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Causal Inference in Randomized and Non-Randomized Studies: the Definition, Identification, and Estimation of Causal Parameters

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Causal Inference in Randomized and Non-Randomized Studies: the Definition, Identification, and Estimation of Causal Parameters 1 Causal Inference in Randomized and Non-Randomized Studies: The Definition, Identification, and Estimation of Causal Parameters Michael E. Sobel INTRODUCTION between the illusory and non-illusory correlations, Yule invented partial correlation The distinction between causation and associ- to ‘control’ for the influence of a common ation has figured prominently in science and factor, arguing in context that because the philosophy for several hundred years at least, relationship between pauperism and out and, more recently, in statistical science as relief did not vanish when ‘controlling’ for well, indeed, since Galton, Pearson and, Yule poverty, this relationship could be deemed developed the theory of correlation. causal. A half century later, philosophers, Statisticians have pioneered two psychologists and social scientists (e.g., approaches to causal inference that have Reichenbach, 1956; Simon 1954; Suppes proven influential in the natural and 1970) rediscovered Yule’s approach to behavioral sciences. The oldest dates back distinguishing between causal and non-causal to Yule (1896), who wrote extensively about relationships, and econometricians (e.g., ‘illusory’ correlations, by which he meant Granger 1969) extended this idea to the correlations that should not be endowed time-series setting. Graphical models, path with a causal interpretation. To distinguish analysis and, more generally, structural 4 DESIGN AND INFERENCE equation models, when these methods are section, ‘Estimation of causal parameters in used to make causal inferences, also rely randomized studies’, discusses estimation. on this type of reasoning. The theory of The section ‘Mediation analyses’ takes up the experimental design, which emerges in topic of mediation, which is of special interest the 1920s and thereafter, and is associated to psychologists and prevention scientists. especially with Neyman (1923) and Fisher I show that the usual approach to mediation, (1925), forms the basis for a second approach which uses structural equation modeling, does to inferring causal relationships. Here, the not yield estimates of causal parameters, use of good design, especially randomization, even in randomized studies. Several other is emphasized, apparently obviating the need approaches to mediation, including principal to worry about spurious relationships. stratification and instrumental variables, are Despite these important contributions, dur- also considered. ing the majority of the twentieth century, most statisticians espoused the view that statistics had little to do with causation. CAUSATION AND PROBABILISTIC But the situation has reversed dramatically CAUSATION in the last 30 years, since Rubin (1974, 1977, 1978, 1980) rediscovered Neyman’s Regularity theories of causation are concerned potential outcomes notation and extended with the full (or philosophical) cause of an the theory of experimental design to obser- effect, by which is meant a set of conditions vational studies. Currently, there is a large that is sufficient (or necessary or necessary and growing inference in statistics on the and sufficient) for the effect to occur. This topic of causal inference and this second type of theory descends from Hume, who approach to inferring causal relationships claimed that causation (as it exists in the is coming to dominate the first approach, real world) consists only of the following: even in disciplines such as economics, (1) temporal priority, i.e., the cause must which rely on observational studies and precede the effect in time; (2) spatiotemporal where the first approach has traditionally contiguity, i.e., the cause and effect are dominated. ‘near’ in time and space; and (3) constant This chapter provides an introduction, conjunction, i.e., if the same circumstances tailored to the concerns of behavioral sci- are repeated, the same outcome will occur. entists, to this second approach to causal Many subsequent writers argued that Hume’s inference. Because causal inference is the analysis cannot distinguish between regular- act of making inferences about the causal ities that are not causal, such as a relation relation and notions of the causal relation between two events brought about by a differ, it is important to understand what common factor, and genuine causation. At the notion of causation is under consideration minimum, this suggests that Hume’s account when such an inference is made. Thus, in is incomplete. A number of philosophers the next section, I briefly review several [e.g., Bunge (1979) and Harré and Madden notions of causation and also briefly examine (1975)] argue that the causal relation is the approach to causal inference that derives generative. While this idea is appealing, from Yule. In the section ‘Unit and average especially to modern scientists who speak of causal effects’ the second approach, which mechanisms, attempts to elaborate this idea is built on the idea that a causal