An Introduction to Causal Inference

May combine two claims.

The algebraic machinery of counterfactual notation can be powerful in refining assumptions, deriving consistent estimands, bounding probabilities of necessary and sufficient causation, and combining experimental and nonexperimental data.

“the algebraic machinery offered by the counterfactual notation, Y x (u), once a problem is properly formalized, can be extremely powerful in refining assumptions (Angrist et al., 1996, Heckman andVytlacil, 2005), deriving consistent estimands (Robins, 1986), bounding probabilities of necessary and sufficient causation (Tian and Pearl, 2000), and combining data from experimental and nonexperimental”

• cites Angrist et al., 1996 sourceNot checked — no full text of the source
• cites Robins, 1986 sourceSupported
• cites Tian and Pearl, 2000 sourceNot checked — no full text of the source

Classified as probably not a substantial claim.

Hypothetical entities Yx are assumed to be connected to observed variables via consistency constraints such as X=x implies Yx=Y.

“They are assumed to be connected to observed variables via consistency constraints (Robins, 1986) such as X = x =⇒ Y x = Y,(32)”

• cites Robins, 1986 sourceCould not verify

  The 'Definition of a treatment' section discusses hypothetical survival histories under treatment, and Section 2C.3 relates planned and observed exposure histories. Neither passage states that a hypothetical outcome equals the observed outcome when the hypothetical treatment matches the treatment received. Searches for consistency and that outcome-equality condition did not find it in our copy. (LLM, conf=0.68)

  Experimental — accuracy not yet measured

May combine two claims.

In the model of Fig. 2(a), P(y|do(x)) is identifiable and equals P(y|x).

“Using a similar derivation, though somewhat more involved, we can show that P(y|do(x)) is identifiable and given by the conditional probability P(y|x).”

May combine two claims.

In the example of Fig. 2(b), the distribution of Y is sensitive to x0 and P(y|do(x0)) = P(y|x0).

“while that of Y is sensitive to x 0 , and is given by P(y|do(x 0 )) = ∑ z P(z,y|do(x 0 )) = ∑ z P(z)P(y|x 0 ) = P(y|x 0 )”

May combine two claims.

In the model of Fig. 2(a), despite ignorance of f_X, f_Y, f_Z and P(u), E(Y|do(x0)) is identifiable and equals E(Y|X=x0).

“We will see that, despite our ignorance of f X , f Y , f Z and P(u), E(Y|do(x 0 )) is nevertheless identifiable and is given by the conditional expectation E(Y|X = x 0 ).”

May combine two claims.

The assumptions of Eqs. (40)-(41) imply the conditional independence Y_x ⊥⊥ Z | {X_z, X} but do not imply Y_x ⊥⊥ Z | X.

“it is hardly straightforward to show that the assumptions of Eqs. ( 40)-( 41) imply the conditional independence (Y x ⊥⊥ Z|{X z ,X}) but do not imply the conditional independence (Y x ⊥⊥ Z|X).”

May combine two claims. The quote only partly states this claim.

In general, the total effect is not the sum of the direct and indirect effects; instead, TE = DE + IE - DE • IE.

“Second, the total effect is not the sum of the direct and indirect effects.Instead, we have:”

May combine two claims.

The approximate method always underestimates the indirect effect, and the distortion is minimal for high values of P(x = 1) and (1-p_x).

“Comparing (59) and ( 54) reveals that the approximate method always underestimates the indirect effect and the distortion is minimal for high values of P(x = 1) and (1p x ).”

• AssumptionThe authors state this as an assumption: The error terms e_x, e_z, and e_xz are three independent Bernoulli variables with probabilities p_x, p_z, and p_xz, respectively.
• AssumptionThe authors state this as an assumption: The assumption of nonconfoundedness permits use of the Mediation Formula for estimating the Z-mediated effect of X on Y.

May combine two claims.

Expression (43) is both valid and identifiable in Markovian models (no unobserved confounders) where each term on the right can be reduced to a do-free expression and estimated by regression.

“In particular, it can be shown (Pearl, 2001) that expression (43) is both valid and identifiable in Markovian models (i.e., no unobserved confounders) where each term on the right can be reduced to a”

• cites Pearl, 2001Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: Certain assumptions of no confounding are valid, or the model is Markovian with no unobserved confounders.

May combine two claims.

