Flowers of the First Law of Causal Inference (2)
Flower 2 — Conditioning on post-treatment variables
In this 2nd flower of the First Law, I share with readers interesting relationships among various ways of extracting information from post-treatment variables. These relationships came up in conversations with readers, students and curious colleagues, so I will present them in a question-answers format.
Question-1
Rule 2 of do-calculus does not distinguish post-treatment from pre-treatment variables. Thus, regardless of the nature of Z, it permits us to replace P (y|do(x), z) with P (y|x, z) whenever Z separates X from Y in a mutilated graph GX (i.e., the causal graph, from which arrows emanating from X are removed). How can this rule be correct, when we know that one should be careful about conditioning on a post treatment variables Z?
Example 1 Consider the simple causal chain X → Y → Z. We know that if we condition on Z (as in case control studies) selected units cease to be representative of the population, and we cannot identify the causal effect of X on Y even when X is randomized. Applying Rule-2 however we get P (y|do(x), z) = P (y|x, z). (Since X and Y are separated in the mutilated graph X Y → Z). This tells us that the causal effect of X on Y IS identifiable conditioned on Z. Something must be wrong here.
[…] But resistance aside, the past two postings lay before readers two miracles of the first law, which I labeled “Flowers”. The first tells us how counterfactuals can be seen in the causal graph (link), and the second clarifies questions concerned with conditioning on post-treatment variables. (link). […]
Pingback by Causal Analysis in Theory and Practice » Winter Greeting from the UCLA Causality Blog — January 27, 2015 @ 7:35 am
I’m glad you directed me to those two articles. Controlling seems like a knee-jerk reaction in research (in addition to education, it shows up in medical journals and even missing data literature). It’s easy to be blindsided by that.
Comment by Dustin Fife — January 29, 2015 @ 9:34 am
[…] our examination of “the flowers of the First Law” (see previous flowers here and here) this posting looks at one of the most crucial questions in causal inference: “How generalizable […]
Pingback by Causal Analysis in Theory and Practice » Flowers of the First Law of Causal Inference (3) — April 24, 2015 @ 10:44 pm