2 Indirect Confounding \u2013 An Example<\/h2>\n
To illustrate indirect confounding, Fig. 1 below depicts the example used in WC\u201808, 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.<\/p>\n
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 \u2192 W \u2190 U \u2192 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)).<\/p>\n
![](\"http:\/\/causality.cs.ucla.edu\/blog\/wp-content\/uploads\/2015\/07\/blog-figure.gif\")
\nFigure 1: An example of \u201cindirect confounding\u201d from WC\u201808. 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\u201808 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).]<\/p>\n
Solution:<\/strong> We are done, because the last expression consists of estimable factors. What makes this problem appear difficult in the linear model treated by WC\u201808 is that the direct effect of X on Y (say \u03b1) 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 G\u03b1. This means that \u03b1 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\u201808 identify \u03b1 using one of these methods, and their solution coincides of course with the general derivation given above.<\/p>\n 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\u201808. 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 \u2013 the derivation is almost the same (see Pearl (2009, p. 122)).<\/p>\n Most importantly, readers of Causality also know that, once we write the problem as \u201cFind P(y|do(x, z))\u201d 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.<\/p>\n 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\u2019s private thoughts. Still, the reluctance shown by Cox and Wermuth seems to be of a different kind. Here are a few examples:<\/p>\n Cox and Wermuth (CW\u201915) write: These comments are puzzling because the do-operator and its associated \u201ccausal calculus\u201d operate not \u201con a probability distribution,\u201d but on a data generating model (i.e., the DAG). Likewise, the calculus is used, not for \u201cinferring causality\u201d (God forbid!!) but for predicting the effects of interventions from causal assumptions that are already encoded in the DAG.<\/p>\n In WC\u201814 we find an even more puzzling description of \u201cvirtual intervention\u201d: \u201cFamiliarity is the mother of acceptance,\u201d 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\u20134 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.<\/p>\n BTW, I have offered this exercise repeatedly to colleagues from the potential outcome camp, and the response was uniform: \u201cwe do not work on toy problems, we work on real-life problems.\u201d Perhaps this note would entice them to join us, mortals, and try a small problem once, just for sport.<\/p>\n Let\u2019s hope,<\/p>\n Judea<\/p>\n Chen, B. and Pearl, J. (2014). Graphical tools for linear structural equation modeling. Tech. Rep. R-432, 1. Introduction This note concerns three papers by Cox and Wermuth (2008; 2014; 2015 (hereforth WC\u201808, WC\u201814 and CW\u201815)) in which they call attention to a class of problems they named \u201cindirect confounding,\u201d where \u201ca much stronger distortion may be introduced than by an unmeasured confounder alone or by a selection bias alone.\u201d We will […]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,10,11,12],"tags":[],"class_list":["post-1545","post","type-post","status-publish","format-standard","hentry","category-causal-effect","category-definition","category-discussion","category-do-calculus"],"_links":{"self":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1545","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=1545"}],"version-history":[{"count":0,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1545\/revisions"}],"wp:attachment":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=1545"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=1545"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=1545"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\u00a0 \u00a0 \u00a0P(y|do(x,z))
\n\u00a0 \u00a0 =P(y|x, do(z))\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(since X is randomized)
\n\u00a0 \u00a0 = ∑w<\/sub> P(Y|x,w,do(z))P(w|x, do(z))\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(by Rule 1 of do-calculus)
\n\u00a0 \u00a0 = ∑w<\/sub> P(Y|x,w,z)P(w|x)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(by Rule 2 and Rule 3 of do-calculus)<\/p>\n3 Conclusions<\/h2>\n
\n\u201c…some of our colleagues have derived a \u2018causal calculus\u2019 for the challenging
\nprocess of inferring causality; see Pearl (2015). In our view, it is unlikely that
\na virtual intervention on a probability distribution, as specified in this calculus,
\nis an accurate representation of a proper intervention in a given real world
\nsituation.\u201d (p. 3)<\/p>\n
\n\u201cThese recorded changes in virtual interventions, even though they are often
\ncalled \u2018causal effects,\u2019 may tell next to nothing about actual effects in real interventions
\nwith, for instance, completely randomized allocation of patients to
\ntreatments. In such studies, independence result by design and they lead to
\nmissing arrows in well-fitting graphs; see for example Figure 9 below, in the last
\nsubsection.\u201d [our Fig. 1]<\/p>\n References<\/h2>\n
\nCox, D. and Wermuth, N. (2015). Design and interpretation of studies: Relevant concepts from the past and some extensions. Observational Studies This issue.
\nPearl, J. (2009). Causality: Models, Reasoning, and Inference. 2nd ed. Cambridge Uni- versity Press, New York.
\nPearl, J. (2015). Trygve Haavelmo and the emergence of causal calculus. Econometric Theory 31 152\u2013179. Special issue on Haavelmo Centennial.
\nPearl, 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\u2013453.
\nRobins, 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 \u201897). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 409\u2013420.
\nWermuth, N. and Cox, D. (2008). Distortion of effects caused by indirect confounding. Biometrika 95 17\u201333.
\nWermuth, N. and Cox, D. (2014). Graphical Markov models: Overview. ArXiv: 1407.7783.<\/p>\n","protected":false},"excerpt":{"rendered":"