In linear systems, the Mediation Formula yields standard expressions for indirect effects, estimable either as a difference in two regression coefficients or a product of two regression coefficients, with Y regressed on both X and Z.

“We thus obtained the standard expressions for indirect effects in linear systems, which can be estimated either as a difference in two regression coefficients (Eq. 53) or a product of two regression coefficients (Eq.52), with Y regressed on both X and Z.”

• cites MacKinnon et al., 2007bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The assumption of nonconfoundedness permits use of the Mediation Formula for estimating the Z-mediated effect of X on Y.

May combine two claims. The quote only partly states this claim.

Applying the linear approximation (57) to data generated by the nonlinear model of Fig. 7 yields a distorted solution, with IE = p_xz P(x=1)p_z(1-p_x), which differs from the correct value p_z p_xz.

“If however we were to apply the approximation in (57) to data generated by the nonlinear model of Fig. 7, a distorted solution would ensue;”

May combine two claims.

If the correct data-generating process was the linear model of (50), the expected estimates are a2 = c_z, E(z|x')-E(z|x) = b_x(x'-x), and IE = b_x c_z (x'-x).

“If the correct data-generating process was the linear model of (50), we would obtain the expected estimates a 2 = c z ,E(z|x )-E(z|x ) = b x (xx) and”

May combine two claims.

The natural direct effect DE_x,x'(Y) involves probabilities of nested counterfactuals and cannot be written in terms of the do(x) operator.

“We see that DE x,x (Y), the natural direct effect of the transition from x to x , involves probabilities of nested counterfactuals and cannot be written in terms of the do(x) operator.”

May combine two claims.

When outcome Y is binary, (1 - IE/TE) represents the fraction of responding individuals who owe their response to direct paths, while (1 - DE/TE) represents the fraction who owe their response to Z-mediated paths.

“When the outcome Y is binary (e.g., recovery, or hiring) the ratio (1 -IE/TE) represents the fraction of responding individuals who owe their response to direct paths, while (1 -DE/TE) represents the fraction who owe their response to Z-mediated paths.”

May combine two claims.

In linear systems, the direct effect is fully specified by the path coefficient attached to the link from X to Y, so the direct effect is independent of the values at which Z is held.

“In linear systems, the direct effect is fully specified by the path coefficient attached to the link from X to Y; therefore, the direct effect is independent of the values at which we hold Z.”

Classified as probably not a substantial claim.

SCM demystifies notions such as confounding, mediation, ignorability, comparability, exchangeability, and superexogeneity within a single conceptual framework.

“Demystifying enigmatic notions such as”

May combine two claims.

The assumptions encoded by missing links in Fig. 1(a) are causal, not associational, and cannot be determined from the joint density of observed variables X and Y.

“Note also that both assumptions are causal, not associational, since none can be determined from the joint density of the observed variables, X and Y; the association between the unobserved terms, U Y and U X , can only be uncovered in an experimental setting; or (in more intricate models, as in Fig. 5) from other causal assumptions.”

Classified as probably not a substantial claim. May combine two claims.

SCM solidifies the mathematical foundations of the potential-outcome model and formulates the counterfactual foundations of structural equation models.

“Solidifying the mathematical foundations of the potential-outcome model, and formulating the counterfactual foundations of structural equation models.”

Classified as probably not a substantial claim.

SCM reduces the evaluation of effects of causes, mediated effects, and causes of effects to an algorithmic level of analysis.

“Reducing the evaluation of”

May combine two claims.

Adding either the double arrow between U_X and U_Y or the arrow from Y to X to Fig. 1(a) would prevent identification of the direct effect beta.

“Had any of these two links been added to the diagram, we would not have been able to identify the direct effect β .”

Classified as probably not a substantial claim.

SCM provides definition, axiomatization, and algorithmization of counterfactuals and joint probabilities of counterfactuals.

“The definition, axiomatization and algorithmization of counterfactuals and joint probabilities of counterfactuals”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The basic elements of SCM were introduced in the mid-1990s and have been adapted widely by epidemiologists, statisticians, and social scientists.

“Although the basic elements of SCM were introduced in the mid 1990's (Pearl, 1995), and have been adapted widely by epidemiologists (Greenland, Pearl, andRobins, 1999, Glymour andGreenland, 2008), statisticians (Cox andWermuth, 2004, Lauritzen, 2001), and social scientists (Morgan and Winship, 2007), its potentials as a comprehensive theory of causation are yet to be fully utilized.”

