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A Crash Course in Good and Bad Controls

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Title: A Crash Course in Good and Bad Controls
Language: English
Authors: Carlos Cinelli (ORCID 0000-0002-2021-7739), Andrew Forney (ORCID 0000-0002-6366-1290), Judea Pearl
Source: Sociological Methods & Research. 2024 53(3):1071-1104.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 34
Publication Date: 2024
Sponsoring Agency: National Science Foundation (NSF), Division of Information and Intelligent Systems (IIS)
Office of Naval Research (ONR) (DOD)
Contract Number: 2106908
N0001417S12091
N000142112351
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Regression (Statistics), Robustness (Statistics), Error of Measurement, Testing Problems, Causal Models
DOI: 10.1177/00491241221099552
ISSN: 0049-1241
1552-8294
Abstract: Many students of statistics and econometrics express frustration with the way a problem known as "bad control" is treated in the traditional literature. The issue arises when the addition of a variable to a regression equation produces an unintended discrepancy between the regression coefficient and the effect that the coefficient is intended to represent. Avoiding such discrepancies presents a challenge to all analysts in the data intensive sciences. This note describes graphical tools for understanding, visualizing, and resolving the problem through a series of illustrative examples. By making this "crash course" accessible to instructors and practitioners, we hope to avail these tools to a broader community of scientists concerned with the causal interpretation of regression models.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1434877
Database: ERIC
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  Value: <anid>AN0178879684;som01aug.24;2024Aug09.05:11;v2.2.500</anid> <title id="AN0178879684-1">A Crash Course in Good and Bad Controls </title> <p>Many students of statistics and econometrics express frustration with the way a problem known as "bad control" is treated in the traditional literature. The issue arises when the addition of a variable to a regression equation produces an unintended discrepancy between the regression coefficient and the effect that the coefficient is intended to represent. Avoiding such discrepancies presents a challenge to all analysts in the data intensive sciences. This note describes graphical tools for understanding, visualizing, and resolving the problem through a series of illustrative examples. By making this "crash course" accessible to instructors and practitioners, we hope to avail these tools to a broader community of scientists concerned with the causal interpretation of regression models.</p> <p>Keywords: causal inference; bad controls; back-door criterion; DAG; regression</p> <hd id="AN0178879684-2">Introduction</hd> <p>Students, data analysts, and empirical social scientists have likely encountered the problem of "bad controls" ([<reflink idref="bib1" id="ref1">1</reflink>], [<reflink idref="bib3" id="ref2">3</reflink>]). The problem arises when an analyst needs to decide whether or not the addition of a variable to a regression equation helps getting estimates closer to the parameter of interest. Analysts have long known that some variables, when added to the regression equation, can produce unintended discrepancies between the regression coefficient and the effect that the coefficient is expected to represent. Such variables have become known as "bad controls," to be distinguished from "good controls" (also known as "confounders" or "deconfounders") which are variables that must be added to the regression equation to eliminate what came to be known as "omitted variable bias"([<reflink idref="bib1" id="ref3">1</reflink>]; [<reflink idref="bib65" id="ref4">65</reflink>]; [<reflink idref="bib17" id="ref5">17</reflink>]).</p> <p>The problem of "bad controls" however, has not received <emph>systematic</emph> attention in the standard statistics and econometrics literature. While most of the widely adopted textbooks discuss the problem of omitting "relevant" variables, they do not provide guidance on deciding which variables are relevant, nor which variables, if included in the regression, could induce, or worsen existing biases.[<reflink idref="bib4" id="ref6">4</reflink>] Researchers exposed only to this literature may get the impression that adding "more controls" to a regression model is always better. The few exceptions that do discuss the problem of "bad controls" unfortunately cover only a narrow aspect of the problem (e.g. [<reflink idref="bib1" id="ref7">1</reflink>], [<reflink idref="bib3" id="ref8">3</reflink>]; [<reflink idref="bib73" id="ref9">73</reflink>]; [<reflink idref="bib36" id="ref10">36</reflink>]; [<reflink idref="bib28" id="ref11">28</reflink>]). Typical is the discussion found in [<reflink idref="bib1" id="ref12">1</reflink>], p.64)</p> <p>Some variables are bad controls and should not be included in a regression model, even when their inclusion might be expected to change the short regression coefficients. Bad controls are variables that are themselves outcome variables in the notional experiment at hand. That is, bad controls might just as well be dependent variables too. Good controls are variables that we can think of having been fixed at the time the regressor of interest was determined.</p> <p>Here, "good controls" are defined as variables that are thought to be unaffected by the treatment, whereas "bad controls" are variables that could be in principle affected by the treatment. Similar discussion can be found in [<reflink idref="bib57" id="ref13">57</reflink>]) and [<reflink idref="bib59" id="ref14">59</reflink>]), for qualifying a variable for inclusion in propensity score analysis, as well as in [<reflink idref="bib72" id="ref15">72</reflink>]). Some authors (e.g, [<reflink idref="bib73" id="ref16">73</reflink>]; [<reflink idref="bib28" id="ref17">28</reflink>]) briefly warn about the potential of bias amplification of certain pre-treatment variables, but do not elaborate further. Although an improvement over an absence of discussion, these conditions are neither necessary nor sufficient for deciding whether a variable is a good control.</p> <p>Recent advances in graphical models have produced simple criteria to distinguish "good" from "bad" controls; these range from necessary and sufficient conditions for deciding which set of variables should be adjusted for to identify the causal effect of interest (e.g, the back-door criterion and adjustment criterion in [<reflink idref="bib43" id="ref18">43</reflink>]), [<reflink idref="bib62" id="ref19">62</reflink>]) and [<reflink idref="bib54" id="ref20">54</reflink>])), to deciding which, among a set of valid adjustment sets, would yield more precise estimates ([<reflink idref="bib39" id="ref21">39</reflink>]; [<reflink idref="bib38" id="ref22">38</reflink>]; [<reflink idref="bib31" id="ref23">31</reflink>]; [<reflink idref="bib69" id="ref24">69</reflink>]; [<reflink idref="bib32" id="ref25">32</reflink>]; [<reflink idref="bib58" id="ref26">58</reflink>]; [<reflink idref="bib70" id="ref27">70</reflink>]). The purpose of this note is to provide practicing analysts a concise, simple, and <emph>visual</emph> summary of these criteria through illustrative examples.</p> <hd id="AN0178879684-3">Preliminaries and basic terminology</hd> <p>Causal diagrams, and more specifically directed acyclic graphs (DAGs), have become popular in the social and health sciences for explaining and resolving difficult problems of causal inference in a rigorous, yet accessible manner. Many introductions to DAGs have now been published in a number of academic fields, such as sociology ([<reflink idref="bib25" id="ref28">25</reflink>]; [<reflink idref="bib42" id="ref29">42</reflink>]), economics ([<reflink idref="bib35" id="ref30">35</reflink>]; [<reflink idref="bib22" id="ref31">22</reflink>]), psychology ([<reflink idref="bib55" id="ref32">55</reflink>]), epidemiology ([<reflink idref="bib29" id="ref33">29</reflink>]; [<reflink idref="bib33" id="ref34">33</reflink>]) and statistics ([<reflink idref="bib51" id="ref35">51</reflink>]; [<reflink idref="bib52" id="ref36">52</reflink>]). Here we assume that readers are familiar with the basic notions of causal inference, DAGs, and in particular "path-blocking" as well as back-door paths. For those who need to refresh these notions, we provide a gentle introduction in the appendix. Still, given the simplicity of our illustrative examples, even the uninitiated reader will be able to understand and benefit from the main lessons of this crash course.</p> <p>Briefly, causal DAGs provide a parsimonious representation of the qualitative aspects of the data generating process. Letters (e.g, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ) represent random variables, and arrows, such as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , denote a (possible) direct causal effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . No assumptions need to be made regarding the functional form of the causal relationships, nor about the distribution of variables. For this crash course, it is important to recall the three main sources of association that form the building blocks of a DAG, and when these are closed or opened:</p> <p></p> <ulist> <item> <emph>Mediators</emph>, or chains, are patterns of the form <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , meaning that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml><emph>causally</emph> affects <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> through the mediator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . Conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in a chain <emph>blocks</emph> (closes) this flow of association.</item> <p></p> <item> <emph>Common causes</emph>, or forks, are patterns of the form <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><mi>Z</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , meaning <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> share a common cause (a confounder) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , thus inducing a <emph>non-causal</emph> association between both variables. Conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in a fork <emph>blocks</emph> this flow of association.</item> <p></p> <item> <emph>Common effects</emph>, or colliders, are patterns of the form <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">←</mo><mi>Y</mi></math> </ephtml> , meaning that both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> share a common effect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . Contrary to the other two variables, by default a common effect does not induce an association between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . However, conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> induces a <emph>non-causal</emph> association between both variables.</item> </ulist> <p>Moreover, one important fact to keep in mind is that controlling for a descendant of a variable is equivalent to "partially" controlling for that variable. Any arbitrary path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> (consisting of a sequence of mediators, common causes, or colliders) will be blocked conditional on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> if, and only if, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a common cause or mediator along the path, or if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> contains a collider and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is <emph>not</emph> that collider, nor any of its descendants. We say that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> </ephtml> -separates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> blocks (closes) all paths from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ; <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> </ephtml> -separation implies that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> are conditionally independent given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> .</p> <p>Note that causal paths from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> are paths of the form <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mo>...