Causal Analysis in Theory and Practice

June 28, 2016

On the Classification and Subsumption of Causal Models

Filed under: Causal Effect,Counterfactual,structural equations — bryantc @ 5:32 pm

From Christos Dimitrakakis:

>> To be honest, there is such a plethora of causal models, that it is not entirely clear what subsumes what, and which one is equivalent to what. Is there a simple taxonomy somewhere? I thought that influence diagrams were sufficient for all causal questions, for example, but one of Pearl’s papers asserts that this is not the case.

Reply from J. Pearl:

Dear Christos,

From my perspective, I do not see a plethora of causal models at all, so it is hard for me to answer your question in specific terms. What I do see is a symbiosis of all causal models in one framework, called Structural Causal Model (SCM) which unifies structural equations, potential outcomes, and graphical models. So, for me, the world appears simple, well organized, and smiling. Perhaps you can tell us what models lured your attention and caused you to see a plethora of models lacking subsumption taxonomy.

The taxonomy that has helped me immensely is the three-level hierarchy described in chapter 1 of my book Causality: 1. association, 2. intervention, and 3 counterfactuals. It is a useful hierarchy because it has an objective criterion for the classification: You cannot answer questions at level i unless you have assumptions from level i or higher.

As to influence diagrams, the relations between them and SCM is discussed in Section 11.6 of my book Causality (2009), Influence diagrams belong to the 2nd layer of the causal hierarchy, together with Causal Bayesian Networks. They lack however two facilities:

1. The ability to process counterfactuals.
2. The ability to handle novel actions.

To elaborate,

1. Counterfactual sentences (e.g., Given what I see, I should have acted differently) require functional models. Influence diagrams are built on conditional and interventional probabilities, that is, p(y|x) or p(y|do(x)). There is no interpretation of E(Y_x| x’) in this framework.

2. The probabilities that annotate links emanating from Action Nodes are interventional type, p(y|do(x)), that must be assessed judgmentally by the user. No facility is provided for deriving these probabilities from data together with the structure of the graph. Such a derivation is developed in chapter 3 of Causality, in the context of Causal Bayes Networks where every node can turn into an action node.

Using the causal hierarchy, the 1st Law of Counterfactuals and the unification provided by SCM, the space of causal models should shine in clarity and simplicity. Try it, and let us know of any questions remaining.


June 21, 2016

Spring Greeting from the UCLA Causality Blog

Filed under: Announcement — bryantc @ 3:13 am

Dear friends in causality research,
This Spring Greeting from UCLA Causality blog contains:
A. News items concerning causality research,
B. New postings, new problems and some solutions.

The American Statistical Association (ASA) has announced recipients of the 2016 “Causality in Statistics Education Award”.
Congratulations go to Onyebuchi Arah and Arvid Sjolander who will receive this Award in July, at the 2016 JSM meeting in Chicago.
For details of purpose and selection criteria, see

I will be giving another tutorial at the 2016 JSM meeting, titled “Causal Inference in Statistics: A Gentle Introduction.”
Details and Abstract can be viewed here:

A3. Causal Inference — A Primer
For the many readers who have inquired, the print version of our new book “Causal Inference in Statistics – A Primer” is now up and running on Amazon and Wiley, and is awaiting your reviews, your questions and suggestions. We have posted a book page for this very purpose, which includes selected excerpts from each chapter, errata and updates, and a sample homework solution manual.

The errata page was updated recently under the diligent eye of Adamo Vincenzo. Thank you Adamo!

The Solution Manual will be available for instructors and will incorporate software solutions based on a DAGitty R package, authored by Johannes Textor.  See

Vol. 4 Issue 2 of the Journal of Causal Inference (JCI) is scheduled to appear in September 2018. The current issue can be viewed here: My own contribution to the current issue discusses Savage’s Sure Thing Principle and its ramifications to causal reasoning.

