Pearl 2014 - Selection Diagrams and Definition of Transportability

Summary

A selection diagram augments the common causal diagram of two populations with square $S$-nodes pointing at exactly those mechanisms (functions or exogenous distributions) that may differ between them. Transportability of a causal relation $R$ means $R({\Pi }^{\ast })$ is uniquely computable from the source experiments, source observations and target observations in every model inducing the diagram. Theorem 1 reduces it to a do-calculus derivation in which $S$ appears only as a conditioning variable in $do$-free terms.

Overview

Section 3 showed transportability is a causal notion; Section 4 builds the representation. The key design decision: encode population differences as local modifications of mechanisms, represented by selection variables, so that switching between populations is conditioning on different values of $S$.

Main Content

4.1 Selection variables

Let $P(v\mid do(x))$ be the distribution of $V$ in the experimental study on $\Pi$ (with $X$ randomized). Let $P^{\ast }(v\mid do(x))$ be the distribution of $V$ if the study were conducted on ${\Pi }^{\ast }$. The difference is attributed to a set $S$ of selection variables:

$$P^{\ast }(v\mid do(x))=P(v\mid do(x),s^{\ast })$$

(footnote 16: alternatively write $P(v\mid do(x),s)$ and $P(v\mid do(x),s^{\ast })$; only the location of $S$ matters). $S$ represents all factors by which populations may differ or that “threaten” transport. Each $S_i$ corresponds to a mechanism that differs. In Fig. 4(a), $P(z)\ne P(z\mid s)$ where $S$ stands for all factors responsible for drawing subjects of age $Z=z$ to NYC rather than LA. Equally important is the absence of an $S$ pointing to $Y$ in Fig. 4(a): the age-specific effects are invariant.

Footnote 15: disparities can also come from design differences (two samples drawn by different criteria from one population), usually called sampling selection bias (Heckman 1979; Hernan et al. 2004; Cole and Stuart 2010; Pearl 2013; Bareinboim, Tian and Pearl 2014). This paper treats only nature-induced, not man-made, disparities. (Man-made selection bias is treated in the 2016 paper: Bareinboim 2016 - Sample Selection Bias and Recoverability.)

Definition 4 (Selection diagram)

Let $⟨M,M^{\ast }⟩$ be a pair of structural causal models (Definition 1) relative to domains $⟨\Pi ,{\Pi }^{\ast }⟩$, sharing a causal diagram $G$. $⟨M,M^{\ast }⟩$ is said to induce a selection diagram $D$ if $D$ is constructed as follows:

1. every edge in $G$ is also an edge in $D$;
2. $D$ contains an extra edge $S_i\to V_i$ whenever there might exist a discrepancy $f_i\ne f_i^{\ast }$ or $P(U_i)\ne P^{\ast }(U_i)$ between $M$ and $M^{\ast }$.

Reading a selection diagram.

• An arrow $S\to V_i$: the mechanism assigning $V_i$ may differ. It is a statement of possible difference, not a commitment to a specific one.
• Absence of an arrow into $V_i$: the mechanism is the same in both populations (a strong, testable-in-principle invariance assumption).
• Extreme case: $S$-nodes on every variable = no shared mechanism = no information can be exchanged. The invariance assumptions are what “open the door” for transporting some findings.
• $S$-variables drawn as squares; latent common causes as dashed bidirected arcs.
• Footnote 18: the assumption of no structural changes between domains can be relaxed by starting with $D=G^{\ast }$ and adding $S$-nodes by the same procedure while enforcing acyclicity. Differences in causal directionality (acyclicity fails) or $G$ a strict edge-superset of $G^{\ast }$ need a more elaborate representation (beyond the paper).
• Footnote 17: Pearl (1993, 2009b), Spirtes et al. (1993), Dawid (2002) use conditioning on auxiliary variables to switch between experimental and observational studies; Dawid further uses them for changes in parameters.

The three examples as selection diagrams (Fig. 4).