• cites Pearl, 1995Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Greenland, Pearl, andRobins, 1999Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Morgan and Winship, 2007Not checked — no full text of the source

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

SCM weeds out myths and misconceptions from outdated traditions.

“Weeding out myths and misconceptions from outdated traditions (Meek and Glymour, 1994, Greenland et al., 1999, Cole and Hernan, 2002, Arah, 2008, Shrier, 2009, Pearl, 2009c).”

• cites Meek and Glymour, 1994 sourceNot checked — no full text of the source
• cites Greenland et al., 1999 sourceNot checked — no full text of the source
• cites Cole and Hernan, 2002 sourceNot checked — no full text of the source

May combine two claims.

Although each causal assumption in isolation cannot be tested, the sum total of all causal assumptions in a model often has testable implications.

“Although each causal assumption in isolation cannot be tested, the sum total of all causal assumptions in a model often has testable implications.”

Classified as probably not a substantial claim.

The structural theory used in this survey satisfies the criteria for a general theory of causation.

“The structural theory that we use in this survey satisfies the criteria above.”

Classified as probably not a substantial claim. May combine two claims.

A missing arrow represents a claim of zero influence, and a missing double arrow represents a claim of zero covariance.

“a missing arrow represents a claim of zero influence, while a missing double arrow represents a claim of zero covariance.”

May combine two claims.

S must be associated with X and, simultaneously, associated with Y, given X.

“In other words, S must be associated with X and, simultaneously, associated with Y, given X.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing). May combine two claims.

Tests for c-equivalence (27) are fairly easy to perform and can be assisted by propensity scores methods.

“Tests for c-equivalence (27) are fairly easy to perform, and they can also be assisted by propensity scores methods.”

May combine two claims.

For two sets Z and T of variables in G containing no descendant of X, a necessary and sufficient condition for Z and T to be c-equivalent is that at least one of the following holds: Z_m = T_m, or Z and T are admissible.

“Theorem 2 (Pearl and Paz, 2009) Let Z and T be two sets of variables in G, containing no descendant of X.A necessary and sufficient conditions for Z and T to be c-equivalent is that at least one of the following conditions holds: 1. Z m = T m , (i.e., the Markov boundary of Z coincides with that of T) 2. Z and T are admissible (i.e., satisfy the back-door condition)”

• cites Pearl and Paz, 2009Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim. May combine two claims.

Correlations may impede estimation of beta from nonexperimental data but do not change its definition as an effect coefficient.

“Correlations may only impede our ability to estimate β from nonexperimental data, but will not change its definition as given in ( 14).”

Classified as probably not a substantial claim.

Berkson's paradox is a general phenomenon whereby observations on a common consequence of two independent causes render those causes dependent.

“This special handling of collision nodes (or colliders, e.g., Z → X←U X ) reflects a general phenomenon known as Berkson's paradox (Berkson, 1946), whereby observations on a common consequence of two independent causes render those causes dependent.”

• cites Berkson, 1946 sourceNot checked — no full text of the source

May combine two claims.

Non-Markovian models permit identification only under certain conditions, which can be determined from the graph structure.

“Non-Markovian models, such as those involving correlated errors (resulting from unmeasured confounders), permit identification only under certain conditions, and these conditions too can be determined from the graph structure (Section 3.3).”

Classified as probably not a substantial claim.

From the post-intervention distribution, one can assess treatment efficacy by comparing aspects of this distribution at different levels of x0.

“From this distribution, one is able to assess treatment efficacy by comparing aspects of this distribution at different levels of x 0 .”

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

The conditional independencies entailed by d-separation constitute the main opening through which assumptions embodied in structural equation models can confront scrutiny of nonexperimental data.

“The conditional independencies entailed by d-separation constitute the main opening through which the assumptions embodied in structural equation models can confront the scrutiny of nonexperimental data.”

Classified as probably not a substantial claim. May combine two claims.

The derivation of E(Y|do(x0)) = E(Y|X=x0) is enabled by two assumptions: Y is a function of X and U_Y only, and U_Y is independent of {U_Z, U_X}, hence of X.