As always, submissions are welcome on all aspects of causal analysis, especially those deemed foundational. Chances of acceptance are inversely proportional to the time it takes a reviewer to figure out what problem the paper attempts to solve. So, please be transparent.

Recollections from the WCE conference at Stanford.

On May 21, Kosuke Imai and I participated in a panel on Mediation, at the annual meeting of the West Coast Experiment Conference, organized by Stanford Graduate School of Business.

Some of my recollections are summarized on our Causality Blog here:

B2. Generalizing Experimental findings
In light of new results concerning generalizability and selection bias, our team has updated the “external validity” entry of wikipedia. Previously, the entry was all about threats to validity, with no word on how those threats can be circumvented. You may wish to check this entry for accuracy and possible extensions.

B3. Causality celebrates its 10,000 citations
According to Google Scholar,, my book Causality (Cambridge, 2000, 2009) has crossed the symbolic mark of 10,000 citations. To celebrate this numerological event, I wish to invite all readers of this blog to an open online party with the beer entirely on me. I dont exactly know how to choreograph such a huge party, or how to make sure that each of you gets a fair share of the inspiration (or beer). So, please send creative suggestions for posting on this blog.

On a personal note: I am extremely gratified by this sign of receptiveness, and I thank readers of Causality for their comments, questions, corrections and reservations which have helped bring this book to its current shape (see


June 20, 2016

Recollections from the WCE conference at Stanford

Filed under: Counterfactual,General,Mediated Effects,structural equations — bryantc @ 7:45 am

On May 21, Kosuke Imai and I participated in a panel on Mediation, at the annual meeting of the West Coast Experiment Conference, organized by Stanford Graduate School of Business The following are some of my recollections from that panel.

We began the discussion by reviewing causal mediation analysis and summarizing the exchange we had on the pages of Psychological Methods (2014)

My slides for the panel can be viewed here:

We ended with a consensus regarding the importance of causal mediation and the conditions for identifying of Natural Direct and Indirect Effects, from randomized as well as observational studies.

We proceeded to discuss the symbiosis between the structural and the counterfactual languages. Here I focused on slides 4-6 (page 3), and remarked that only those who are willing to solve a toy problem from begining to end, using both potential outcomes and DAGs can understand the tradeoff between the two. Such a toy problem (and its solution) was presented in slide 5 (page 3) titled “Formulating a problem in Three Languages” and the questions that I asked the audience are still ringing in my ears. Please have a good look at these two sets of assumptions and ask yourself:

a. Have we forgotten any assumption?
b. Are these assumptions consistent?
c. Is any of the assumptions redundant (i.e. does it follow logically from the others)?
d. Do they have testable implications?
e. Do these assumptions permit the identification of causal effects?
f. Are these assumptions plausible in the context of the scenario given?

As I was discussing these questions over slide 5, the audience seemed to be in general agreement with the conclusion that, despite their logical equivalence, the graphical language  enables  us to answer these questions immediately while the potential outcome language remains silent on all.

I consider this example to be pivotal to the comparison of the two frameworks. I hope that questions a,b,c,d,e,f will be remembered, and speakers from both camps will be asked to address them squarely and explicitly .

The fact that graduate students made up the majority of the participants gives me the hope that questions a,b,c,d,e,f will finally receive the attention they deserve.

As we discussed the virtues of graphs, I found it necessary to reiterate the observation that DAGs are more than just “natural and convenient way to express assumptions about causal structures” (Imbens and Rubin , 2013, p. 25). Praising their transparency while ignoring their inferential power misses the main role that graphs play in causal analysis. The power of graphs lies in computing complex implications of causal assumptions (i.e., the “science”) no matter in what language they are expressed.  Typical implications are: conditional independencies among variables and counterfactuals, what covariates need be controlled to remove confounding or selection bias, whether effects can be identified, and more. These implications could, in principle, be derived from any equivalent representation of the causal assumption, not necessarily graphical, but not before incurring a prohibitive computational cost. See, for example, what happens when economists try to replace d-separation with graphoid axioms

Following the discussion of representations, we addressed questions posed to us by the audience, in particular, five questions submitted by Professor Jon Krosnick (Political Science, Stanford).