</mo><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , namely, those consisting of a sequence of (possibly empty) mediators. All other paths are non-causal, and may induce "spurious" associations between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . In particular, for a given variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , we call "back-door" paths those confounding paths that begin with an arrow pointing into <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> . If we are interested, then, in estimating the causal effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , our task is conceptually simple: we must block all spurious paths between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , and we must not perturb any of the causal paths between them. This will be our guiding principle for deciding whether or not <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> should be included in the regression equation, and it characterizes the essence of the graphical conditions known as the <emph>back-door</emph> criterion and the <emph>adjustment</emph> criterion ([<reflink idref="bib43" id="ref37">43</reflink>]; [<reflink idref="bib62" id="ref38">62</reflink>]; [<reflink idref="bib54" id="ref39">54</reflink>]). Readers can find the formal statements of these graphical criteria in the appendix.</p> <hd id="AN0178879684-4">Illustrative examples</hd> <p>In the following set of models, the target of our analysis is the <emph>average causal effect</emph> (ACE) of a treatment <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on an outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , which stands for the expected increase of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> in response to a unit increase in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> due to an <emph>intervention</emph>. Observed variables will be designated by black dots and unobserved variables by white empty circles. Variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , highlighted in red, will represent the variable whose inclusion in the regression equation is to be decided, with "good control" standing for bias reduction, "bad control" standing for bias increase, and "neutral control" when the addition of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> neither increases nor decreases the asymptotic bias. For this last case, we will also make brief remarks about how <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> could affect the precision of the ACE estimate. Readers accustomed with the potential outcomes framework should know that deciding whether <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a "good control" is equivalent to deciding whether ignorability of treatment assignment holds, conditional on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . Readers who prefer to see algebraic derivations can find in the appendix analytical expressions for each graph, under the assumption of linearity[<reflink idref="bib5" id="ref40">5</reflink>] (the problem of "bad controls" is, however, non-parametric, i.e, it holds regardless of functional form assumptions).</p> <hd id="AN0178879684-5">Models 1, 2 and 3 – Good Controls (blocking back-door paths)</hd> <p>In Model 1 (Figure 1a), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> stands for a common cause of both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . Once we control for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , we block the back-door path from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , producing an unbiased estimate of the ACE. In Models 2 and 3, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is not a common cause of both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , and therefore, not a traditional "confounder" as in Model 1. Nevertheless, controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> blocks the back-door path from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> due to the unobserved confounder <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> , and again, produces an unbiased estimate of the ACE.</p> <hd id="AN0178879684-6">Models 4, 5 and 6 – Good Controls (blocking back-door paths)</hd> <p>When thinking about possible threats of confounding, modelers need to keep in mind that common causes of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and any mediator (between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ) also confound the effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . Therefore, Models 4, 5 and 6 (Figure 2) are analogous to Models 1, 2 and 3—controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> blocks the back-door path from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> and produces an unbiased estimate of the ACE.</p> <hd id="AN0178879684-7">Model 7 – Bad Control (M-bias)</hd> <p>We now encounter our first "bad control" (Figure 3). Here <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is correlated with the treatment and the outcome and it is also a "pre-treatment" variable. Traditional econometrics textbooks usually deem pre-treatment variables "good controls" ([<reflink idref="bib1" id="ref41">1</reflink>], [<reflink idref="bib3" id="ref42">3</reflink>]; [<reflink idref="bib36" id="ref43">36</reflink>]). Careful analysis, however, reveals that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a "bad control." Controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> will induce bias by opening the back-door path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><mi>U</mi><mn>1</mn><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">←</mo><mi>U</mi><mn>2</mn><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , thus spoiling a previously unbiased estimate of the ACE. This structure is known as the "M-bias," and has spurred several controversies. Readers can find further discussion in [<reflink idref="bib44" id="ref44">44</reflink>], p. 186), [<reflink idref="bib63" id="ref45">63</reflink>]), [<reflink idref="bib46" id="ref46">46</reflink>]), [<reflink idref="bib64" id="ref47">64</reflink>]), [<reflink idref="bib59" id="ref48">59</reflink>]), [<reflink idref="bib23" id="ref49">23</reflink>]), and [<reflink idref="bib50" id="ref50">50</reflink>]).</p> <hd id="AN0178879684-8">Undecidable—"Damned if you do, damned if you don't."</hd> <p>Consider a variation of Model 7 (Figure 4) such that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> has a direct effect on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , as the one presented in Figure 4. Note that now we have an open back-door path, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><msub><mi>U</mi><mn>1</mn></msub><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , and the unadjusted estimate is no longer unbiased. While adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> closes this back-door path, it also opens back-door the path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><msub><mi>U</mi><mn>1</mn></msub><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">←</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , as we had in our previous example. In either case, the causal effect is not identified, and whether adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> reduces or increases the absolute value of the bias cannot be determined without further assumptions (see appendix). In this case, progress can be made with sensitivity analyses, by, for instance, positing plausible bounds on the the strength of the direct effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , or on the strength of the effects of the latent variables ([<reflink idref="bib19" id="ref51">19</reflink>]; [<reflink idref="bib17" id="ref52">17</reflink>]).</p> <p>Graph: Figure 1. (a) Model 1; (b) Model 2; (c) Model 3.</p> <p>Graph: Figure 2. (a) Model 4; (b) Model 5; (c) Model 6.</p> <p>Graph: Figure 3. Model 7</p> <p>Graph: Figure 4. Variation of Model 7</p> <p>Graph: Figure 5. Model 8</p> <hd id="AN0178879684-9">Model 8 – Neutral Control (possibly good for precision)</hd> <p>In Model 8 (Figure 5), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is not a confounder nor does it block any back-door paths. Likewise, controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> does not open any back-door paths from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . Thus, in terms of asymptotic bias, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a "neutral control." Analysis shows, however, that controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> reduces the variation of the outcome variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , and helps to improve the precision of the ACE estimate in finite samples ([<reflink idref="bib31" id="ref53">31</reflink>]; [<reflink idref="bib69" id="ref54">69</reflink>]; [<reflink idref="bib32" id="ref55">32</reflink>]; [<reflink idref="bib58" id="ref56">58</reflink>]).</p> <hd id="AN0178879684-10">Model 9 – Neutral Control (possibly bad for precision)</hd> <p>Similar to the previous case, in Model 9 (Figure 6) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is "neutral" in terms of bias reduction. However, controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> will reduce the variation of the treatment variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and so may hurt the precision of the estimate of the ACE in finite samples ([<reflink idref="bib32" id="ref57">32</reflink>], Corollary 3.4). As a general rule of thumb, parents of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> which are not necessary for identification are harmful for the asymptotic variance of the estimator; on the other hand, parents of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> which do not spoil identification are beneficial. See [<reflink idref="bib32" id="ref58">32</reflink>]) for recent developments in graphical criteria for efficient estimation via adjustment in linear models. Remarkably, these conditions also have been shown to hold in non-parametric models for a broad class of non-parametric estimators ([<reflink idref="bib58" id="ref59">58</reflink>]).</p> <hd id="AN0178879684-11">Model 10 – Bad Control (bias amplification)</hd> <p>We now encounter our second "pre-treatment" "bad control" (Figure 7), due to a phenomenon called "bias amplification" ([<reflink idref="bib9" id="ref60">9</reflink>]; [<reflink idref="bib71" id="ref61">71</reflink>]; [<reflink idref="bib48" id="ref62">48</reflink>], [<reflink idref="bib47" id="ref63">47</reflink>], [<reflink idref="bib49" id="ref64">49</reflink>]; [<reflink idref="bib41" id="ref65">41</reflink>]; [<reflink idref="bib65" id="ref66">65</reflink>]). Naive control for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in this model will not only fail to deconfound the effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , but, in linear models, will amplify any existing bias.</p> <hd id="AN0178879684-12">Models 11 and 12 – Bad Controls (overcontrol bias)</hd> <p>If our target quantity is the ACE, we want to leave all channels through which the causal effect flows "untouched." In Model 11 (Figure 8a), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a mediator of the causal effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . Controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> will block the very effect we want to estimate (the <emph>total</emph> effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ), thus biasing our estimates (this is usually known as "overcontrol bias"). In Model 12 (Figure 8b), although <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is not itself a mediator of the causal effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is equivalent to partially controlling for the mediator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> , and will thus bias our estimates. Models 11 and 12 violate the back-door criterion ([<reflink idref="bib44" id="ref67">44</reflink>]), which excludes controls that are descendants of the treatment along paths to the outcome. Note that the same conclusions would hold if we had an extra direct causal path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> .