### Indirect Confounding and Causal Calculus (On three papers by Cox and Wermuth)

## 1. Introduction

This note concerns three papers by Cox and Wermuth (2008; 2014; 2015 (hereforth WC‘08, WC‘14 and CW‘15)) in which they call attention to a class of problems they named “indirect confounding,” where “a much stronger distortion may be introduced than by an unmeasured confounder alone or by a selection bias alone.” We will show that problems classified as “indirect confounding” can be resolved in just a few steps of derivation in do-calculus.

This in itself would not have led me to post a note on this blog, for we have witnessed many difficult problems resolved by formal causal analysis. However, in their three papers, Cox and Wermuth also raise questions regarding the capability and/or adequacy of the do-operator and do-calculus to accurately predict effects of interventions. Thus, a second purpose of this note is to reassure students and users of do-calculus that they can continue to apply these tools with confidence, comfort, and scientifically grounded guarantees.

Finally, I would like to invite the skeptic among my colleagues to re-examine their hesitations and accept causal calculus for what it is: A formal representation of interventions in real world situations, and a worthwhile tool to acquire, use and teach. Among those skeptics I must include colleagues from the potential-outcome camp, whose graph-evading theology is becoming increasing anachronistic (see discussions on this blog, for example, here).

## 2 Indirect Confounding – An Example

To illustrate indirect confounding, Fig. 1 below depicts the example used in WC‘08, which involves two treatments, one randomized (X), and the other (Z) taken in response to an observation (W) which depends on X. The task is to estimate the direct effect of X on the primary outcome (Y), discarding the effect transmitted through Z.

As we know from elementary theory of mediation (e.g., Causality, p. 127) we cannot block the effect transmitted through Z by simply conditioning on Z, for that would open the spurious path X → W ← U → Y , since W is a collider whose descendant (Z) is instantiated. Instead, we need to hold Z constant by external means, through the do-operator do(Z = z). Accordingly, the problem of estimating the direct effect of X on Y amounts to finding P(y|do(x, z)) since Z is the only other parent of Y (see Pearl (2009, p. 127, Def. 4.5.1)).

Figure 1: An example of “indirect confounding” from WC‘08. Z stands for a treatment taken in response to a test W, whose outcome depend ends on a previous treatment X. U is unobserved. [WC‘08 attribute this example to Robins and Wasserman (1997); an identical structure is treated in Causality, p. 119, Fig. 4.4, as well as in Pearl and Robins (1995).]

**Solution:**

P(y|do(x,z))

=P(y|x, do(z)) (since X is randomized)

= ∑_{w} P(Y|x,w,do(z))P(w|x, do(z)) (by Rule 1 of do-calculus)

= ∑_{w} P(Y|x,w,z)P(w|x) (by Rule 2 and Rule 3 of do-calculus)

We are done, because the last expression consists of estimable factors. What makes this problem appear difficult in the linear model treated by WC‘08 is that the direct effect of X on Y (say α) cannot be identified using a simple adjustment. As we can see from the graph, there is no set S that separates X from Y in Gα. This means that α cannot be estimated as a coefficient in a regression of Y on X and S. Readers of Causality, Chapter 5, would not panic by such revelation, knowing that there are dozens of ways to identify a parameter, going way beyond adjustment (surveyed in Chen and Pearl (2014)). WC‘08 identify α using one of these methods, and their solution coincides of course with the general derivation given above.

The example above demonstrates that the direct effect of X on Y (as well as Z on Y ) can be identified nonparametrically, which extends the linear analysis of WC‘08. It also demonstrates that the effect is identifiable even if we add a direct effect from X to Z, and even if there is an unobserved confounder between X and W – the derivation is almost the same (see Pearl (2009, p. 122)).

Most importantly, readers of Causality also know that, once we write the problem as “Find P(y|do(x, z))” it is essentially solved, because the completeness of the do-calculus together with the algorithmic results of Tian and Shpitser can deliver the answer in polynomial time, and, if terminated with failure, we are assured that the effect is not estimable by any method whatsoever.