I summarize them in the following slide:

Krosnick’s Questions to Panel
1) Do you think an experiment has any value without mediational analysis?
2) Is a separate study directly manipulating the mediator useful? How is the second study any different from the first one?
3) Imai’s correlated residuals test seems valuable for distinguishing fake from genuine mediation. Is that so? And how it is related to traditional mediational test?
4) Why isn’t it easy to test whether participants who show the largest increases in the posited mediator show the largest changes in the outcome?
5) Why is mediational analysis any “worse” than any other method of investigation?
My answers focused on question 2, 4 and 5, which I summarize below:

Q. Is a separate study directly manipulating the mediator useful?
Answer: Yes, it is useful if physically feasible but, still, it cannot give us an answer to the basic mediation question: “What percentage of the observed response is due to mediation?” The concept of mediation is necessarily counterfactual, i.e. sitting on the top layer of the causal hierarchy (see “Causality” chapter 1). It cannot be defined therefore in terms of population experiments, however clever. Mediation can be evaluated with the help of counterfactual assumptions such as “conditional ignorability” or “no interaction,” but these assumptions cannot be verified in population experiments.

Q. Why isn’t it easy to test whether participants who show the largest increases in the posited mediator show the largest changes in the outcome?
Answer: Translating the question to counterfactual notation the test suggested requires the existence of monotonic function f_m such that, for every individual, we have Y_1 – Y_0 =f_m (M_1 – M_0)

This condition expresses a feature we expect to find in mediation, but it cannot be taken as a DEFINITION of mediation. This condition is essentially the way indirect effects are defined in the Principal Strata framework (Frangakis and Rubin, 2002) the deficiencies of which are well known. See

In particular, imagine a switch S controlling two light bulbs L1 and L2. Positive correlation between L1 and L2 does not mean that L1 mediates between the switch and L2. Many examples of incompatibility are demonstrated in the paper above.

The conventional mediation tests (in the Baron and Kenny tradition) suffer from the same problem; they test features of mediation that are common in linear systems, but not the essence of mediation which is universal to all systems, linear and nonlinear, continuous as well as categorical variables.

Q. Why is mediational analysis any “worse” than any other method of investigation?
Answer: The answer is closely related to the one given to question 3). Mediation is not a “method” but a property of the population which is defined counterfactually, and therefore requires counterfactual assumption for evaluation. Experiments are not sufficient; and in this sense mediation is “worse” than other properties under investigation, eg., causal effects, which can be estimated entirely from experiments.

About the only thing we can ascertain experimentally is whether the (controlled) direct effect differs from the total effect, but we cannot evaluate the extent of mediation.

Another way to appreciate why stronger assumptions are needed for mediation is to note that non-confoundedness is not the same as ignorability. For non-binary variables one can construct examples where X and Y are not confounded ( i.e., P(y|do(x))= P(y|x)) and yet they are not ignorable, (i.e., Y_x is not independent of X.) Mediation requires ignorability in addition to nonconfoundedness.

Overall, the panel was illuminating, primarily due to the active participation of curious students. It gave me good reasons to believe that Political Science is destined to become a bastion of modern causal analysis. I wish economists would follow suit, despite the hurdles they face in getting causal analysis to economics education.


June 10, 2016

Post-doc Causality and Machine Learning

Filed under: Announcement — bryantc @ 7:58 am

We received the following announcement from Isabelle Guyon (UPSud/INRIA):

The Machine Learning and Optimization (TAO) group of the Laboratory of Research in Informatics (LRI) is seeking a postdoctoral researcher for working at the interface of machine learning and causal modeling to support scientific discovery and computer assisted decision making using big data. The researcher will work with an interdisciplinary group including Isabelle Guyon (UPSud/INRIA), Cecile Germain UPSud), Balazs Kegl (CNRS), Antoine Marot (RTE), Patrick Panciatici (RTE), Marc Schoenauer (INRIA), Michele Sebag (CNRS), and Olivier Teytaud (INRIA).