“We see that the derivation of (13) was enabled by two assumptions; first, Y is a function of X and U Y only, and, second, U Y is independent of {U Z ,U X }, hence of X.”

Classified as probably not a substantial claim.

The back-door criterion allows writing Eq. (25) directly by selecting a sufficient set S from the diagram without manipulating the truncated factorization formula.

“The back-door criterion allows us to write Eq. ( 25) directly, by selecting a sufficient set S directly from the diagram, without manipulating the truncated factorization formula.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The formula for the causal effect in Fig. 3 corresponds precisely to what is commonly called adjusting for Z1, Z2, and Z3.

“Note that this formula corresponds precisely to what is commonly called”

May combine two claims.

An extension of SEM based on simulating hypothetical interventions was proposed in prior work and has led to new ways of defining and estimating causal effects in nonlinear and nonparametric models.

“Such an extension, based on simulating hypothetical interventions in the model, was proposed in (Haavelmo, 1943, Strotz and Wold, 1960, Spirtes, Glymour, and Scheines, 1993, Pearl, 1993a, 2000a, Lindley, 2002) and has led to new ways of defining and estimating causal effects in nonlinear and nonparametric models (that is, models in which the functional form of the equations is unknown).”

• cites Haavelmo, 1943 sourceNot checked — no full text of the source
• cites Strotz and Wold, 1960Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Spirtes, Glymour, and Scheines, 1993 sourceNot checked — no full text of the source

May combine two claims.

The chain model of Fig. 2(a) encodes seven causal assumptions, none testable in isolation, yet their totality implies that Z is unassociated with Y in every stratum of X.

“The chain model of Fig. 2(a), for example, encodes seven causal assumptions, each corresponding to a missing arrow or a missing double-arrow between a pair of variables.None of those assumptions is testable in isolation, yet the totality of all those assumptions implies that Z is unassociated with Y in every stratum of X.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The G-computation formula dictates an adjustment for covariates that might be affected by previous treatments.

“As noted by Robins, the formula dictates an adjustment for covariates (e.g., Z 3 ) that might be affected by previous treatments (e.g., Z 2 ).”

• cites RobinsNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The graphical tools presented unveil the character of covariates that may increase bias and show what covariates should and should not be included in the conditioning set for propensity-score matching.

“The graphical tools presented in this section unveil the character of these covariates and show precisely what covariates should, and should not be included in the conditioning set for propensity-score matching (see also (Pearl andPaz, 2009, Pearl, 2009e)).”

• cites Pearl andPaz, 2009Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Pearl, 2009eNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

Queries about causal effects are phrased as queries about the marginal distribution of the counterfactual variable of interest.

“Queries about causal effects (written P(y|do(x)) in the structural analysis) are phrased as queries about the marginal distribution of the counterfactual variable of interest, written P * (Y x = y).”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

Counterfactual independence assumptions are cast in a language so far removed from ordinary understanding that they cannot be comprehended or ascertained by ordinary mortals.

“Such assumptions are cast in a language so far removed from ordinary understanding of scientific theories that, for all practical purposes, they cannot be comprehended or ascertained by ordinary mortals.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Identification of counterfactual queries often requires a symbiosis of algebraic and graphical techniques.

“Identification of counterfactual queries, on the other hand, often require a symbiosis of both algebraic and graphical techniques.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Equation (34) generalizes (32) to cases where Z is held fixed at z.

“Equation ( 34) generalizes (32) to cases where Z is held fixed, at z.”

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

Causal conclusions are only as valid as the causal assumptions upon which they rest.

“causal conclusions are only as valid as the causal assumptions upon which they rest.”

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

Even the most experienced potential-outcome expert would be unable to discern whether any subset Z of covariates in Fig. 4 would satisfy conditional ignorability.

“Even the most experienced potential-outcome expert would be unable to discern whether any subset Z of covariates in Fig. 4 would satisfy this conditional independence condition.”

May combine two claims.

The potential-outcome framework is subsumed by the structural theory and excludes ordinary cause-effect relationships from its assumption vocabulary.

“This framework is not a contender for a comprehensive theory of causation for it is subsumed by the structural theory and excludes ordinary cause-effect relationships from its assumption vocabulary.”

May combine two claims.

Shpitser and Pearl (2006) extended Theorem 3 by presenting a necessary and sufficient condition for identification and extending the condition from causal effects to any counterfactual expression.