</p> <hd id="AN0178879684-13">Total versus direct effects</hd> <p>The previous considerations assume the researcher is interested in the <emph>total</emph> effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , as given by the ACE. If, instead, interest lies in the <emph>controlled direct effect</emph> (CDE) of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> (i.e, the effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> while holding <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> constant by intervention, see [<reflink idref="bib44" id="ref68">44</reflink>], [<reflink idref="bib48" id="ref69">48</reflink>]) as well as the appendix), then adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in Model 11 (Figure 8a) would indeed be appropriate. However, consider a variation of Model 11 (Figure 9) with an unobserved confounder of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> , as shown in Figure 9. First notice that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml><emph>does not</emph> confound the effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , and thus our ACE estimate remains unbiased as it were in Model 11, so long as we do not adjust for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . On the other hand, here adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> now opens the colliding path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">←</mo><mi>U</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , thus biasing the CDE estimate.</p> <p>Graph: Figure 6. Model 9</p> <p>Graph: Figure 7. Model 10</p> <p>Graph: Figure 8. (a) Model 11; (b) Model 12</p> <p>Graph: Figure 9. Variation of Model 11</p> <p>Graph: Figure 10. Model 13</p> <p>Graph: Figure 11. (a) Model 14; (b) Model 15</p> <p>Graph: Figure 12. (a) Model 16; (b) Model 17</p> <p>Graph: Figure 13. Model 18</p> <hd id="AN0178879684-14">Model 13 – Neutral Control (possibly good for precision)</hd> <p>At first look, Model 13 (Figure 10) might seem similar to Model 12, and one may think that adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> would bias the effect estimate, by restricting variations of the mediator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> . However, the key difference here is that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a cause, not an effect, of the mediator (and, consequently, also a cause of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ). Thus, Model 13 is analogous to Model 8, and so controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> will be neutral in terms of bias and may increase the precision of the ACE estimate in finite samples. Readers can find further discussion of this case in [<reflink idref="bib49" id="ref70">49</reflink>]).</p> <hd id="AN0178879684-15">Models 14 and 15 – Neutral Controls (possibly helpful in the case of selection bias)</hd> <p>Contrary to folklore, not all "post-treatment" variables are inherently bad controls. In Models 14 and 15 (Figure 11) controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> does not open any confounding paths between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . Thus, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is neutral in terms of bias. However, controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> does reduce the variation of the treatment variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and so may hurt the precision of the ACE estimate in finite samples. Additionally, in Model 15, suppose one has only samples with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><mi>w</mi></math> </ephtml> recorded (a case of selection bias[<reflink idref="bib6" id="ref71">6</reflink>], which we explain next). In this case, controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> can help to obtain the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> </ephtml> -specific effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> by blocking the colliding path due to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> </ephtml> . In linear models, controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> actually fully recovers the ACE (see appendix).</p> <hd id="AN0178879684-16">Models 16 and 17 – Bad Controls (selection bias)</hd> <p>Contrary to Models 14 and 15, here (Figure 12) controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is no longer harmless, and induces what is classically known as "selection bias" or "collider stratification bias." Adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in Model 16 opens the colliding path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">←</mo><mi>U</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> and so biases the ACE. In Model 17, adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> not only opens the path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">←</mo><mi>Y</mi></math> </ephtml> , but also the colliding path due to the latent parents of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , thus biasing the ACE and motivating our final example.</p> <hd id="AN0178879684-17">Model 18 – Bad Control (case-control bias)</hd> <p>In our last example (Figure 13), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is not in the causal pathway from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is not a direct cause of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is connected to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . Thus, one might surmise that, as in Model 8, controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is harmless for identification, and perhaps beneficial for finite sample efficiency. However, controlling for the effects of the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> will induce bias in the estimate of the ACE, even without the direct arrow <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Z</mi></math> </ephtml> , thus making <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> a "bad control." This happens because <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is in fact a descendant of a collider: the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> itself. A visual explanation of this phenomenon using "virtual colliders" can be found in [<reflink idref="bib44" id="ref72">44</reflink>], Sec. 11.3). The same phenomenon can also be explained by explicitly drawing the potential outcomes on the DAG (see both explanations in the appendix). Model 18 is special case of selection bias usually known as "case-control" bias. Finally, although controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> will generally bias numerical estimates of the ACE, it does have an exception when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> has no causal effect on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> . In this scenario, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> is still <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> </ephtml> -separated from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> even after conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . Thus, adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is valid for testing whether the effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> is <emph>zero</emph>.</p> <hd id="AN0178879684-18">Bad controls in applied research</hd> <p>Despite their simplicity, these illustrative examples should provide practitioners with a principled framework to understand many problems found in real world applications. To demonstrate, we now briefly present three cases of bad controls discussed in applied research, coming from diverse areas such as epidemiology, sociology, and economics.</p> <p>The birth-weight paradox ([<reflink idref="bib34" id="ref73">34</reflink>]). Infants born to smokers were found to have higher risks of mortality than infants born to non-smokers. However, among infants with low birth-weight (LBW), this relationship was reversed. This reversal of effects has created many controversies in epidemiology—does it mean that maternal smoking is beneficial for LBW infants? A plausible reason for such a finding could simply be collider stratification bias, as shown in Model 16. Here <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> is maternal smoking, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> infant mortality, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> birth-weight, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> stands for unobserved risk-factors (such as birth-defects and malnutrition), that could also affect birth-weight. Note that stratifying the analysis by birth-weight would induce a spurious association between smoking and mortality due to the competing risk-factors. LBW infants of non-smokers need to have alternative causes for their LBW (such as malnutrition), and such causes could also lead to higher mortality.</p> <p>Homophily bias in social network analysis ([<reflink idref="bib26" id="ref74">26</reflink>]). An important task in the causal inference of social networks is to estimate the causal effects of social contagion, also known as "interpersonal effects." However, social ties in the analysis of social networks may be pre-treatment colliders as exemplified in the "M-bias" structure of Model 7. Suppose we are interested in assessing whether the civic engagement of individual <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ) leads to the civic engagement of individual <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> </ephtml> in the subsequent time period ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ). Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> denote whether such individuals are friends, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>2</mn></msub></math> </ephtml> denote the personal characteristics (such as altruism) of individuals 1 and 2, respectively. Here, the social tie <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a collider, and computing the association of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> between friends ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mn>1</mn></math> </ephtml> ) would bias the interpersonal causal effects in civic engagement.</p> <p>The Antebellum Puzzle ([<reflink idref="bib60" id="ref75">60</reflink>]). An interesting puzzle of economic history is the fact that, during the nineteenth century in Britain and the United States, the average height of adult men fell even though the economic conditions of these countries improved alongside childhood nutrition. One possible explanation for such a paradoxical finding is selection bias in the forms of Models 17 and 18 wherein researchers using data from individuals enlisted in the military or in prison are effectively conditioning on colliders. For military records, consider Model 18, and let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> denote childhood nutrition, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> adult height, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> an indicator of whether the individual was enlisted in the military. The causal path from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> represents the fact that taller men may have better opportunities in the civilian market, and thus shorter men were more likely to enlist. Restricting the analysis to those enlisted in the military is therefore equivalent to controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , and leads to selection bias. Now for prison records, consider Model 17, and let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> be an indicator of whether the individual was arrested. Here one could argue that both childhood nutrition and adult height have pathways to committing a crime through socio-economic opportunities, thus again leading to selection bias.</p> <p>These examples are by no means exhaustive. Readers can find other interesting cases across applied sciences, such as: the threats of collider bias in understanding risk factors of COVID-19 ([<reflink idref="bib30" id="ref76">30</reflink>]); the "Obesity paradox," in which obesity appears to benefit individuals who survive heart failure ([<reflink idref="bib6" id="ref77">6</reflink>]); and examples of "bad controls" due to adjustment of mediators and colliders in multigenerational mobility ([<reflink idref="bib12" id="ref78">12</reflink>]), anesthesiology research ([<reflink idref="bib27" id="ref79">27</reflink>]) or animal science ([<reflink idref="bib8" id="ref80">8</reflink>]). Further discussion of bad controls in theoretical and applied works can be found in [<reflink idref="bib53" id="ref81">53</reflink>]).</p> <hd id="AN0178879684-19">Multiple controls</hd> <p>When considering multiple controls, the status of a single control as "good" or "bad" may change depending on the context of the other variables under consideration. Nevertheless, the main lessons from our illustrative examples remain. A set of control variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> will be "good" if: (i) it blocks all non-causal paths from the treatment to the outcome; (ii) it leaves any mediating paths from the treatment to the outcome "untouched" (since we are interested in the total effect); and, (iii) it does not open new spurious paths between the treatment and the outcome (e.g., due to colliders). As to efficiency considerations, we should give preference to those variables "closer" to the outcome, in opposition to those closer to the treatment—so long as, of course, this does not spoil identification.