## 3 Conclusions

It is hard to explain why tools of causal inference encounter slower acceptance than tools in any other scientific endeavor. Some say that the difference comes from the fact that humans are born with strong causal intuitions and, so, any formal tool is perceived as a threatening intrusion into one’s private thoughts. Still, the reluctance shown by Cox and Wermuth seems to be of a different kind. Here are a few examples:

Cox and Wermuth (CW’15) write:

“…some of our colleagues have derived a ‘causal calculus’ for the challenging

process of inferring causality; see Pearl (2015). In our view, it is unlikely that

a virtual intervention on a probability distribution, as specified in this calculus,

is an accurate representation of a proper intervention in a given real world

situation.” (p. 3)

These comments are puzzling because the do-operator and its associated “causal calculus” operate not “on a probability distribution,” but on a data generating model (i.e., the DAG). Likewise, the calculus is used, not for “inferring causality” (God forbid!!) but for predicting the effects of interventions from causal assumptions that are already encoded in the DAG.

In WC‘14 we find an even more puzzling description of “virtual intervention”:

“These recorded changes in virtual interventions, even though they are often

called ‘causal effects,’ may tell next to nothing about actual effects in real interventions

with, for instance, completely randomized allocation of patients to

treatments. In such studies, independence result by design and they lead to

missing arrows in well-fitting graphs; see for example Figure 9 below, in the last

subsection.” [our Fig. 1]

“Familiarity is the mother of acceptance,” say the sages (or should have said). I therefore invite my colleagues David Cox and Nanny Wermuth to familiarize themselves with the miracles of do-calculus. Take any causal problem for which you know the answer in advance, submit it for analysis through the do-calculus and marvel with us at the power of the calculus to deliver the correct result in just 3–4 lines of derivation. Alternatively, if we cannot agree on the correct answer, let us simulate it on a computer, using a well specified data-generating model, then marvel at the way do-calculus, given only the graph, is able to predict the effects of (simulated) interventions. I am confident that after such experience all hesitations will turn into endorsements.

BTW, I have offered this exercise repeatedly to colleagues from the potential outcome camp, and the response was uniform: “we do not work on toy problems, we work on real-life problems.” Perhaps this note would entice them to join us, mortals, and try a small problem once, just for sport.

Let’s hope,

Judea

## References

Chen, B. and Pearl, J. (2014). Graphical tools for linear structural equation modeling. Tech. Rep. R-432,

Cox, D. and Wermuth, N. (2015). Design and interpretation of studies: Relevant concepts from the past and some extensions. Observational Studies This issue.

Pearl, J. (2009). Causality: Models, Reasoning, and Inference. 2nd ed. Cambridge Uni- versity Press, New York.

Pearl, J. (2015). Trygve Haavelmo and the emergence of causal calculus. Econometric Theory 31 152–179. Special issue on Haavelmo Centennial.

Pearl, J. and Robins, J. (1995). Probabilistic evaluation of sequential plans from causal models with hidden variables. In Uncertainty in Artificial Intelligence 11 (P. Besnard and S. Hanks, eds.). Morgan Kaufmann, San Francisco, 444–453.

Robins, J. M. and Wasserman, L. (1997). Estimation of effects of sequential treatments by reparameterizing directed acyclic graphs. In Proceedings of the Thirteenth Conference on Uncertainty in Artificial Intelligence (UAI ‘97). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 409–420.

Wermuth, N. and Cox, D. (2008). Distortion of effects caused by indirect confounding. Biometrika 95 17–33.

Wermuth, N. and Cox, D. (2014). Graphical Markov models: Overview. ArXiv: 1407.7783.

It had puzzled me why WC felt that their problem of indirect confounding (a special case of ‘intermediate confounding’, ‘exposure-induced confounding’, ‘time-dependent confounding’ as discussed in the causal inference literature?) required its own solution, and when discussing this with them, I was not really able to understand their objection to your work (I am not claiming that their objections are not valid, merely that I could not follow them). So what you have written above is useful to me; thank you!

I don’t want to risk misrepresenting anyone’s views, and so the rest of my comment relates to the objections I have heard statisticians raise to the do-calculus in general; this is not specific to Profs Wermuth and Cox, whose objections may well be quite different.

1. The first is a trivial matter of semantics, but the fact that Pr(y|do(x)) is written with the usual “|” sign for conditioning seems to lure people into thinking that it is a conditional probability distribution. So I find that my first task is to point out that Pr(y|do(x)) is the marginal distribution of Y in a world in which we could intervene on X and set it to x. It might be more immediately understandable to people if it were written instead as Pr_{do(x)}(y), with “do(x)” indexing the distribution. But am I missing something here?

