Causal inference in statistics: An overview

1 Statistics SurveysVol. 3 (2009) 96 146 ISSN: 1935-7516 inference in statistics: An overview Judea PearlComputer Science DepartmentUniversity of California, Los Angeles, CA 90095 review presents empirical researcherswith recent advancesin Causal inference , and stresses the paradigmatic shifts that must be un-dertaken in moving from traditional statistical analysis to Causal analysis ofmultivariate data. Special emphasis is placed on the assumptions that un-derly all Causal inferences, the languages used in formulating those assump-tions, the conditional nature of all Causal and counterfactual claims, andthe methods that have been developed for the assessment of such advances are illustrated using a general theory of causation basedon the Structural Causal Model (SCM) described inPearl(2000a), whichsubsumes and unifies other approaches to causation, and provides a coher-ent mathematical foundation for the analysis of causes and particular, the paper surveys the development of mathematical tools forinferring (from a combination of data and assumptions) answers to threetypes of Causal queries.

2 (1) queries about the effects of potential interven-tions, (also called Causal effects or policy evaluation ) (2) queries aboutprobabilities of counterfactuals, (including assessmentof regret, attri-bution or causes of effects ) and (3) queries about direct and indirecteffects (also known as mediation ). Finally, the paper defines the formaland conceptual relationships between the structural and potential-outcomeframeworks and presents tools for a symbiotic analysis thatuses the strongfeatures of and phrases:Structuralequation models, confounding,graph-ical methods, counterfactuals, Causal effects, potential-outcome, mediation,policy evaluation, causes of September Introduction ..972 From association to causation .. The basic distinction: Coping with change .. Formulating the basic distinction .. Ramifications of the basic distinction .. Two mental barriers: Untested assumptions and new notation.

3 101 Portions of this paper are based on my bookCausality(Pearl, 2000, 2nd edition 2009),and have benefited appreciably from conversations with readers, students, and colleagues. This research was supported in parts by an ONR grant #N000-14-09-1-0665. This paper was accepted by Elja Arjas, Executive Editor for the Pearl/ Causal inference in statistics973 Structural models, diagrams, Causal effects, and counterfactuals .. Introduction to structural equation models .. From linear to nonparametric models and graphs .. Representing interventions .. Estimating the effect of interventions .. Causal effects from data and graphs .. Coping with unmeasured confounders .. Covariate selection the back-door criterion .. General control of confounding .. From identification to estimation .. Bayesianism and causality , or where do the probabilitiescome from? .. Counterfactual analysis in structural models.

4 An example: Non-compliance in clinical trials .. Defining the target quantity .. Formulating the assumptions Instrumental variables .. Bounding Causal effects .. Testable implications of instrumental variables ..1254 The potential outcome framework .. The Black-Box missing-data paradigm .. Problem formulation and the demystification of ignorability .. Combining graphs and potential outcomes ..1315 Counterfactuals at work .. Mediation: Direct and indirect effects .. Direct versus total effects: .. Natural direct effects .. Indirect effects and the Mediation Formula .. Causes of effects and probabilities of causation ..1366 Conclusions ..139 References ..1391. IntroductionThe questions that motivate most studies in the health, social and behavioralsciences are not associational but Causal in nature. For example, what is theefficacy of a given drug in a given population?

5 Whether data canprove anemployer guilty of hiring discrimination? What fraction ofpast crimes couldhave been avoided by a given policy? What was the cause of death of a givenindividual, in a specific incident? These arecausalquestions because they requiresome knowledge of the data-generating process; they cannotbe computed fromthe data alone, nor from the distributions that govern the , although much of the conceptual framework and algorithmictools needed for tackling such problems are now well established, they are hardlyknown to researchers who could put them into practical use. The main reason iseducational. Solving Causal problems systematically requires certain extensionsJ. Pearl/ Causal inference in statistics98in the standard mathematical language of statistics, and these extensions are notgenerally emphasized in the mainstream literature and education. As a result,large segments of the statistical research community find ithard to appreciateand benefit from the many results that Causal analysis has produced in the pasttwo decades.