“Shpitser and Pearl (2006) have further extended this theorem by (1) presenting a necessary and sufficient condition for identification, and (2) extending the condition from causal effects to any counterfactual expression.”

• cites Shpitser and Pearl (2006)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

When e includes all factors u, there is deterministic unit-level causation; when e contains only a few known attributes while others are assigned probabilities, a population-level analysis ensues.

“When e includes all factors u, we have a deterministic, unit-level causation on our hand; when e contains only a few known attributes (e.g., age, income, occupation etc.) while others are assigned probabilities, a population-level analysis ensues.”

Classified as probably not a substantial claim.

Structural equation models, in their counterfactual reading, have removed the difficulty of lacking a language for causal assumptions.

“Structural equation models, in their counterfactual reading, have removed this lingering difficulty by providing the needed language for causal analysis.”

May combine two claims.

Only a tiny fraction of any kernel lends itself to statistical tests; the bulk must remain untestable.

“More importantly, investigators must keep in mind that only a tiny fraction of any kernel lends itself to statistical tests, the bulk of it must remain untestable, at the mercy of scientific judgment.”

Classified as probably not a substantial claim.

To communicate scientific knowledge, the potential-outcome analyst must express assumptions as constraints on P*, usually as conditional independence assertions involving counterfactual variables.

“To communicate scientific knowledge, the potential-outcome analyst must express assumptions as constraints on P * , usually in the form of conditional independence assertions involving counterfactual variables.”

Classified as probably not a substantial claim.

The main drawback of the potential-outcome black-box approach surfaces in problem formulation, where a researcher must articulate the science or causal assumptions behind the problem.

“The main drawback of this black-box approach surfaces in problem formulation, namely, the phase where a researcher begins to articulate the”

May combine two claims.

The potential-outcome approach requires that the problem be reformulated to fit the theory, and all background knowledge must first be translated into counterfactual language before analysis can commence.

“Instead of bringing the theory to the problem, the problem must be reformulated to fit the theory; all background knowledge pertaining to a given problem must first be translated into the language of counterfactuals (e.g., ignorability conditions) before analysis can commence.”

Probably not a substantial claim: may not pass S2 (checkable).

When counterfactual variables are not viewed as byproducts of a deeper process-based model, it is hard to ascertain whether all relevant judgments have been articulated, whether they are redundant, or whether they are self-consistent.

“When counterfactual variables are not viewed as byproducts of a deeper, process-based model, it is hard to ascertain whether all relevant judgments have been articulated, whether the judgments articulated are redundant, or whether those judgments are self-consistent.”

May combine two claims.

Not all questions of causal character can be encoded in P(y|do(x)) type expressions, implying that not all causal questions can be answered from experimental studies.

“Not all questions of causal character can be encoded in P(y|do(x)) type expressions, thus implying that not all causal questions can be answered from experimental studies.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The corresponding unbiased estimands for these causal quantities are readable directly from the diagram.

“The corresponding unbiased estimands for these causal quantities are readable directly from the diagram.”

May combine two claims.

Statistical tests can reject certain kernels in rare cases with testable implications, but the lion's share of supporting causal claims falls on untested causal assumptions.

“Statistical test can be used for rejecting certain kernels, in the rare cases where such kernels have testable implications, but the lion's share of supporting causal claims falls on the shoulders of untested causal assumptions.”

Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

A major impediment to progress in mediation analysis has been the lack of notational facility for expressing the notion of holding mediating variables fixed in the definition of direct effect.

“A major impediment to progress in mediation analysis has been the lack of notational facility for expressing the key notion of”

May combine two claims.

Conditional ignorability qualifies Z as an admissible covariate for adjustment and mirrors the back-door criterion.

“We see that the assumption of conditional ignorability (35) qualifies Z as an admissible covariate for adjustment; it mirrors therefore the”

• AssumptionThe authors state this as an assumption: In Fig. 4, a set Z of covariates satisfies the conditional independence Y_x ⊥⊥ X | Z (conditional ignorability).

Classified as probably not a substantial claim.

The potential-outcome analysis treats the observed distribution as the marginal distribution of an augmented probability function over observed and counterfactual variables.

“The potential-outcome analysis proceeds by treating the observed distribution P(x 1 ,...,x n ) as the marginal distribution of an augmented probability function P * defined over both observed and counterfactual variables.”