Fig.	Story	Where $S$ points	Formula dictated
4(a)	populations differ in age distribution (Example 1)	$S\to Z$ ($Z$ = age); no arrow into $Y$	(3.1)
4(b)	populations differ in how $Z$ (reading skill) depends on unmeasured age; age distribution the same (Example 2)	$S\to Z$, $\text{age}\to Z$	(3.2) direct transport
4(c)	populations differ in how $Z$ depends on $X$ (Example 3)	$S\to Z$ with $X\to Z\to Y$	(3.4)

All three also contain dashed arcs $X⇠⇢Y$ for latent confounders. The location of $S$ makes the difference: the same observed $P(z)\ne P^{\ast }(z)$ yields (3.1) in 4(a) and (3.2) in 4(b).

4.2 Transportability

Definition 5 (Transportability)

Let $D$ be a selection diagram relative to domains $⟨\Pi ,{\Pi }^{\ast }⟩$. Let $⟨P,I⟩$ be the pair of observational and interventional distributions of $\Pi$, and $P^{\ast }$ the observational distribution of ${\Pi }^{\ast }$. The causal relation $R({\Pi }^{\ast })=P^{\ast }(y\mid do(x),z)$ is transportable from $\Pi$ to ${\Pi }^{\ast }$ in $D$ if $R({\Pi }^{\ast })$ is uniquely computable from $P,P^{\ast },I$ in any model that induces $D$.

Two connections to identifiability:

1. All identifiable causal relations in $D$ are transportable, since they can be computed directly from $P^{\ast }$ with no experimental information from $\Pi$.
2. Given a causal diagram $G$, one can produce $D$ such that identifiability in $G$ equals transportability in $D$: set $D=G$ and add selection nodes pointing to all variables, meaning the target shares no mechanism with the source. Then the only way to achieve transportability is to identify $R$ from scratch in the target.

Proofs of nontransportability are more involved than proofs of non-identifiability, because they require demonstrating the nonexistence of two competing models compatible with $D$, agreeing on $\{P,P^{\ast },I\}$ and disagreeing on $R({\Pi }^{\ast })$. Definition 5 is declarative; Theorem 1 gives a method.

Theorem 1 (do-calculus criterion for transportability)

Let $D$ be the selection diagram characterizing $\Pi$ and ${\Pi }^{\ast }$, and $S$ a set of selection variables in $D$. The relation $R=P^{\ast }(y\mid do(x),z)$ is transportable from $\Pi$ to ${\Pi }^{\ast }$ if the expression $P(y\mid do(x),z,s)$ is reducible, using the rules of do-calculus, to an expression in which $S$ appears only as a conditioning variable in do-free terms.

Proof

Every relation satisfying the condition can be written as an algebraic combination of two kinds of terms: those involving $S$ and those not. The former can be written as $P^{\ast }$-terms and are therefore estimable from observations on ${\Pi }^{\ast }$, as Definition 5 requires. All other terms, especially those involving $do$-operators, do not contain $S$; they are experimentally identifiable in $\Pi$. $■$

Status. The criterion is sufficient and necessary for causal effects $R=P^{\ast }(y\mid do(x))$ (Bareinboim and Pearl 2012). Theorem 1 is procedural: it does not say which sequence of rules reaches the reduction. Bareinboim and Pearl (2013b) gave a complete procedural solution based on graphical methods developed in Tian and Pearl (2002) and Shpitser and Pearl (2006), but it is non-trivial, so the paper takes an alternative route to a simple transparent procedure guided by two subgoals: trivial and direct transportability.

Definition 6 (Trivial transportability)

A causal relation $R$ is trivially transportable from $\Pi$ to ${\Pi }^{\ast }$ if $R({\Pi }^{\ast })$ is identifiable from $(G^{\ast },P^{\ast })$.

An ordinary graphical identifiability test (Definition 2) on the target diagram; it estimates $R({\Pi }^{\ast })$ directly from observational studies on ${\Pi }^{\ast }$, unaided by causal information from $\Pi$.