</p> <p>Finally, we remind readers that, when considering models with more complicated structures, one can always resort to specialized computer programs. Open-source software implementing algorithms for selecting adjustment sets can be found in the R packages pcalg ([<reflink idref="bib37" id="ref82">37</reflink>]), dagitty ([<reflink idref="bib66" id="ref83">66</reflink>])[<reflink idref="bib7" id="ref84">7</reflink>], and causaleffect ([<reflink idref="bib68" id="ref85">68</reflink>]). Users familiar with the software SAS may find the procedure CAUSALGRAPH useful ([<reflink idref="bib67" id="ref86">67</reflink>]). A web application implementing the methods discussed in [<reflink idref="bib7" id="ref87">7</reflink>]) is also available.[<reflink idref="bib8" id="ref88">8</reflink>] In other words, given a causal diagram, the problem of deciding which variables are good or bad controls has been automatized.</p> <hd id="AN0178879684-20">Beyond adjustment</hd> <p>Here we have focused on the identification of causal effects through simple covariate adjustment, classifying <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> as a "good" or "bad" control according to this criterion. However, other identification opportunities may be available. For instance, going back again to Model 10, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is what is usually known as an "instrumental variable." In this case, while <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is indeed a "bad control," it can still be used as an instrument to bound or point identify causal effects under certain parametric assumptions, albeit using a different formula ([<reflink idref="bib74" id="ref89">74</reflink>]; [<reflink idref="bib11" id="ref90">11</reflink>]; [<reflink idref="bib4" id="ref91">4</reflink>]; [<reflink idref="bib2" id="ref92">2</reflink>]; [<reflink idref="bib5" id="ref93">5</reflink>]; [<reflink idref="bib13" id="ref94">13</reflink>]). More generally, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>o</mi></math> </ephtml> -calculus provides a complete solution for the task of non-parametric identification of treatment effects in causal DAGs, beyond the simple adjustment formula ([<reflink idref="bib43" id="ref95">43</reflink>]; [<reflink idref="bib61" id="ref96">61</reflink>]; [<reflink idref="bib44" id="ref97">44</reflink>]). In certain instances, such as the "front-door" criterion, this allows exploiting post-treatment variables for identification ([<reflink idref="bib44" id="ref98">44</reflink>]). Further details on the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>o</mi></math> </ephtml> -calculus should be the topic of a separate crash course.</p> <hd id="AN0178879684-21">Sensitivity analysis</hd> <p>In real world applications, it can be the case that the causal effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> cannot be identified from the DAG structure alone. When that happens, without further assumptions, it is usually not possible to determine whether including <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in the regression equation will reduce or increase the absolute value of the bias, as we have seen in the example of Figure 4. In such cases, claims about the causal effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> must rely on knowledge beyond the constraints of the DAG, such as plausibility judgments (i) on the direct effect of observed variables, (ii) on the strength of association of latent variables with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , or (iii) on the relative importance of unobserved confounders as compared to observed confounders ([<reflink idref="bib20" id="ref99">20</reflink>]; [<reflink idref="bib19" id="ref100">19</reflink>]; [<reflink idref="bib17" id="ref101">17</reflink>]; [<reflink idref="bib76" id="ref102">76</reflink>]). A suite of sensitivity analysis tools to examine the robustness of linear regression estimates to omitted variable biases (OVB) can be found in the package sensemakr for R, Stata and Python ([<reflink idref="bib16" id="ref103">16</reflink>]; [<reflink idref="bib40" id="ref104">40</reflink>]). An interactive web application is also available.[<reflink idref="bib9" id="ref105">9</reflink>] Generalization of OVB results to fully nonparametric models, using Debiased Machine Learning, is developed in [<reflink idref="bib15" id="ref106">15</reflink>]).</p> <hd id="AN0178879684-22">Concluding remarks</hd> <p>In this note, we demonstrated through illustrative examples how simple graphical criteria can be used to decide when a variable should (or should not) be included in a regression equation—and thus whether it can be deemed a "good" or "bad" control. Many of these examples act as cautionary notes against prevailing practices: for instance, Models 7 to 10 reveal that one should be cautious of the general recommendation, usually derived from propensity score logic, of conditioning on <emph>all</emph> pre-treatment predictors of the treatment assignment[<reflink idref="bib10" id="ref107">10</reflink>]; whereas Models 14 and 15 show that not all "post-treatment" variables are "bad-controls," and some may even help with identification.</p> <p>In all cases, structural knowledge is indispensable for deciding whether a variable is a good or bad control, and graphical models provide a natural language for articulating such knowledge, as well as efficient tools for examining its logical ramifications. We have found that an example-based approach to "bad controls," such as the one presented here, can serve as a powerful instructional device to supplement more extended and formal discussions of the problem. By making this "crash course" accessible to instructors and practitioners, we hope to avail these tools to a broader community of scientists concerned with the causal interpretation of regression models.</p> <hd id="AN0178879684-23">Appendix</hd> <p>This appendix provides a short introduction to the notions of causal models, causal diagrams and "path-blocking" for the identification of causal effects via adjustment. Readers can find more extensive discussions in [<reflink idref="bib44" id="ref108">44</reflink>]); [<reflink idref="bib52" id="ref109">52</reflink>]) and [<reflink idref="bib53" id="ref110">53</reflink>]).</p> <hd id="AN0178879684-24">Structural causal models and causal diagrams</hd> <p>In order to decide whether there is a discrepancy between a certain regression equation (an associational quantity), and a target "causal effect" (a causal quantity), we need to mathematically <emph>define</emph> what this causal effect is. And to do that, we first need the concept of a <emph>causal model</emph>. We briefly introduce <emph>structural causal models</emph> (SCM) ([<reflink idref="bib44" id="ref111">44</reflink>]) with an example.</p> <p>Consider the SCM <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> shown in Figure 14a. The variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">V</mi></mrow><mo>=</mo><mo fence="false" stretchy="false">{</mo><mi>Z</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo fence="false" stretchy="false">}</mo></math> </ephtml> are called the <emph>endogenous</emph> variables, and stand for those variables that the investigator chose to model their cause-effect relationships; the variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">U</mi></mrow><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>U</mi><mi>z</mi></msub><mo>,</mo><msub><mi>U</mi><mi>x</mi></msub><mo>,</mo><msub><mi>U</mi><mi>y</mi></msub><mo fence="false" stretchy="false">}</mo></math> </ephtml> are called the <emph>exogenous</emph> variables and represent everything else that the investigator chose <emph>not</emph> to explicitly model (these are also usually called <emph>disturbances</emph>). The functions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">F</mi></mrow><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>f</mi><mi>z</mi></msub><mo>,</mo><msub><mi>f</mi><mi>x</mi></msub><mo>,</mo><msub><mi>f</mi><mi>y</mi></msub><mo fence="false" stretchy="false">}</mo></math> </ephtml> are called <emph>structural equations</emph>, and each function represents a causal process that <emph>assigns</emph> to its respective endogenous variable a value based on the values of the other variables. We use the assignment symbol ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">←</mo></math> </ephtml> ) to emphasize the <emph>asymmetry</emph> in a causal relationship, flowing from cause to effect. Finally, the exogenous variables have an associated probability distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">U</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> summarizing their uncertainty. In this particular example, we assume the exogenous variables are mutually independent (but in general, this need not be the case). The SCM <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> induces a joint distribution on the endogenous variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">V</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> , which we denote by <emph>observational distribution</emph>. In observational studies, the investigator only has access to samples of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">V</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>Graph: Figure 14. Structural Causal Model M and its associated graph G</p> <p>Every SCM has an associated graph <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> , usually called its <emph>causal diagram</emph>. In the types of models we consider here, which do not exhibit cycles, the causal diagram will be a directed acyclic graph (DAG). The causal diagram of our example is shown in Figure 14b. The graph <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> contains one node for each variable in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> , and a directed arrow <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub><mo stretchy="false">→</mo><msub><mi>V</mi><mi>j</mi></msub></math> </ephtml> whenever <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub></math> </ephtml> appears in the structural equation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>j</mi></msub></math> </ephtml> , meaning that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub></math> </ephtml> is a <emph>direct cause</emph> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>j</mi></msub></math> </ephtml> . Here we explicitly show the exogenous variables, but, conventionally, these are omitted from the graph for brevity. When the exogenous variables are omitted from the diagram, a dashed bidirected arrow <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub><mo stretchy="false">↔</mo><msub><mi>V</mi><mi>j</mi></msub></math> </ephtml> should be added whenever the exogenous variables entering <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><msub><mi>v</mi><mi>i</mi></msub></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><msub><mi>v</mi><mi>j</mi></msub></mrow></msub></math> </ephtml> are <emph>not</emph> independent.</p> <hd id="AN0178879684-25">Interventions and causal effects</hd> <p>Interventions are modeled by modifying mechanisms of the SCM. For example, the act <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>o</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> in the model of Figure 14a amounts to replacing the original mechanism <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><msub><mi>f</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><msub><mi>U</mi><mi>x</mi></msub><mo stretchy="false">)</mo></math> </ephtml> with a new mechanism in which <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> is externally forced to attain the value <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> , i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><mi>x</mi></math> </ephtml> . This results in the modified SCM <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>x</mi></msub></math> </ephtml> of Figure 15a.