Classified as probably not a substantial claim.

The formal definition of causal effect P(y|do(x)) is universally applicable to all models, parametric and nonparametric, through the formation of a submodel Mx.

“For example, the formal definition of the causal effect P(y|do(x)), as given in Eq. ( 7), is universally applicable to all models, parametric as well as nonparametric, through the formation of a submodel M x .”

May combine two claims.

Pearl (2000a, Chapter 7) shows that it is legitimate to treat counterfactuals as jointly distributed random variables, that consistency constraints like (32) are automatically satisfied in the structural interpretation, and that no additional constraints are needed except (33) and (34).

“Pearl (2000a, Chapter 7) shows, using the structural interpretation of Y x (u), that it is indeed legitimate to treat counterfactuals as jointly distributed random variables in all respects, that consistency constraints like (32) are automatically satisfied in the structural interpretation and, moreover, that investigators need not be concerned about any additional constraints except the following”

• cites Pearl (2000a, Chapter 7)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

The structural theory dictates a four-step methodology for every investigation involving causal relationships.

“The methodology dictates that every investigation involving causal relationships (and this entails the vast majority of empirical studies in the health, social, and behavioral sciences) should be structured along the following four-step process:”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

The longevity and natural appeal of structural equations stem from their permitting investigators to communicate causal assumptions formally in the same vocabulary as scientific knowledge.

“Today we understand that the longevity and natural appeal of structural equations stem from the fact that they permit investigators to communicate causal assumptions formally and in the very same vocabulary in which scientific knowledge is stored.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The main conceptual difference between structural and potential-outcome approaches is that structural views do(x) as changing a distribution while keeping variables the same, whereas potential-outcome views Y under do(x) as a different variable Yx.

“The main conceptual difference between the two approaches is that, whereas the structural approach views the intervention do(x) as an operation that changes a distribution but keeps the variables the same, the potential-outcome approach views the variable Y under do(x) to be a different variable, Y x , loosely connected to Y through relations such as (32), but remaining unobserved whenever X = x.”

Classified as probably not a substantial claim.

A collection of constraints of this type might sometimes be sufficient to permit a unique solution to the query of interest.

“A collection of constraints of this type might sometimes be sufficient to permit a unique solution to the query of interest.”

May combine two claims.

Propensity score methods are merely efficient estimators of the right hand side of (25); they entail the same asymptotic bias and cannot be expected to reduce bias if the set S does not satisfy the back-door criterion.

“It should be emphasized, however, that contrary to conventional wisdom (e.g., (Rubin, 2007(Rubin, , 2009))), propensity score methods are merely efficient estimators of the right hand side of (25); they entail the same asymptotic bias, and cannot be expected to reduce bias in case the set S does not satisfy the back-door criterion (Pearl, 2000a(Pearl, , 2009c,d),d).”

• cites Pearl, 2000aNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites 2009c,dNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The distribution P(uY,uX,uZ) induces a well-defined probability on the counterfactual event Yx0=y and on joint counterfactual events.

“Clearly, the distribution P(u Y ,u X ,u Z ) induces a well defined probability on the counterfactual event Y x 0 = y, as well as on joint counterfactual events, such as 'Y x 0 = y AND Y x 1 = y ,' which are, in principle, unobservable if x 0 = x 1 .”

Probably not a substantial claim: may not pass S4 (based).

The notion of holding mediating variables fixed must be interpreted as setting intermediate variables to constants by physical intervention, not by analytical means such as selection, regression, conditioning, matching or adjustment.

“Clearly, this notion must be interpreted as (hypothetically) setting the intermediate variables to constants by physical intervention, not by analytical means such as selection, regression, conditioning, matching or adjustment.”

May combine two claims.

Questions of attribution and susceptibility cannot be answered from experimental studies and cannot be expressed in P(y|do(x)) notation.

“For example, questions of attribution (e.g., what fraction of death cases are due to specific exposure?)or of susceptibility (what fraction of the healthy unexposed population would have gotten the disease had they been exposed?)cannot be answered from experimental studies, and naturally, this kind of questions cannot be expressed in P(y|do(x)) notation.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The surgical definition of counterfactuals provides the conceptual and formal basis for the Neyman-Rubin potential-outcome framework.

