies if this is a little sloppy. The best resource to learn more is probably Judea Pearl's Causality. I hope that this will help answer your question anyway!

Confounders

Say we have a simple DAG depicting variables $X$ and $Y$ confounded by a third variable $W$, where $X$ does not cause $Y$, or vice versa. This would look like $X \leftarrow W \rightarrow Y$. What this implies is that $X$ is generated according to $W$ and some exogenous error, $Y$ is generated according to $W$ and some exogenous error, and $W$ is generated only according to an exogenous error. The error terms of each variable are independent from the error terms of the other variables. For example, error in $X$ causes stochasticity in $X$, but is fully independent of $Y$ and $W$. We can write the corresponding (unknown) functions for our DAG as

$w = f_1(u_1)$

$x = f_2(w, u_2)$

$y = f_3(w, u_3)$

which simply states that each realization $w$ depends on no other variable (only some stochastic error), while the corresponding values for $x$ and $y$ depend on that particular $w$ in addition to the exogenous error. You could intervene on this causal model by setting $W$ to different values. This would change the input for the functions $f_2(w, u_2)$ and $f_3(w, u_3)$ each time you try a new $w$. In other words, changing $w$ also changes $x$ and $y$, creating an association between them. We have an open path.

What if we condition on $W$ now to close this open path? Maybe we simply set $W = 1$. Our functions for $x$ and $y$ become

$x = f_2(1, u_2)$

$y = f_3(1, u_3)$

$X$ and $Y$ can now only change through their own errors $U_2$ and $U_3$. Since these are assumed to be independent, $X$ and $Y$ are independent and we have closed the open path.

Colliders

If $W$ was a collider instead, our DAG would change to $X \rightarrow W \leftarrow Y$ and the functions of our structural model would be

$w = f_4(u_4, x, y)$

$x = f_5(u_5)$

$y = f_6(u_6)$

We now have the same situation that we had in the confounder example after conditioning on $W$: since $X$ and $Y$ only depend on independent errors, they are independent. There is no open path.

Explaning what happens when $W$ is conditioned on is a bit less intuitive. For some intuition, assume that $X$, $Y$, and $W$ are binary variables. Also suppose we knew that the function generating each realization of $W$ was $w = I(x = 1 \lor y = 1)$. In other words, $w$ takes the value $1$ if and only if either $x$ takes the value $1$, $y$ takes the value $1$, or both take the value $1$. A possible example for such a data-generating process is one where everyone older than 50 years ($x = 1$) or female ($y = 1$) is invited to a cancer screening ($w = 1$).

What if we condition on $W$ now? We cause selection bias! For example, if we set as condition that $w$ equals $1$, we make any observation inadmissible that does not have $x = 1$ or $y = 1$ because no such observation can cause $w$ to be $1$. This is strictly not possible given our structural model. Alternatively, if we look only among those that were invited to a cancer screening, no individuals that are not female or older than 50 will be observed. There is an association between $X$ and $Y$ now; we have opened a path. This might be a little easier to see the other way around: If we condition on $W = 0$, we know that $X = 0$ and $Y = 0$ and the two variables $X$ and $Y$ are perfectly correlated.

actual-garlic
  • 51