</p> <p>Graph: Figure 15. Effect of intervention do(X=x)</p> <p>The model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>x</mi></msub></math> </ephtml> induces an <emph>interventional distribution</emph> on the endogenous variables, denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">V</mi></mrow><mo>∣</mo><mi>d</mi><mi>o</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> </ephtml> . With the concept of an intervention in mind, we can now define the average causal effect (i.e, the expected increase of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> in response to a unit increase in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> due to an <emph>intervention</emph>) as the average contrast of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> under two distinct interventions: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">E</mi><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo>∣</mo><mi mathvariant="normal">do</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo>+</mo>1<mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mi mathvariant="normal">E</mi><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo>∣</mo><mi mathvariant="normal">do</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>In general the ACE varies depending on levels of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> , but in linear models, as we show below, the ACE reduces to a single number.</p> <p>Other causal effects can be defined with the same model modification logic. For instance, the controlled direct effect (or CDE, i.e, the expected increase of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> per unit of a controlled increase in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , while holding <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> constant) is defined as the difference: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">CDE</mi></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo>,</mo><mi mathvariant="normal">z</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">E</mi><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo>∣</mo><mi mathvariant="normal">do</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo>+</mo>1<mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="normal">do</mi><mo stretchy="false">(</mo><mi mathvariant="normal">z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mi mathvariant="normal">E</mi><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo>∣</mo><mi mathvariant="normal">do</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="normal">do</mi><mo stretchy="false">(</mo><mi mathvariant="normal">z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <hd id="AN0178879684-26">Potential outcomes</hd> <p>Potential outcomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold">V</mi></mrow><mi>x</mi></msub></math> </ephtml> are defined as the solution of the endogenous variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">V</mi></mrow></math> </ephtml> in the modified model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>x</mi></msub></math> </ephtml> . Thus, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">V</mi></mrow><mo>∣</mo><mi>d</mi><mi>o</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> </ephtml> can be equivalently written as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold">V</mi></mrow><mi>x</mi></msub><mo stretchy="false">)</mo></math> </ephtml> (likewise, we could have written all variables in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>x</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>x</mi></msub></math> </ephtml> as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Z</mi><mi>x</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>x</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>x</mi></msub></math> </ephtml> ). As such, the ACE can be equivalently written as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>C</mi><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>x</mi><mo>+</mo>1</mrow></msub><mo stretchy="false">]</mo><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>x</mi></msub><mo stretchy="false">]</mo></math> </ephtml> .</p> <hd id="AN0178879684-27">Causal and non-causal paths: chains, forks and colliders</hd> <p>Concretely, let us suppose that the structural equations of our example are linear, that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo stretchy="false">←</mo><msub><mi>U</mi><mi>z</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><mi>Z</mi><mo>+</mo><msub><mi>U</mi><mi>x</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo stretchy="false">←</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mi>X</mi><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mi>Z</mi><mo>+</mo><msub><mi>U</mi><mi>y</mi></msub></math> </ephtml> . Further assume that the disturbances <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">U</mi></mrow></math> </ephtml> are normally distributed, and that the random variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>,</mo><mi>Z</mi><mo>,</mo><mi>Y</mi></math> </ephtml> have mean zero and unit variance. Then the ACE evaluates to: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">E</mi><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo>∣</mo><mi mathvariant="normal">do</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo>+</mo>1<mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mi mathvariant="normal">E</mi><mo stretchy="false">[</mo><mi mathvariant="normal">Y</mi><mo>∣</mo><mi mathvariant="normal">do</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>To contrast, now let us compute the regression coefficient of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></math> </ephtml><ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><mrow><mfrac><mrow><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Y</mi><mo>,</mo><mi mathvariant="normal">X</mi><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">Var</mi></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>Note how the regression coefficient <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml> differs from the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> . This happens because the observed association of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> mixes both the <emph>causal</emph> association (the path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> ), and the <emph>non-causal</emph> association due to the confounder <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> (the path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><mi>Z</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> ). We call such confounding paths, that start with an arrow pointing to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , "back-door paths." Note, however, that the regression coefficient of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> (denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub></math> </ephtml> ) evaluates to (a derivation is provided later, in Equations A.6 to A.10) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>That is, controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in this model effectively blocks the back-door path, and recovers the ACE.</p> <p>In general, how does path blocking work in a graphical model? To answer this question, we need to understand the three main patterns of a causal diagram, which help us characterize when paths (consisting of sequences of the following triplets) of the graph are blocked or open.</p> <p></p> <ulist> <item> <bold> Chains (mediators). </bold> Chains are patterns of the form <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , meaning that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> causally affects <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> through the mediator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . Conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in a chain blocks this flow of association.</item> <p></p> <item> <bold> Forks (common causes). </bold> Forks are patterns of the form <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">←</mo><mi>Z</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> , meaning <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> share a common cause (a confounder) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , thus inducing a <emph>non-causal</emph> association between both variables. Conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> in a fork blocks this flow of association.</item> <p></p> <item> <bold> Colliders (common effects). </bold> Colliders are patterns of the form <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Z</mi><mo stretchy="false">←</mo><mi>Y</mi></math> </ephtml> , meaning that both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> share a common effect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . Contrary to the other two patterns, this path is closed by default—conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml><emph>opens</emph> the path and induces a <emph>non-causal</emph> association between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> .</item> </ulist> <p>A final rule to keep in mind is that controlling for a descendant of a variable is equivalent to "partially" controlling for that variable. Thus, controlling for a descendant of a mediator or a confounder partially blocks the flow of association, whereas controlling for a descendant of a collider partially opens the flow of association.</p> <p>We can now judge whether any path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> in a graph, no matter how complicated, is blocked by a set <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> . This happens if, and only if: (i) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> contains a chain or a fork, such that the middle node is in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> ; <emph>or</emph>, (ii) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> contains a collider, such that neither the middle node, nor any of its descendants, are in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> .</p> <hd id="AN0178879684-28">The back-door and the adjustment criteria</hd> <p>Armed with these tools, the DAG reveals which set of variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> blocks the correct paths for valid estimation of the ACE. We would like to find a set <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> , such that,</p> <p></p> <ulist> <item> it blocks all spurious paths from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ;</item> <p></p> <item> it <emph>does not</emph> (partially) block any of the causal paths from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ; and,</item> <p></p> <item> it does not (partially) open other spurious paths.</item> </ulist> <p>The above conditions characterize the so-called <emph>back-door</emph> criterion, later generalized by the <emph>adjustment</emph> criterion ([<reflink idref="bib43" id="ref112">43</reflink>]; [<reflink idref="bib62" id="ref113">62</reflink>]; [<reflink idref="bib54" id="ref114">54</reflink>]). If we can find such a set of controls <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>Z</mi>1</msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mi>Z</mi><mi>k</mi></msub><mo fence="false" stretchy="false">}</mo></math> </ephtml> , then the interventional expectation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> can be computed from the observational distribution as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo>∣</mo><mi>d</mi><mi>o</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo>∣</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mrow><mi mathvariant="bold">Z</mi></mrow><mo stretchy="false">]</mo><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>(A.1) Readers accustomed to potential outcomes should note that, if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> satisfies the adjustment criterion, then conditional ignorability holds, ie., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>x</mi></msub><mrow><mo>⊥</mo><mo>⊥</mo></mrow><mi>X</mi><mo>∣</mo><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> .</p> <p> <bold>Formal statements.</bold> For completeness, we now provide the formal statements of the back-door and adjustment criteria.