Some research directions we want to pursue include: extending the formulation of causal discovery as a pattern recognition problem (developed through the ChaLearn cause-effect pairs challenge) to times series and spatio-temporal data; combining feature learning using deep learning methods with the creation of cause-effect explanatory models; furthering the unification of structural equation models and reinforcement learning approaches; and developing interventional learning algorithms.

As part of the exciting applications we are working on, we will be leveraging a long term collaboration with the company RTE (French Transmission System Operator for electricity). With the current limitations on adding new transportation lines, the opportunity to use demand response, and the advent of renewable energies interfaced through fast power electronics to the grid, there is an urgent need to adapt the historical way to operate the electricity power grid. The candidate will have the opportunity to use a combination of historical data (several years of data for the entire RTE network sampled every 5 minutes) and very accurate simulations (precise at the MW level), to develop causal models capable of identifying strategies to prevent or to mitigate the impact of incidents on the network as well as inferring what would have happened if we had intervened (i.e., counterfactual).Other applications we are working on with partner laboratories include epidemiology studies about diabetes and happiness in the workplace, modeling embryologic development, modeling high energy particle collision, analyzing human behavior in videos, and game playing.

The candidate will also be part of the Paris-Saclay Center of Data Science and will be expected to participate in the mission of the center through its activities, including organizing challenges on machine learning, and help advising PhD students.

We are accepting candidates with background in machine learning, reinforcement learning, causality, statistics, scientific modeling, physics, and other neighboring disciplines. The candidate should have the ability of working on cross-disciplinary problems, have a strong math background, and the experience or strong desire to work on practical problems.

The TAO group ( conducts interdisciplinary research in theory, algorithms, and applications of machine learning and optimization and it has also strong ties with AppStat the physics machine learning group of the Linear Accelerator Laboratory ( Both laboratories are part of the University Paris-Saclay, located in the outskirts of Paris. The position is available for a period of three years, starting in (the earliest) September, 2016. The monthly salary is around 2500 Euros per month. Interested candidates should send a motivation letter, a CV, and the names and addresses of three referees to Isabelle Guyon.

Contact: Isabelle Guyon (
Deadline: June 30, 2016, then every in 2 weeks until the position is filled.

February 12, 2016

Winter Greeting from the UCLA Causality Blog

Friends in causality research,
This greeting from the UCLA Causality blog contains:

A. An introduction to our newly published book, Causal Inference in Statistics – A Primer, Wiley 2016 (with M. Glymour and N. Jewell)
B. Comments on two other books: (1) R. Klein’s Structural Equation Modeling and (2) L Pereira and A. Saptawijaya’s on Machine Ethics.
C. News, Journals, awards and other frills.

Our publisher (Wiley) has informed us that the book “Causal Inference in Statistics – A Primer” by J. Pearl, M. Glymour and N. Jewell is already available on Kindle, and will be available in print Feb. 26, 2016.

This book introduces core elements of causal inference into undergraduate and lower-division graduate classes in statistics and data-intensive sciences. The aim is to provide students with the understanding of how data are generated and interpreted at the earliest stage of their statistics education. To that end, the book empowers students with models and tools that answer nontrivial causal questions using vivid examples and simple mathematics. Topics include: causal models, model testing, effects of interventions, mediation and counterfactuals, in both linear and nonparametric systems.

The Table of Contents, Preface and excerpts from the four chapters can be viewed here:
A book website providing answers to home-works and interactive computer programs for simulation and analysis (using dagitty)  is currently under construction.

We are in receipt of the fourth edition of Rex Kline’s book “Principles and Practice of Structural Equation Modeling”,

This book is unique in that it treats structural equation models (SEMs) as carriers of causal assumptions and tools for causal inference. Gone are the inhibitions and trepidation that characterize most SEM texts in their treatments of causation.