6 These results rest on contemporary advances infour areas:1. Counterfactual analysis2. Nonparametric structural equations3. Graphical models4. Symbiosis between counterfactual and graphical survey aims at making these advances more accessible tothe general re-search community by, first, contrasting Causal analysis with standard statisticalanalysis, second, presenting a unifying theory, called structural, within whichmost (if not all) aspects of causation can be formulated, analyzed and compared,thirdly, presenting a set of simple yet effective tools, spawned by the structuraltheory, for solving a wide variety of Causal problems and, finally, demonstratinghow former approaches to Causal analysis emerge as special cases of the generalstructural this end, Section2begins by illuminating two conceptual barriers that im-pede the transition from statistical to Causal analysis.

7 (i) coping with untestedassumptions and (ii) acquiring new mathematical these bar-riers, introduces the fundamentals of the structural theoryof causation, with emphasis on the formal representation ofcausal assump-tions, and formal definitions of Causal effects, counterfactuals and joint prob-abilities of counterfactuals. these modeling fundamentals torepresent interventions and develop mathematical tools for estimating causaleffects ( ) and counterfactual quantities ( ). These tools aredemonstrated by attending to the analysis of instrumental variables and theirrole in bounding treatment effects in experiments marred by noncompliance( ).The tools described in this section permit investigators tocommunicate causalassumptions formally using diagrams, then inspect the diagram and1. Decide whether the assumptions made are sufficient for obtaining consis-tent estimates of the target quantity;2.

8 Derive (if the answer to item1is affirmative) a closed-form expression forthe target quantity in terms of distributions of observed quantities; and3. Suggest (if the answer to item 1 is negative) a set of observations and ex-periments that, if performed, would render a consistent estimate these tools to those used in the potential-outcome frame-work, and offers a formal mapping between the two frameworks and a symbiosis( ) that exploits the best features of both. Finally, the benefit of thissymbiosis is demonstrated in Section5, in which the structure-based logic ofcounterfactuals is harnessed to estimate Causal quantities that cannot be de-fined within the paradigm of controlled randomized experiments. These includedirect and indirect effects, the effect of treatment on the treated, and ques-J. Pearl/ Causal inference in statistics99tions of attribution, , whether one event can be deemed responsible From association to The basic distinction: Coping with changeThe aim of standard statistical analysis, typified by regression, estimation, andhypothesis testing techniques, is to assess parameters of adistribution fromsamples drawn of that distribution.

9 With the help of such parameters, one caninfer associations among variables, estimate beliefs or probabilities of past andfuture events, as well as update those probabilities in light of new evidenceor new measurements. These tasks are managed well by standard statisticalanalysis so long as experimental conditions remain the same. Causal analysisgoes one step further; its aim is to infer not only beliefs or probabilities understatic conditions, but also the dynamics of beliefs underchanging conditions,for example, changes induced by treatments or external distinction implies that Causal and associational concepts do not is nothing in the joint distribution of symptoms and diseases to tell usthat curing the former would or would not cure the latter. More generally, thereis nothing in a distribution function to tell us how that distribution would differif external conditions were to change say from observational to experimentalsetup because the laws of probability theory do not dictatehow one propertyof a distribution ought to change when another property is modified.

10 This in-formation must be provided by Causal assumptions which identify relationshipsthat remain invariant when external conditions considerations imply that the slogan correlation does not imply cau-sation can be translated into a useful principle: one cannot substantiate causalclaims from associations alone, even at the population level behind everycausal conclusion there must lie some Causal assumption that is not testablein observational Formulating the basic distinctionA useful demarcation line that makes the distinction between associational andcausal concepts crisp and easy to apply, can be formulated asfollows. An as-sociational concept is any relationship that can be defined in terms of a jointdistribution of observed variables, and a Causal concept isany relationship thatcannot be defined from the distribution alone. Examples of associational con-cepts are: correlation, regression, dependence, conditional independence, like-lihood, collapsibility, propensity score, risk ratio, odds ratio, marginalization,1 The methodology of Causal discovery (Spirtes et al.)