</p> <hd id="AN0178879684-29">Definition 1 (Back-door criterion (Pearl, 2009a))</hd> <p>A set of variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> satisfies the back-door criterion relative to ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ) in a DAG <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> if:</p> <p></p> <ulist> <item> No node in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a descendant of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ; and,</item> <p></p> <item> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> blocks every path between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> that contains an arrow into <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> .</item> </ulist> <p>The adjustment criterion was later devised to explicitly handle cases in which <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> may contain descendants of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> .</p> <hd id="AN0178879684-30">Definition 2 (Adjustment criterion (Shpitser, VanderWeele and Robins, 2012 ; Perkovic et al.,...</hd> <p>A set of variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> satisfies the adjustment criterion relative to ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ) in a DAG <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> if:</p> <p></p> <ulist> <item> In the dag <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mrow><mrow><mi>x</mi><mo stretchy="false">¯</mo></mrow></mrow></msub></math> </ephtml> (the DAG <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> with the arrows incoming into <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> removed), no element in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a descendant of any <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>∉</mo><mi>X</mi></math> </ephtml> which lies on a proper causal path from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> .</item> <p></p> <item> All proper non-causal paths in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> are blocked by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> .</item> </ulist> <hd id="AN0178879684-31">Linear versus non-linear models</hd> <p>The previous identification result is non-parametric, and it involves <emph>two</emph> expectations. First we compute the conditional expectation <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo>∣</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mrow><mi mathvariant="bold">Z</mi><mo mathvariant="bold">=</mo><mi mathvariant="bold">z</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> , then we <emph>average</emph> this conditional expectation over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">Z</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> . If, however, the conditional expectation function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo>∣</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mrow><mi mathvariant="bold">Z</mi><mo mathvariant="bold">=</mo><mi mathvariant="bold">z</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> is linear, the expression simplifies to <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo>∣</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo>,</mo><mrow><mi mathvariant="bold">Z</mi></mrow><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mi>α</mi><mo>+</mo><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mrow><mi mathvariant="bold">z</mi></mrow></mrow></msub><mi>x</mi><mo>+</mo><munderover><mo>∑</mo><mrow><mspace width=".1em" /><mi>j</mi><mo>=</mo>1</mrow><mi>k</mi></munderover><msub><mi>β</mi><mrow><mi>y</mi><msub><mi>z</mi><mi>j</mi></msub><mo>.</mo><mi>x</mi><mrow><msub><mi mathvariant="bold">z</mi><mrow><mo mathvariant="bold">−</mo><mi mathvariant="bold">j</mi></mrow></msub></mrow></mrow></msub><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mspace width=".1em" /><mi>j</mi></mrow></msub><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>Where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> is a constant and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold">Z</mi><mrow><mo mathvariant="bold">−</mo><mi mathvariant="bold">j</mi></mrow></msub></mrow></math> </ephtml> denotes the set <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> excluding <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Z</mi><mi>j</mi></msub></math> </ephtml> . Therefore, under the parameteric assumption of linearity, the ACE simply equals the regression coefficient <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mrow><mi mathvariant="bold">z</mi></mrow></mrow></msub></math> </ephtml> , and no averaging over the distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> is necessary (similar result can be obtained if the conditional expectation is linearly separable on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ). If, however, the conditional expectation is not linear, the regression coefficient <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mrow><mi mathvariant="bold">z</mi></mrow></mrow></msub></math> </ephtml> targets a different causal quantity, which may be an incomplete summary of the ACE (see, e.g., [<reflink idref="bib1" id="ref115">1</reflink>]). In such cases, users should resort back to the proper adjustment formula as given by Equation A.1.</p> <hd id="AN0178879684-32">Virtual colliders and d-separation</hd> <p>Finally, we explain both d-separation and virtual colliders using the case of Model 18. Rewrite Model 18 showing the exogenous variables explicitly, as in Figure 16a. We can now clearly see the colliding path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Y</mi><mo stretchy="false">←</mo><msub><mi>U</mi><mi>y</mi></msub></math> </ephtml> . Conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , a descendant of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , thus partially opens this path, and creates a spurious association between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>y</mi></msub></math> </ephtml> , the disturbance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , making <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> a "bad control." Another approach to see why <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> is a bad control is to explicitly draw the potential outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>x</mi></msub></math> </ephtml> in the DAG, as shown in Figure 16b. As explained, recall that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>x</mi></msub></math> </ephtml> is the solution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> in the modified model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>x</mi></msub></math> </ephtml> . This results in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>x</mi></msub><mo>=</mo><msub><mi>f</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>U</mi><mi>y</mi></msub><mo stretchy="false">)</mo></math> </ephtml> , a function of the random variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>y</mi></msub></math> </ephtml> . Therefore, conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , a descendant of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , partially opens the path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Y</mi><mo stretchy="false">←</mo><msub><mi>U</mi><mi>y</mi></msub><mo stretchy="false">→</mo><msub><mi>Y</mi><mi>x</mi></msub></math> </ephtml> , and thus <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>x</mi></msub><mrow><mpadded width="0"><mtext>⧸</mtext></mpadded></mrow><mrow><mo>⊥</mo><mo>⊥</mo></mrow><mi>X</mi><mo>∣</mo><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> .</p> <p>Now let us consider the case in which the arrow <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> is removed (zero causal effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ). First recall that two nodes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> are <emph>d-separated</emph> conditional on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> if the set <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> blocks every path from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> in the graph. If <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> are <emph>d-separated</emph> conditional on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> , this implies the conditional independence <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>⊥</mo><mo>⊥</mo><mi>X</mi><mo>∣</mo><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> . In Model 18, when there is no path from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> also does not open any other paths between these two variables. Hence, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> is still <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> </ephtml> -separated from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> even after conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , and the conditional independence <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mrow><mo>⊥</mo><mo>⊥</mo></mrow><mi>X</mi><mo>∣</mo><mrow><mi mathvariant="bold">Z</mi></mrow></math> </ephtml> holds.</p> <p>Graph: Figure 16. Model 18 explained</p> <hd id="AN0178879684-33">Analytical expressions for linear models</hd> <p>Here we provide algebraic derivations for each illustrative model under the assumption of <emph>linearity</emph> of the structural equations. Before proceeding, we remind readers that adjusting for "bad controls" still lead to bias in non-parametric models. Furthermore, overt selection (rather than adjustment for) bad controls will also lead to bias (although the size and sign of the bias may differ).</p> <p>Without loss of generality, we assume random variables have been standardized to have mean zero and unit variance. We use <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></math> </ephtml> to denote the covariance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and, like before, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub></math> </ephtml> to denote the partial regression coefficient of the regression of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . The partial regression coefficient <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub></math> </ephtml> can be written in terms of the covariances as ([<reflink idref="bib21" id="ref116">21</reflink>]), <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>−</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.2) In linear structural causal models, each edge <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub><mo stretchy="false">→</mo><msub><mi>V</mi><mi>j</mi></msub></math> </ephtml> of a causal DAG can be mapped to a single structural coefficient <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mrow><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub></mrow></msub></math> </ephtml> representing the strength of the direct effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>j</mi></msub></math> </ephtml> . We can use Wright's path-tracing rules ([<reflink idref="bib75" id="ref117">75</reflink>]) to equate the covariance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub></mrow></msub></math> </ephtml> of any two variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>j</mi></msub></math> </ephtml> , to the sum of products of structural coefficients along <emph>unblocked</emph> paths between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>j</mi></msub></math> </ephtml> .</p> <p>For instance, in Model 1, path-tracing results in the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.3) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.4) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.5) With the help of Wright's rules and Equation A.2, one can easily proceed with the algebraic derivation for each model. For the sake of brevity, we provide the full derivation for Model 1, and for the remaining models (except Model 15) only the final result is presented (since the derivations would be very similar).</p> <p> <bold>Model 1.</bold> In Model 1, the average causal effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> equals <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> . The unadjusted regression coefficient equals <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml> , and the partial regression coefficient equals, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>−</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.6) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mo>−</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.7) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mo>−</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mo>−</mo><msubsup><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow>2</msubsup><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.8) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow>2</msubsup><mo stretchy="false">)</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.9) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.10)<bold>Model 2.