“The surgical definition of counterfactuals given by ( 29), provides the conceptual and formal basis for the Neyman-Rubin potential-outcome framework, an approach to causation that takes a controlled randomized trial (CRT) as its ruling paradigm, assuming that nothing is known to the experimenter about the science behind the data.”

Classified as probably not a substantial claim.

Replacing ignorability with the back-door condition in a graphical model permits researchers to understand what conditions covariates must fulfill before they eliminate bias, what to watch for, and what experiments can test covariate selection knowledge.

“Replacing”

Classified as probably not a substantial claim. May combine two claims.

Researchers in the graph-less potential-outcome camp rarely use conditional ignorability to guide covariate choice; they view it as a hoped-for miracle of nature rather than a target of reasoned design.

“researchers in the graph-less potential-outcome camp rarely use”

May combine two claims.

No death case can be tested twice, with and without treatment, which is the reason for a fundamental limitation in attribution.

“The reason for this fundamental limitation is that no death case can be tested twice, with and without treatment.”

Classified as probably not a substantial claim.

P(Y = y|do(X = x)) is equivalent to P(Y_x = y).

“Clearly, P(Y = y|do(X = x)) is equivalent to P(Y x = y).”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

The Mediation Formula represents the average increase in outcome Y that the transition from X=x to X=x' is expected to produce absent any direct effect of X on Y.

“The Mediation Formula represents the average increase in the outcome Y that the transition from X = x to X = x is expected to produce absent any direct effect of X on Y.”

Probably not a substantial claim: may not pass S2 (checkable).

When unified and synthesized, the algebraic and graphical components offer statistical investigators a powerful and comprehensive methodology for empirical research.

“When unified and synthesized, the two components offer statistical investigators a powerful and comprehensive methodology for empirical research.”

May combine two claims.

ETT can be evaluated from observational studies whenever a sufficient set of covariates can be measured that satisfies the back-door criterion, and more generally in a wide class of graphs that permit identification of conditional interventions.

“It can also be evaluated from observational studies whenever a sufficient set of covariates can be measured that satisfies the back-door criterion and, more generally, in a wide class of graphs that permit the identification of conditional interventions.”

• cites Shpitser and Pearl (2009)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim. May combine two claims.

The Mediation Formula of Eq. (49) explicates the target quantity for indirect effect formally and casts it in terms of estimable quantities.

“The Mediation Formula of Eq. ( 49) explicates this target quantity formally, and casts it in terms of estimable quantities.”

Probably not a substantial claim: may not pass S4 (based).

The algebraic component of the structural language coincides with the potential-outcome framework.

“The algebraic component of the structural language coincides with the potential-outcome framework”

Probably not a substantial claim: may not pass S2 (checkable).

The structural theory of causation provides a unifying perspective for recent advances in causal analysis.

“This paper surveys recent advances in causal analysis from the unifying perspective of the structural theory of causation”

Probably not a substantial claim: may not pass S3 (load-bearing).

The graphical component of the structural language embraces Wright's method of path diagrams.

“and its graphical component embraces Wright's method of path diagrams.”

Classified as probably not a substantial claim. May combine two claims.

Causal inference requires two additional ingredients beyond traditional statistics: a science-friendly language for articulating causal knowledge, and a mathematical machinery for processing that knowledge, combining it with data, and drawing new causal conclusions about a phenomenon.

“Causal inference requires two additional ingredients: a science-friendly language for articulating causal knowledge, and a mathematical machinery for processing that knowledge, combining it with data and drawing new causal conclusions about a phenomenon.”

May combine two claims.

In a controlled experiment with X randomized, the distribution of Y is estimated for each level x of X, which is what we normally assess as P(Y = y|do(X = x)).

“This is what we normally assess in a controlled experiment, with X randomized, in which the distribution of Y is estimated for each level x of X.”

May combine two claims.

Counterfactual quantities like DE and IE, which cannot be expressed in do(x) operators, can under certain conditions be estimated from empirical studies and serve to guide policies.

“It is remarkable that counterfactual quantities like DE and IE that could not be expressed in terms of do(x) operators, and appear therefore void of empirical content, can, under certain conditions be estimated from empirical studies, and serve to guide policies.”

Classified as probably not a substantial claim.

The scientific approach to causation models the activities of Nature rather than those of the experimenter.