To the best of my knowledge, Chapter 8 in Kline’s book is the first SEM text to introduce graphical criteria for parameter identification — a long overdue tool
in a field that depends on identifiability for model “fitting”. Overall, the book elevates SEM education to new heights and promises to usher a renaissance for a field that, five decades ago, has pioneered causal analysis in the behavioral sciences.

Much has been written lately on computer ethics, morality, and free will. The new book “Programming Machine Ethics” by Luis Moniz Pereira and Ari Saptawijaya formalizes these concepts in the language of logic programming. See book announcement As a novice to the literature on ethics and morality, I was happy to find a comprehensive compilation of the many philosophical works on these topics, articulated in a language that even a layman can comprehend. I was also happy to see the critical role that the logic of counterfactuals plays in moral reasoning. The book is a refreshing reminder that there is more to counterfactual reasoning than “average treatment effects”.

C. News, Journals, awards and other frills.
Nominations are Invited for the Causality in Statistics Education Award (Deadline is February 15, 2016).

The ASA Causality in Statistics Education Award is aimed at encouraging the teaching of basic causal inference in introductory statistics courses. Co-sponsored by Microsoft Research and Google, the prize is motivated by the growing importance of introducing core elements of causal inference into undergraduate and lower-division graduate classes in statistics. For more information, please see .

Nominations and questions should be sent to the ASA office at . The nomination deadline is February 15, 2016.

Issue 4.1 of the Journal of Causal Inference is scheduled to appear March 2016, with articles covering all aspects of causal analysis. For mission, policy, and submission information please see:

Finally, enjoy new results and new insights posted on our technical report page:


UAB’s Nutrition Obesity Research Center — Causal Inference Course

Filed under: Announcement,Uncategorized — bryantc @ 1:03 am
We received the following announcement from Richard F. Sarver (UAB):

UAB’s Nutrition Obesity Research Center invite you to join them at one or both of our five-day short courses at the University of Alabama at Birmingham.

June: The Mathematical Sciences in Obesity Research The mathematical sciences including engineering, statistics, computer science, physics, econometrics, psychometrics, epidemiology, and mathematics qua mathematics are increasingly being applied to advance our understanding of the causes, consequences, and alleviation of obesity. These applications do not merely involve routine well-established approaches easily implemented in widely available commercial software. Rather, they increasingly involve computationally demanding tasks, use and in some cases development of novel analytic methods and software, new derivations, computer simulations, and unprecedented interdigitation of two or more existing techniques. Such advances at the interface of the mathematical sciences and obesity research require bilateral training and exposure for investigators in both disciplines. July: Strengthening Causal Inference in Behavioral Obesity Research Identifying causal relations among variables is fundamental to science. Obesity is a major problem for which much progress in understanding, treatment, and prevention remains to be made. Understanding which social and behavioral factors cause variations in adiposity and which other factors cause variations is vital to producing, evaluating, and selecting intervention and prevention strategies. In addition, developing a greater understanding of obesity’s causes, requires input from diverse disciplines including statistics, economics, psychology, epidemiology, mathematics, philosophy, and in some cases behavioral or statistical genetics. However, applying techniques from these disciplines does not involve routine well-known ‘cookbook’ approaches but requires an understanding of the underlying principles, so the investigator can tailor approaches to specific and varying situations. For full details of each of the courses, please refer to our websites below: Mon 6/13/2016 – Fri 6/17/2016: The Mathematical Sciences in Obesity, Mon 7/25/2016 – Fri 7/29/2016: Strengthening Causal Inference in Behavioral Obesity Research, Limited travel scholarships are available to young investigators. Please apply by Fri 4/1/2016 and be notified of acceptance by Fri 4/8/2016. Women, members of underrepresented minority groups and individuals with disabilities are strongly encouraged to apply. We look forward to seeing you in Birmingham this summer!

