Abstract
This chapter discusses the use of directed acyclic graphs (DAGs) for causal inference in the observational social sciences. It focuses on DAGs’ main uses, discusses central principles, and gives applied examples. DAGs are visual representations of qualitative causal assumptions: They encode researchers’ beliefs about how the world works. Straightforward rules map these causal assumptions onto the associations and independencies in observable data. The two primary uses of DAGs are (1) determining the identifiability of causal effects from observed data and (2) deriving the testable implications of a causal model. Concepts covered in this chapter include identification, d-separation, confounding, endogenous selection, and overcontrol. Illustrative applications then demonstrate that conditioning on variables at any stage in a causal process can induce as well as remove bias, that confounding is a fundamentally causal rather than an associational concept, that conventional approaches to causal mediation analysis are often biased, and that causal inference in social networks inherently faces endogenous selection bias. The chapter discusses several graphical criteria for the identification of causal effects of single, time-point treatments (including the famous backdoor criterion), as well identification criteria for multiple, time-varying treatments.
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Notes
- 1.
Identification is also relative to the set of observed variables. Identification may be possible for one set of observed variables, but not for another set. Mimicking the logic of secondary data analysis, here I assume that the analyst is given a set of observed variables (and hence that all other variables are unobserved). Identification analysis can also be used to ask what sets of variables should be observed to achieve identification.
- 2.
A detailed tutorial for reading counterfactuals (including nested counterfactuals) off a DAG is presented in Section 4.4 of Pearl (2012a).
- 3.
By contrast, marginally correlated error terms must be explicitly included in the causal DAG, since they represent common causes.
- 4.
These assumptions are, first, the causal Markov assumption, which states that a variable is independent of its nondescendants given its parents, and second, stability or faithfulness, which, among other things, rules out exact cancelation of positive and negative effects. In this chapter, I mostly use weak faithfulness, which is the reason for interpreting arrows as possible rather than certain direct effects. Glymour and Greenland (2008) give an accessible summary. See Pearl (2009) and Spirtes et al. ([1993] 2001) for technical details.
- 5.
Common cause confounding by unobserved variables is sometimes represented by a bi-headed dashed arrow.
- 6.
Terminology is in flux. The name “endogenous selection bias” highlights that the problem originates from conditioning on an endogenous variable. Others prefer “selection bias” (Hernán et al. 2004), “collider stratification bias” (Greenland 2003), “M-bias” (Greenland 2003), “Berkson’s [1946] bias,” “explaining away effect” (Kim and Pearl 1983), or “conditioning bias” (Morgan and Winship 2007). Simpson’s paradox (Hernán et al. 2011) and the Monty-Hall dilemma (Burns and Wieth 2004) involve specific examples of endogenous selection. The shared structure of some examples of endogenous selection bias has been known in the social sciences at least since Heckman (1976). For a comprehensive treatment, see Elwert and Winship (forthcoming).
- 7.
Endogenous selection bias is guaranteed if one assumes that positive and negative arrows do not cancel each other out exactly, i.e. if the DAG is faithful. Faithfulness is a mild assumption since exact cancelation is exceedingly unlikely in practice.
- 8.
Occasionally, a variable may be both a collider and a common cause. In that case, conditioning on the variable may eliminate confounding bias but induce endogenous selection bias, whereas not conditioning on the variable would lead to confounding bias yet eliminate endogenous selection bias (Greenland 2003). Nevertheless, the definitions of confounding and endogenous selection remain distinct.
- 9.
D-Connectedness necessarily implies statistical dependence if the DAG is faithful.
- 10.
- 11.
- 12.
The requirement not to condition on a descendant of a variable on a causal path is explained in the discussion of Fig. 13.9 below.
- 13.
Pearl (1995, 2009) and others use so-called do-operator notation to write \( P\left( {{y^T}} \right) \) as \( P\left( {Y=y\,|\,\mathrm{do}(T=t)} \right) \). The do-operator do(T = t) emphasizes that T is set to t by intervention (“doing”). \( P(Y=y\,|\,\mathrm{do}(T=t)) \) gives the post-intervention distribution of Y if one intervened on T to set it to some specific value t, that is, the counterfactual distribution of T.
- 14.
D 5 is a descendant of the collider T→D 2←e 2 (recall the implied existence of idiosyncratic error terms), which opens the noncausal path T---e 2→D 2→Y.
- 15.
The difference between Fig. 13.13a, b illustrates why identifying the magnitude of a causal effect is more difficult than testing the null of no effect. If one could condition on W R in Fig. 13.13a, then the absence of an association between M and W 0 conditional on E and W R would imply the absence of a causal effect M→W 0—the null can be tested. But if there is an effect M→W 0, as in Fig. 13.13b, then the observed association between M and W 0 given E and W R is biased for the causal effect M→W 0—the magnitude of the effect cannot be measured.
- 16.
- 17.
Elwert and Christakis (2008) use additional knowledge of the network topology to gage and remove the bias from residual confounding (i.e., if conditioning on H does not solve the problem).
- 18.
- 19.
DAGs for triadic networks would usually include separate variables for the characteristics of all three members of a generic triad. Obviously, the complexity of a DAG increases with the complexity of social structure. This is one reason why causal inference in social networks is a difficult problem.
- 20.
Here, we focus on causal effects of time-varying treatments that contrast predetermined treatment sequences. For two binary unit treatments, we can define six causal effects corresponding to the six pairwise contrasts between the four possible predetermined treatment sequences, here, (math, math), (math, English), (English, math), and (English, English). Note that some of these causal effects, such as (math, English) vs. (English, English), equal so-called controlled direct effects (Pearl 2001; Robins and Greenland 1992). The identification criteria discussed in this section apply to all causal effects of predetermined treatment sequences and hence to all controlled direct causal effects. See Bollen and Pearl (Chap. 15, this volume) and Wang and Sobel (Chap. 12, this volume) for mediation formulae and the identification of other types of (“natural” or “pure”) direct and indirect effects. See Robins and Richardson (2011) and Pearl (2012b) for graphical identification conditions of path-specific effects. See Robins and Hernán (2009) for yet other types of time-varying treatments, especially the distinction between static and dynamic time-varying treatment effects.
- 21.
Note that the joint causal effect of A 0 and A 1 is not the same as the total causal effect of A 0 plus the total causal effect of A 1, as is sometimes incorrectly thought.
- 22.
A minimally sufficient set is a sufficient set with the smallest number of variables. There may be multiple minimally sufficient sets.
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Acknowledgments
I thank Stephen Morgan, Judea Pearl, Tyler VanderWeele, Xiaolu Wang, Christopher Winship, and my students in Soc 952 at the University of Wisconsin for discussions and advice. Janet Clear and Laurie Silverberg provided editorial assistance. All errors are mine.
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Elwert, F. (2013). Graphical Causal Models. In: Morgan, S. (eds) Handbook of Causal Analysis for Social Research. Handbooks of Sociology and Social Research. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6094-3_13
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