“Most of all, they demonstrate the effectiveness and viability of the scientific approach to causation whereby the dominant paradigm is to model the activities of Nature, rather than those of the experimenter.”

Classified as probably not a substantial claim.

The concepts explicated by the structural theory include randomization, intervention, direct and indirect effects, confounding, counterfactuals, and attribution.

“These include: randomization, intervention, direct and indirect effects, confounding, counterfactuals, and attribution.”

Probably not a substantial claim: may not pass S4 (based).

The Mediation Formula is readily estimable by regression.

“Moreover, the formula is readily estimable by regression.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The derivation of the Mediation Formula was facilitated by the four steps of the structural methodology together with the graphical-counterfactual-structural symbiosis spawned by the surgical interpretation of counterfactuals.

“The derivation of the Mediation Formula was facilitated by taking seriously the four steps of the structural methodology (Section 4) together with the graphicalcounterfactual-structural symbiosis spawned by the surgical interpretation of counterfactuals (Eq.( 29)).”

May combine two claims.

In nonlinear systems, the values at which Z is held would in general modify the effect of X on Y and thus should be chosen carefully to represent the target policy.

“In nonlinear systems, those values would, in general, modify the effect of X on Y and thus should be chosen carefully to represent the target policy under analysis.”

Probably not a substantial claim: may not pass S2 (checkable).

Non-parametric structural equations models provide a formal and meaningful language for defining causal quantities, formulating causal assumptions, testing identifiability, and explicating many concepts used in causal discourse.

“The theory invokes non-parametric structural equations models as a formal and meaningful language for defining causal quantities, formulating causal assumptions, testing identifiability, and explicating many concepts used in causal discourse.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The Mediation Formula calls for a two-step regression which can in principle be performed non-parametrically.

“It calls therefore for a two-step regression which, in principle, can be performed non-parametrically.”

May combine two claims.

Nonparametric estimation is not always practical; when Z consists of a vector of several mediators, dimensionality prohibits estimation of E(Y|x,z) for every (x,z) cell, requiring parametric approximation.

“Nonparametric estimation is not always practical.When Z consists of a vector of several mediators, the dimensionality of the problem would prohibit the estimation of E(Y|x,z) for every (x,z) cell, and the need arises to use parametric approximation.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Signal sensing is more fundamental to the notion of causation than manipulation; manipulation is a crude way of stimulating signal sensing in experimental setup.

“Pearl (2001) has suggested therefore that signal sensing is more fundamental to the notion of causation than manipulation; the latter being but a crude way of stimulating the former in experimental setup.”

• cites Pearl (2001)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

The second term in (61) represents the correction needed to account for confounding bias.

“The second term represents the correction needed to account for confounding bias, that is, P(y|do(x )) = P(y|x ).”

• AssumptionThe authors state this as an assumption: Y is monotonic relative to X, i.e., Y1(u) ≥ Y0(u).

Probably not a substantial claim: may not pass S3 (load-bearing).

This precludes from the analysis all typical families, in which a father and a grandfather have simultaneous, complementary influences on children's upbringing.

“This precludes from the analysis all typical families, in which a father and a grandfather have simultaneous, complementary influences on children's upbringing.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In linear systems, the direct effect would be undefined whenever indirect paths exist from the cause to its effect.

“In linear systems, to take an even sharper example, the direct effect would be undefined whenever indirect paths exist from the cause to its effect.”

Probably not a substantial claim: may not pass S3 (load-bearing).

This means a grandfather would be deemed to have no direct effect on his grandson's behavior in families where he has had some effect on the father.

“This means that a grandfather would be deemed to have no direct effect on his grandson's behavior in families where he has had some effect on the father.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The first term on the r.h.s. of (61) is the familiar excess risk ratio (ERR) that epidemiologists have been using as a surrogate for PN in court cases.

“The first term on the r.h.s. of ( 61) is the familiar excess risk ratio (ERR) that epidemiologists have been using as a surrogate for PN in court cases (Cole, 1997, Robins andGreenland, 1989b).”

• cites Cole, 1997Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Robins and Greenland, 1989bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: Y is monotonic relative to X, i.e., Y1(u) ≥ Y0(u).

Probably not a substantial claim: may not pass S3 (load-bearing).

In the potential-outcome viewpoint, the direct effect is definable only in units absent of indirect effects.

“For example, the direct effect is definable only in units absent of indirect effects.”