February 1, 2016

Workshop on Statistical Causal Inference and its Applications to Genetics

Filed under: Announcement — bryantc @ 11:50 pm

We received the following announcement from Robin Evans (University of Oxford):

Statistical Causal Inference and its Applications to Genetics, to be held at CRM in Montreal, July 2529 2016.

Additional information can be found here:

Dear Colleagues,

We are very excited to announce a week long workshop in Statistical Causal Inference and its Applications to Genetics, to be held at CRM in Montreal, July 2529 2016.

We seek participants from Statistics and Biology to discuss the cutting-edge inferential causal problems in the discipline. Points for discussion will include

– modern datasets in genetics,
– methods to deal with huge quantities of data from multiple experimental settings,
– hypothesis generation from limited experimental data,
– efficient experimental design,
– incorporation of prior information in a computationally tractable way,
– causal methods for time series data,
– Mendelian randomization,

We strongly encourage the participation of junior researchers, and invite the submission of abstracts for oral and poster presentations. To register your interest in participating or presenting please visit our website.

Invited speakers include:

Elias Bareinboim (Purdue University)
Tom Claassen (Radboud University Nijmegen)
Denver Dash (University of Pittsburgh)
Philip Dawid (University of Cambridge)
Vanessa Didelez (University of Bristol)
Frederick Eberhardt (Caltech)
Michael Eichler (Maastricht University)
Julien Gagneur (LMU, Gene Center)
Celia Greenwood (Lady Davis Institute for Medical Research)
Niels Richard Hansen (University of Copenhagen)
Dominik Janzing (Max-Planck-Institute for Intelligent Systems)
Samantha Kleinberg (Stevens Institute of Technology)
Aurélie Labbe (McGill University)
Steffen Lauritzen (University of Oxford)
Po-Ling Loh (University of Pennsylvania)
Sisi Ma (New York University)
Daniel Marbach (Université de Lausanne)
John Marioni (EMBL-EBI)
Lawrence McCandless (Simon Fraser University)
Joris Mooij (AMLab, University of Amsterdam)
Dana Pe’er (Columbia University )
Jonas Peters (MPI for Intelligent Systems)
Garvesh Raskutti (University of Wisconsin-Madison)
Thomas S. Richardson (University of Washington)
James Robins (Harvard School of Public Health)
Olli Saarela (University of Toronto)
Karen Sachs (Stanford University)
Shohei Shimizu (Osaka University)
Ricardo Silva (UCL)
George Davey Smith (University of Bristol)
Peter Spirtes (Carnegie Mellon University)
Oliver Stegle (EMBL-EBI)
Simon Tavare (University of Cambridge )
Jin Tian (Iowa State University)
Achim Tresch (Max Planck Institute)
Ioannis Tsamardinos (ICS – FORTH)

Best regards,

The organisers:

Robin Evans, University of Oxford
Chris Holmes, University of Oxford
Marloes Maathuis, ETH Zurich
Erica Moodie, McGill
Ilya Shpitser, Johns Hopkins
David Stephens, McGill
Caroline Uhler, MIT

We’re very grateful to the workshop sponsors: CRM, CANSSI and PIMS.

November 1, 2015

System Reconfiguration

Filed under: Announcement — bryantc @ 1:38 am

Sorry there will be no entries this Fall due to system reconfiguration.  Please bear with us, we will be back in February 2016.

August 11, 2015

Mid-Summer Greeting from the UCLA Causality Blog

Filed under: Announcement,Causal Effect,Counterfactual,General — moderator @ 6:09 pm

Friends in causality research,

This mid-summer greeting of UCLA Causality blog contains:
A. News items concerning causality research
B. Discussions and scientific results

1. The next issue of the Journal of Causal Inference is scheduled to appear this month, and the table of content can be viewed here.

2. A new digital journal “Observational Studies” is out this month (link) and its first issue is dedicated to the legacy of William Cochran (1909-1980).