Example 4

Let $R=P^{\ast }(y\mid do(x))$ and let the selection diagram be $X\to Y\leftarrow S$. Then $R$ is trivially transportable since $R({\Pi }^{\ast })=P^{\ast }(y\mid x)$ (no confounding of $X$ and $Y$ in the target).

Definition 7 (Direct transportability)

A causal relation $R$ is directly transportable from $\Pi$ to ${\Pi }^{\ast }$ if $R({\Pi }^{\ast })=R(\Pi )$.

Graphical test. For $R=P^{\ast }(y\mid do(x),z)$, direct transportability follows from do-calculus and reads: $(S\perp Y\mid X,Z{)}_{G_{\overline{X}}}$. In words, $X$ blocks all paths from $S$ to $Y$ once arrows pointing to $X$ are removed and $Z$ is conditioned on. It is satisfied in Fig. 4(a): the $z$-specific effect is the same in both populations, hence directly transportable.

Remark: Manski's external validity = direct transportability

The notion of “external validity” defined by Manski (2007) (footnote 1 in Pearl 2014 - Threats vs Licenses and SCM Preliminaries) corresponds to Direct Transportability: $R$ retains its validity without adjustment, as in Eq. (3.2). Such conditions preclude using information from ${\Pi }^{\ast }$ to recalibrate $R$.

Example 5

Let $R$ be the causal effect of $X$ on $Y$, and let $D$ have a single $S$-node pointing to $X$. Then $R$ is directly transportable, because causal effects are independent of the selection mechanism (Pearl 2009b, pp. 72-73): differences in how treatment is assigned do not matter, since $do(x)$ overrides the assignment mechanism.

Example 6

Let $R=P^{\ast }(y\mid do(x),z)$ where $Z$ is a set of variables and $P,P^{\ast }$ differ only in the conditional probabilities $P(z\mid pa(Z))$ and $P^{\ast }(z\mid pa(Z))$, with $(Z\perp Y\mid pa(Z))$, as in Fig. 4(b). Then $R$ is not directly transportable. However, the $pa(Z)$-specific effects $P^{\ast }(y\mid do(x),pa(Z))$ are directly transportable, and so is $P^{\ast }(y\mid do(x))$. Due to confounding arcs, none of these quantities is identifiable (in ${\Pi }^{\ast }$ alone).

Hierarchy of transport notions (summary).

Notion	Condition	Needs target data?	Needs source experiment?
Trivial (Def. 6)	$R$ identifiable from $(G^{\ast },P^{\ast })$	yes, observational only	no
Direct (Def. 7)	$R({\Pi }^{\ast })=R(\Pi )$ ($(S\perp Y\mid X,Z{)}_{G_{\overline{X}}}$ for $z$-specific effects)	no	yes
General (Def. 5)	reducible per Thm 1	observational $P^{\ast }$	yes

Examples

Marketing reading of Definitions 6 and 7

Trivial: if the new market’s ad exposure is effectively randomized by a platform lottery (no confounding of $X$ and $Y$), you can read the effect off the target market’s own data; no transport needed. Direct: if the only difference between test and rollout markets is how media buyers allocated spend (an $S$-node into $X$), Example 5 says the experiment’s effect transfers without adjustment, because the experiment’s randomization washed away the allocation rule.

Connections

• Motivation: Pearl 2014 - Motivating Examples and Transport Formulas. Next, the usable theorems: Pearl 2014 - S-Admissibility and the Graphical Criterion.
• Selection-diagram syntax is reused for sampling selection in Bareinboim 2016 - Sample Selection Bias and Recoverability and for multiple sources in Bareinboim 2016 - Transportability and Data Fusion.
• Estimator-level cousin of ”$S\to Z$ with no $S\to Y$”: the conditional-exchangeability-for-selection assumption $Y^a\perp S\mid X$ in Degtiar 2023 - Identifying Assumptions.
• Identification background: DAGs and Causal Identification.

See Also

• Pearl Bareinboim 2014 - External Validity and Transportability - Overview