2. Because introductions to causal inference (for sensible reasons of simplicity) focus on the target Pr(y|do(x)), or, from the PO school, the related ACE=E(Y_1)-E(Y_0) – both marginal quantities – there is sometimes a misconception that the causal inference world puts a strong emphasis on marginal effects. This, I think, can seem counterintuitive to statisticians who traditionally view conditioning to be the tool that takes them “closer to causality”*. Given only the traditional statistical language of associations, then indeed conditional quantities are usually “closer to causality” than marginal ones. Given covariates C – issues of M-bias and the like aside – Pr(y|x,c) is a better candidate for causal interpretation (wrt X) than Pr(y|x). However, the beauty of the do-notation, or any causal notation, of course, is that we can write either Pr(y|do(x)) or Pr(y|do(x),c). Both are causal wrt X, one is marginal and one is conditional, and we can choose which is our target of inference depending on the context (indeed, it is also useful, of course, to be able to distinguish between Pr(y|do(x),c) and Pr(y|do(x,c))!). I get the impression that some statisticians feel that an equation such as:

Pr(y|do(x))=sum_c Pr(y|x,c)Pr(c)

serves to hide the conditional associations in Pr(y|x,c) in a black box by marginalising them out over C. They want to “see the dependencies” that the data suggest by studying Pr(y|x,c) directly instead of Pr(y|do(x)).

I’m not saying that this is my view at all; I’m just trying to express where I think the do-calculus gets lost in translation when some statisticians encounter it. Some may (mistakenly) feel that they are being drawn into marginal inferences when they prefer conditional ones. Perhaps related to this is that *some* causal inference methods, like inverse weighting, do indeed target marginal effects.

Sorry if this is not sufficiently related to the work by WC to be justified as a comment to this post. But reading about WC’s objections to the do-calculus reminded me of these two problems that I have often encountered when trying to convince statisticians of the usefulness of causal inference methods in general.

Best wishes,

Rhian

* There are many caveats here, I realise: A. conditioning can be harmful eg M-bias, B. epidemiologists have always been keen on standardisation, so the notion of estimating a marginal causal effect from an observational study is not at all new, and C. marginal causal effects are often the target of inference in a randomised study.

Comment by Rhian Daniel — August 13, 2015 @ 4:38 am

Rhian,

I have no clue why Wermuth and Cox (WC) would pose this example as a hard problem, or why they are still objecting to

causal calculus. Rather than speculating on the psychology of my colleagues (I have made too many enemies

that way) I invite WC to join the simulation game proposed above and demonstrate their objections in action.

I think part of the problem is that few people are willing to take a stand

and correct such authors unequivocally: “Your problem is easy and your objections are INVALID.” Not “if”, nor

“maybe”, but plain INVALID. It has to be said, otherwise new students entering statistics would continue to believe

that causal inference is “controversial”. It is not.

And we cant let progress in causal inference be impeded by authors who toss objections from a distance.

Thanks for your thoughtful comments on why newcomers might have difficulties with the do-operator.

1. The indexing notation P_{x}(y) has some advantages over the conditional P(y|do(x)), and I am using it in the

first chapter of my book. But I have found the conditioning notation to be advantageous in a long run. Partly because

it reminds us of the interventional interpretation of x, and partly because it meshes so harmoniously with the

see(z) operator in the do-calculus.

2. I did not think about this perception, that conditional quantities are usually expected to be “closer to causality”.

We should indeed work out more examples where P(y|do(x), c) is the research question of interest,

rather than P(y|do(x)), to assure readers that c-specific causal effects are no less interesting

than the population-averaged effects.

Thanks for pointing this to me.

Comment by Judea Pearl — August 14, 2015 @ 7:13 am

Hi Judea, in what sense is the second step in your derivation an application of rule 1? Shouldn’t this step be always allowed by basic probability theory? Best wishes, Julian

Comment by Julian — September 13, 2016 @ 10:09 am

Dear Judea

This is a great post to help students (like me) who are struggling with graduate analysis.

Do you have any advice to tackle problems of the type: Prove or disprove the following statements … So instead of attacking these problems directly, we have to think twice whether we are heading in the right direction. And usually it’s hard to think of suitable counterexamples during exams!

Comment by StyleExpert — October 15, 2017 @ 11:20 am