My contribution to this issue can be viewed here:

See also comment 1 below.

3. A video recording of my Cassel Lecture at the SER conference, June 2015, Denver, CO, can be viewed here:

4. A video of a conversation with Robert Gould concerning the teaching of causality can be viewed on Wiley’s Statistics Views, link (2 parts, scroll down).

5. We are informed of the upcoming publication of a new book, Rex Kline “Principles and Practice of Structural Equation Modeling, Fourth Edition (link). Judging by the chapters I read, this book promises to be unique; it treats structural equation models for what they are: carriers of causal assumptions and tools for causal inference. Kudos, Rex.

6. We are informed of another book on causal inference: Imbens, Guido W.; Rubin, Donald B. “Causal Inference in Statistics, Social, and Biomedical Sciences: An Introduction” Cambridge University Press (2015). Readers will quickly realize that the ideas, methods, and tools discussed on this blog were kept out of this book. Omissions include: Control of confounding, testable implications of causal assumptions, visualization of causal assumptions, generalized instrumental variables, mediation analysis, moderation, interaction, attribution, external validity, explanation, representation of scientific knowledge and, most importantly, the unification of potential outcomes and structural models.

Given that the book is advertised as describing “the leading analysis methods” of causal inference, unsuspecting readers will get the impression that the field as a whole is facing fundamental obstacles, and that we are still lacking the tools to cope with basic causal tasks such as confounding control and model testing. I do not believe mainstream methods of causal inference are in such state of helplessness.

The authors’ motivation and rationale for this exclusion were discussed at length on this blog. See
“Are economists smarter than epidemiologists”

and “On the First Law of Causal Inference”

As most of you know, I have spent many hours trying to explain to leaders of the potential outcome school what insights and tools their students would be missing if not given exposure to a broader intellectual environment, one that embraces model-based inferences side by side with potential outcomes.

This book confirms my concerns, and its insularity-based impediments are likely to evoke interesting public discussions on the subject. For example, educators will undoubtedly wish to ask:

(1) Is there any guidance we can give students on how to select covariates for matching or adjustment?.

(2) Are there any tools available to help students judge the plausibility of ignorability-type assumptions?

(3) Aren’t there any methods for deciding whether identifying assumptions have testable implications?.

I believe that if such questions are asked often enough, they will eventually evoke non-ignorable answers.

7. The ASA has come up with a press release yesterday, recognizing Tyler VanderWeele’s new book “Explanation in Causal Inference,” winner of the 2015 Causality in Statistics Education Award

Congratulations, Tyler.

Information on nominations for the 2016 Award will soon be announced.

8. Since our last Greetings (Spring, 2015) we have had a few lively discussions posted on this blog. I summarize them below:

8.1. Indirect Confounding and Causal Calculus
(How getting too anxious to criticize do-calculus may cause you to miss an easy solution to a problem you thought was hard).
July 23, 2015

8.2. Does Obesity Shorten Life? Or is it the Soda?
(Discusses whether it was the earth that caused the apple to fall? or the gravitational field created by the earth?.)
May 27, 2015

8.3. Causation without Manipulation
(Asks whether anyone takes this mantra seriously nowadays, and whether we need manipulations to store scientific knowledge)
May 14, 2015

8.4. David Freedman, Statistics, and Structural Equation Models
(On why Freedman invented “response schedule”?)
May 6, 2015

8.5. We also had a few breakthroughs posted on our technical report page

My favorites this summer are these two:
because they deal with the tough and long-standing problem:
“How generalizable are empirical studies?”

Enjoy the rest of the summer

July 23, 2015

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

Filed under: Causal Effect,Definition,Discussion,do-calculus — eb @ 4:52 pm

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).]

    =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,



Chen, B. and Pearl, J. (2014). Graphical tools for linear structural equation modeling. Tech. Rep. R-432, , Department of Com- puter Science, University of California, Los Angeles, CA. Forthcoming, Psychometrika.
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.

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