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.11) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.12) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.13) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 3.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.14) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.15) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.16) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 4.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.17) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.18) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.19) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xm</mi></mrow></msub><msub><mi>λ</mi><mrow><mi mathvariant="normal">my</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 5.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.20) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.21) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.22) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xm</mi></mrow></msub><msub><mi>λ</mi><mrow><mi mathvariant="normal">my</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 6.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.23) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.24) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.25) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xm</mi></mrow></msub><msub><mi>λ</mi><mrow><mi mathvariant="normal">my</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 7.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.26) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.27) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>2</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>2</msub><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.28) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>−</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>2</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>2</msub><mi>y</mi></mrow></msub></mrow><mrow>1<mo>−</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub><mo stretchy="false">)</mo>2</mrow></mfrac></mrow></math> </ephtml> .</p> <p> <bold>Variation of Model 7.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.29) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.30) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>2</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>2</msub><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.31) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>−</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>2</msub><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>2</msub><mi>y</mi></mrow></msub></mrow><mrow>1<mo>−</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><msub><mi>u</mi>1</msub><mi>z</mi></mrow></msub><mo stretchy="false">)</mo>2</mrow></mfrac></mrow></math> </ephtml> . Thus, depending on the parameterization of the model, the absolute value of the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub></math> </ephtml> can be greater than that of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 8.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.32) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo>0</math> </ephtml></p> <p>Graph</p> <p>(A.33) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.34) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 9.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.35) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.36) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.37) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 10.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.38) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.39) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.40) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mrow><mi>u</mi><mi>x</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml> . Since <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">0<mo><</mo>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>z</mi><mi>x</mi></mrow>2</msubsup><mo><</mo>1</math> </ephtml> , the absolute value of the bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub></math> </ephtml> is always greater than that of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 11.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.41) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.42) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.43) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xz</mi></mrow></msub><msub><mi>λ</mi><mrow><mi mathvariant="normal">zy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">CDE</mi></mrow><mo>=</mo>0</math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo>0</math> </ephtml> .</p> <p> <bold>Model 12.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.44) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.45) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>m</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.46) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xm</mi></mrow></msub><msub><mi>λ</mi><mrow><mi mathvariant="normal">my</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">CDE</mi></mrow><mo>=</mo>0</math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><mrow><mfrac><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>m</mi><mi>z</mi></mrow>2</msubsup></mrow><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow>2</msubsup><msubsup><mi>λ</mi><mrow><mi>m</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow><mo stretchy="false">)</mo></math> </ephtml> .</p> <p> <bold>Variation of Model 11.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.47) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.48) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.49) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xz</mi></mrow></msub><msub><mi>λ</mi><mrow><mi mathvariant="normal">zy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">CDE</mi></mrow><mo>=</mo>0</math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><mo>−</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml> .</p> <p> <bold>Model 13.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.50) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo>0</math> </ephtml></p> <p>Graph</p> <p>(A.51) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.52) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xm</mi></mrow></msub><msub><mi>λ</mi><mrow><mi mathvariant="normal">my</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>m</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>m</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 14.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.53) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.54) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.55) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> .</p> <p> <bold>Model 15.</bold> Model 15 requires a more elaborate derivation since we need to consider four variables. Here we only have samples with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><mi>w</mi></math> </ephtml> recorded. In linear models, this is equivalent to having adjusted for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> </ephtml> in a regression model. Thus, this means the researcher does not have access to the regression coefficients <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></math> </ephtml> nor <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub></math> </ephtml> , but rather <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>w</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>w</mi><mi>z</mi></mrow></msub></math> </ephtml> (that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> </ephtml> is <emph>always</emph> conditioned on, due to sample selection). Path-tracing leads to the following covariances (there is no need to compute all of them to solve this problem, but we show them here for completeness), <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.56) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.57) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.58) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.59) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>w</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.60) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.61) The average causal effect is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> . By Equation A.2, the regression coefficient without adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>w</mi></mrow></msub></math> </ephtml> , equals <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>−</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>w</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></mrow><mrow>1<mo>−</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><mo stretchy="false">)</mo>2</mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.62) Now we must compute the regression coefficient adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>w</mi><mi>z</mi></mrow></msub></math> </ephtml> . Following [<reflink idref="bib21" id="ref118">21</reflink>]) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>w</mi><mi>z</mi></mrow></msub></math> </ephtml> can be written as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>w</mi><mi>z</mi></mrow></msub><mo>=</mo><mrow><mfrac><mrow><msub><mi>ρ</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi><mo>.</mo><mi>z</mi></mrow></msub><msub><mi>ρ</mi><mrow><mi>y</mi><mi>w</mi><mo>.</mo><mi>z</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi><mo>.</mo><mi>z</mi></mrow>2</msubsup><mo stretchy="false">)</mo><mrow>1<mo>/</mo>2</mrow></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><msub><mi>σ</mi><mrow><mi>y</mi><mo>.</mo><mi>z</mi></mrow></msub><msub><mi>σ</mi><mrow><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.63) Where <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>−</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup><mo stretchy="false">)</mo><mrow>1<mo>/</mo>2</mrow><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow>2</msubsup><mo stretchy="false">)</mo><mrow>1<mo>/</mo>2</mrow></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.64) denotes the partial correlation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> after adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> and <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow>2</msubsup><mo stretchy="false">)</mo><mrow>1<mo>/</mo>2</mrow></math> </ephtml></p> <p>Graph</p> <p>denotes the partial standard deviation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> after adjusting for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . Since <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> </ephtml> -separated from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> </ephtml> given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , we know that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo>0</math> </ephtml> (we can also verify this by checking that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>x</mi><mi>w</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo>0</math> </ephtml> ). Also note that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></math> </ephtml> . We thus obtain <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>w</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>ρ</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>×</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow>2</msubsup><mo stretchy="false">)</mo><mrow>1<mo>/</mo>2</mrow></mrow><mrow><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup><mo stretchy="false">)</mo><mrow>1<mo>/</mo>2</mrow></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.65) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>−</mo><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup><mo stretchy="false">)</mo><mrow>1<mo>/</mo>2</mrow><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow>2</msubsup><mo stretchy="false">)</mo><mrow>1<mo>/</mo>2</mrow></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow>2</msubsup></mrow><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.66) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>(A.67) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mrow><mfrac><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow><mo>=</mo><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.68)<bold>Model 16.