relation have not been entirely successful. Another should sustain a counterfactual conditional approach (examined later) is to require statement, is introduced, and a number of causal relationships to sustain counterfactual estimands of interest are defined. The section conditional statements. ‘Identification of causal parameters under Hume’s analysis is also deterministic, ignorable treatment assignment’discusses the and the literature on probabilistic causation identification of causal effects and the next that descends from Yule can be viewed as CAUSAL INFERENCE IN RANDOMIZED AND NON-RANDOMIZED STUDIES 5 an attempt to both relax this feature and whether causation is viewed as probabilistic distinguish between causal and non-causal in some inherent sense or if probability arises regularities. The basic idea is as follows. First, in some other way. The deficiencies of this there is a putative cause Z prior in some approach are evident in the psychological sense to an outcome Y. Further, Z and Y literature on casual modeling, where a variety are associated (correlated). However, if the of extra-mathematical considerations (such as Z − Y association vanishes when a (set of) model specification) are used to suggest that variable(s) X prior to Z is conditioned on model coefficients can be endowed with a (or in some accounts, if such a set exists), causal interpretation (see, for example, Sobel this is taken to mean that Z ‘does not cause’ 1995 on this point). Y, that is, the relationship is ‘spurious’. To By way of contrast to regularity theo- complete the picture, various examples where ries, manipulability theories view causes this criterion would seem to work well have as variables that can be manipulated, with been constructed. the outcome depending on the state of the Granger causation and structural equa- manipulated variable. Here, as opposed to tion models use this type of reasoning to specifying all the variables and the functional distinguish between empirical relationships relationship between these and the outcome that are regarded as causal and not causal. (which would constitute a successful causal For example, consider a structural equation account of a phenomenon under a regularity model, with X a vector of variables, Z an theory), the goal is more modest, to examine outcome occurring after X, and Y an outcome the ‘effect’ of a particular variable. See Sobel after Z. If the ‘direct effect’ of Y on Z is 0 (1995) for a reconciliation of these two (not 0), Z is not viewed (viewed) as a cause approaches. of Y. A sufficient condition for this direct Manipulability theories require the causal effect to be 0 is that Y and Z are conditionally relation to sustain a counterfactual conditional independent, given X. Of course, this same statement (e.g., eating the poison caused John kind of reasoning can be extended to the to die means that John ate the poison and consideration of other types of direct and died, but had John not eaten the poison, indirect effects. For example, consider the he would not have died). This is closer to case of a variable X associated with Z, with the way an experimentalist thinks of cau- X and Y conditionally independent, given Z, sation. However, many philosophers regard implying the ‘direct effect of X on Y is 0, and manipulability theories as anthropomorphic. Z and Y not conditionally independent given Further, many questions that scientists ask X, implying the ‘direct effect’ of Z on Y is are not amenable to experimentation, e.g, non-zero. Here there is an effect of X on Y, the effect of education on longevity or but the effect is indirect, through Z. the effect of marriage on happiness. This There are a number of problems with this would appear to seriously limit the value of approach. First and foremost, it confounds this approach for addressing real scientific causation with the act of inferring causation, questions. as evidenced by the fact that the criteria However, even without manipulating a per- above for inferring causation are typically son’s level of education, one might imagine put forth independently of any explicit notion that had this person’s level of education of the causal relation. As notions of the taken on a different value than that actually causal relation vary, this method of inferring realized, this person might also have a causation may be appropriate for some different outcome. This suggests adopting the notions of causation, for example, the case broader view that it is not the manipulation where causation is regarded as a predictive per se , but the idea that the causal variable relationship among variables, but not for could take on a different value than it others. Nor (because the nature of the casual actually takes on, which is key. This is the relation is not explicitly considered), is it clear idea underlying counterfactual theories of 6 DESIGN AND INFERENCE causation (e.g., Lewis 1973), where the closest show that these methods rest on a number of possible world to the one we live in serves as implausible assumptions. the basis for the counterfactual. Counterfactual theories also have their difficulties, both theoretical and practical. UNIT AND AVERAGE CAUSAL EFFECTS One criticism goes under the rubric of ‘preemption’.
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