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.69) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.70) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.71) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>−</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>u</mi><mi>y</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml> . <bold>Model 17.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.72) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.73) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>+</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.74) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>×</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo>1<mo>−</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow><mo>−</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub></mrow><mrow>1<mo>−</mo><msubsup><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml> .</p> <p> <bold>Model 18.</bold> Path-tracing leads to the following covariances, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.75) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><msub><mi>λ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.76) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>y</mi><mi>z</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>(A.77) We have: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ACE</mi></mrow><mo>=</mo><msub><mi>λ</mi><mrow><mi mathvariant="normal">xy</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><mi>y</mi><mi>x</mi><mo>.</mo><mi>z</mi></mrow></msub><mo>=</mo><msub><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>×</mo><mrow><mfrac><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>y</mi><mi>z</mi></mrow>2</msubsup></mrow><mrow>1<mo>−</mo><msubsup><mi>λ</mi><mrow><mi>x</mi><mi>y</mi></mrow>2</msubsup><msubsup><mi>λ</mi><mrow><mi>y</mi><mi>z</mi></mrow>2</msubsup></mrow></mfrac></mrow></math> </ephtml> .</p> <ref id="AN0178879684-34"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Angrist J., Pischke J.-S.2009. 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PMLR.</bibtext> </blist> </ref> <ref id="AN0178879684-35"> <title> Footnotes </title> <blist> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> This research was supported in parts by grants from the National Science Foundation [#IIS-2106908], Office of Naval Research [#N00014-17-S-12091 and #N00014-21-1-2351], and Toyota Research Institute of North America [#PO000897].</bibtext> </blist> <blist> <bibtext> Carlos Cinelli https://orcid.org/0000-0002-2021-7739 Andrew Forney https://orcid.org/0000-0002-6366-1290</bibtext> </blist> <blist> <bibtext> See Chen and Pearl (2013) for a critical appraisal of econometrics textbooks, and Bollen and Pearl (2013) for eight misconceptions that still prevail in statistics and the social sciences. Warnings regarding the adjustment of post-treatment variables date back to at least Rosenbaum (1984), but a systematic solution to the problem of covariate selection was not available before the development of causal graphical models.</bibtext> </blist> <blist> <bibtext> R code with numerical simulations for all examples can be found in: https://<ulink href="http://www.kaggle.com/code/carloscinelli/crash-course-in-good-and-bad-controls-linear-r">www.kaggle.com/code/carloscinelli/crash-course-in-good-and-bad-controls-linear-r</ulink>.</bibtext> </blist> <blist> <bibtext> Some economists may denote confounding bias as "selection bias," meaning preferential selection to treatment (Angrist and Pischke, 2009, 2014). Here selection bias means preferential selection into the available data.</bibtext> </blist> <blist> <bibtext> Also available online in <ulink href="http://www.dagitty.net">http://www.dagitty.net</ulink>.</bibtext> </blist> <blist> <bibtext> Available online at <ulink href="http://www.causalfusion.net">http://www.causalfusion.net</ulink>.</bibtext> </blist> <blist> <bibtext> Available online at https://carloscinelli.shinyapps.io/robustness_value/.</bibtext> </blist> <blist> <bibtext> For instance, examples of such recommendations can be found in Rosenbaum (2002, p.76), Rubin (2009), Imbens and Rubin (2015, p.265), Dorie et al. (2016, p.3453).</bibtext> </blist> </ref> <aug> <p>By Carlos Cinelli; Andrew Forney and Judea Pearl</p> <p>Reported by Author; Author; Author</p> <p></p> <p>Carlos Cinelli is an assistant professor of statistics at the University of Washington. His research focuses on developing new causal and statistical methods for transparent and robust causal claims in the empirical sciences.</p> <p>Andrew Forney is currently an assistant professor of computer science at Loyola Marymount University. His research primarily focuses on the intertwining of topics in reinforcement learning and causal inference towards crafting increasingly dynamic and cognitive autonomous agents. With backgrounds in both computing and psychology, he hopes to concert the tools of each discipline to better understand human thought and to likewise better approximate it.</p> <p>Judea Pearl is Chancellor's professor of computer science and statistics at the UCLA. He is a graduate of the Technion, Israel, and has joined the faculty of the UCLA in 1970, where he currently directs the Cognitive Systems Laboratory and conducts research in artificial intelligence, human cognition and philosophy of science. Pearl has authored Heuristics (1983), Probabilistic Reasoning (1988) and Causality (2000, 2009), winner of the London School of Economics Lakatosh Award. In 2018, he authored The Book of Why (with D. Mackenzie), which brings causal reasoning to the general public. Pearl is a member of the National Academy of Sciences and a fellow of the cognitive science society and the Association for the Advancement of Artificial Intelligence. In 2012, he won the Technion's Harvey Prize and the ACM Alan Turing Award.</p> </aug> <nolink nlid="nl1" bibid="bib65" firstref="ref4"></nolink> <nolink nlid="nl2" bibid="bib17" firstref="ref5"></nolink> <nolink nlid="nl3" bibid="bib73" firstref="ref9"></nolink> <nolink nlid="nl4" bibid="bib36" firstref="ref10"></nolink> <nolink nlid="nl5" bibid="bib28" firstref="ref11"></nolink> <nolink nlid="nl6" bibid="bib57" firstref="ref13"></nolink> <nolink nlid="nl7" bibid="bib59" firstref="ref14"></nolink> <nolink nlid="nl8" bibid="bib72" firstref="ref15"></nolink> <nolink nlid="nl9" bibid="bib43" firstref="ref18"></nolink> <nolink nlid="nl10" bibid="bib62" firstref="ref19"></nolink> <nolink nlid="nl11" bibid="bib54" firstref="ref20"></nolink> <nolink nlid="nl12" bibid="bib39" firstref="ref21"></nolink> <nolink nlid="nl13" bibid="bib38" firstref="ref22"></nolink> <nolink nlid="nl14" bibid="bib31" firstref="ref23"></nolink> <nolink nlid="nl15" bibid="bib69" firstref="ref24"></nolink> <nolink nlid="nl16" bibid="bib32" firstref="ref25"></nolink> <nolink nlid="nl17" bibid="bib58" firstref="ref26"></nolink> <nolink nlid="nl18" bibid="bib70" firstref="ref27"></nolink> <nolink nlid="nl19" bibid="bib25" firstref="ref28"></nolink> <nolink nlid="nl20" bibid="bib42" firstref="ref29"></nolink> <nolink nlid="nl21" bibid="bib35" firstref="ref30"></nolink> <nolink nlid="nl22" bibid="bib22" firstref="ref31"></nolink> <nolink nlid="nl23" bibid="bib55" firstref="ref32"></nolink> <nolink nlid="nl24" bibid="bib29" firstref="ref33"></nolink> <nolink nlid="nl25" bibid="bib33" firstref="ref34"></nolink> <nolink nlid="nl26" bibid="bib51" firstref="ref35"></nolink> <nolink nlid="nl27" bibid="bib52" firstref="ref36"></nolink> <nolink nlid="nl28" bibid="bib44" firstref="ref44"></nolink> <nolink nlid="nl29" bibid="bib63" firstref="ref45"></nolink> <nolink nlid="nl30" bibid="bib46" firstref="ref46"></nolink> <nolink nlid="nl31" bibid="bib64" firstref="ref47"></nolink> <nolink nlid="nl32" bibid="bib23" firstref="ref49"></nolink> <nolink nlid="nl33" bibid="bib50" firstref="ref50"></nolink> <nolink nlid="nl34" bibid="bib19" firstref="ref51"></nolink> <nolink nlid="nl35" bibid="bib71" firstref="ref61"></nolink> <nolink nlid="nl36" bibid="bib48" firstref="ref62"></nolink> <nolink nlid="nl37" bibid="bib47" firstref="ref63"></nolink> <nolink nlid="nl38" bibid="bib49" firstref="ref64"></nolink> <nolink nlid="nl39" bibid="bib41" firstref="ref65"></nolink> <nolink nlid="nl40" bibid="bib34" firstref="ref73"></nolink> <nolink nlid="nl41" bibid="bib26" firstref="ref74"></nolink> <nolink nlid="nl42" bibid="bib60" firstref="ref75"></nolink> <nolink nlid="nl43" bibid="bib30" firstref="ref76"></nolink> <nolink nlid="nl44" bibid="bib12" firstref="ref78"></nolink> <nolink nlid="nl45" bibid="bib27" firstref="ref79"></nolink> <nolink nlid="nl46" bibid="bib53" firstref="ref81"></nolink> <nolink nlid="nl47" bibid="bib37" firstref="ref82"></nolink> <nolink nlid="nl48" bibid="bib66" firstref="ref83"></nolink> <nolink nlid="nl49" bibid="bib68" firstref="ref85"></nolink> <nolink nlid="nl50" bibid="bib67" firstref="ref86"></nolink> <nolink nlid="nl51" bibid="bib74" firstref="ref89"></nolink> <nolink nlid="nl52" bibid="bib11" firstref="ref90"></nolink> <nolink nlid="nl53" bibid="bib13" firstref="ref94"></nolink> <nolink nlid="nl54" bibid="bib61" firstref="ref96"></nolink> <nolink nlid="nl55" bibid="bib20" firstref="ref99"></nolink> <nolink nlid="nl56" bibid="bib76" firstref="ref102"></nolink> <nolink nlid="nl57" bibid="bib16" firstref="ref103"></nolink> <nolink nlid="nl58" bibid="bib40" firstref="ref104"></nolink> <nolink nlid="nl59" bibid="bib15" firstref="ref106"></nolink> <nolink nlid="nl60" bibid="bib10" firstref="ref107"></nolink> <nolink nlid="nl61" bibid="bib21" firstref="ref116"></nolink> <nolink nlid="nl62" bibid="bib75" firstref="ref117"></nolink>
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  Data: A Crash Course in Good and Bad Controls
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  Data: <searchLink fieldCode="AR" term="%22Carlos+Cinelli%22">Carlos Cinelli</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-2021-7739">0000-0002-2021-7739</externalLink>)<br /><searchLink fieldCode="AR" term="%22Andrew+Forney%22">Andrew Forney</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-6366-1290">0000-0002-6366-1290</externalLink>)<br /><searchLink fieldCode="AR" term="%22Judea+Pearl%22">Judea Pearl</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2024 53(3):1071-1104.
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  Data: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
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  Data: 34
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  Data: 2024
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  Data: National Science Foundation (NSF), Division of Information and Intelligent Systems (IIS)<br />Office of Naval Research (ONR) (DOD)
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  Data: Journal Articles<br />Reports - Descriptive
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  Data: <searchLink fieldCode="DE" term="%22Regression+%28Statistics%29%22">Regression (Statistics)</searchLink><br /><searchLink fieldCode="DE" term="%22Robustness+%28Statistics%29%22">Robustness (Statistics)</searchLink><br /><searchLink fieldCode="DE" term="%22Error+of+Measurement%22">Error of Measurement</searchLink><br /><searchLink fieldCode="DE" term="%22Testing+Problems%22">Testing Problems</searchLink><br /><searchLink fieldCode="DE" term="%22Causal+Models%22">Causal Models</searchLink>
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  Data: 10.1177/00491241221099552
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  Data: 0049-1241<br />1552-8294
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  Data: Many students of statistics and econometrics express frustration with the way a problem known as "bad control" is treated in the traditional literature. The issue arises when the addition of a variable to a regression equation produces an unintended discrepancy between the regression coefficient and the effect that the coefficient is intended to represent. Avoiding such discrepancies presents a challenge to all analysts in the data intensive sciences. This note describes graphical tools for understanding, visualizing, and resolving the problem through a series of illustrative examples. By making this "crash course" accessible to instructors and practitioners, we hope to avail these tools to a broader community of scientists concerned with the causal interpretation of regression models.
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      – TitleFull: A Crash Course in Good and Bad Controls
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