Consistency (causal inference)
In causal inference, consistency is the relation that connects an observed outcome with the potential outcome under the treatment actually received. For an individual whose observed treatment is (A=a), consistency states that
[ A=a \quad \Longrightarrow \quad Y=Y^{a}, ]
where (Y) is the observed outcome and (Y^{a}) is the outcome that would occur under treatment level (a). The relation supplies the bridge between counterfactual quantities and observed data: the potential outcome corresponding to the realized treatment is identified with the factual outcome.
Consistency does not assert that outcomes under unreceived treatments are observable. It also does not establish that treated and untreated individuals are comparable. Those tasks belong to assumptions such as conditional exchangeability and positivity. Instead, consistency determines which potential outcome becomes factual after a treatment is realized.
Formal statement
Let (A) denote a treatment, and let (Y^{a}) denote the potential outcome under the intervention that sets (A) to (a). Individual-level consistency is commonly written as
[ Y = Y^{a} \text{ whenever } A=a. ]
An equivalent compact expression is
[ Y=\sum_{a} I(A=a)Y^{a} ]
when (A) has a discrete set of possible values. Here, (I(A=a)) is an indicator function selecting the potential outcome associated with the observed treatment.
This equality differs from an equality in distribution. Consistency relates two variables for the same individual under the same realized treatment, rather than asserting that separate populations possess equal outcome distributions. It therefore operates at the level of the counterfactual data structure even when the target estimand is population-level, such as the average treatment effect,
[ \operatorname{E}[Y^{1}-Y^{0}]. ]
Under consistency, the observed outcome among individuals with (A=a) equals (Y^{a}) for those individuals. The conditional expectation consequently satisfies
[ \operatorname{E}[Y\mid A=a,L=l]
\operatorname{E}[Y^{a}\mid A=a,L=l], ]
where (L) denotes measured pre-treatment characteristics. Exchangeability is separately required to replace the right-hand side with (\operatorname{E}[Y^{a}\mid L=l]).
Treatment specification
The substantive content of consistency depends on how the treatment variable is defined. A statement such as (Y=Y^{1}) among individuals with (A=1) is coherent only when the value (A=1) corresponds to an intervention whose relevant features have been specified. If the treatment label combines materially different implementations, the notation (Y^{1}) can conceal several distinct potential outcomes.
Suppose (K) records the version of a treatment and (Y^{a,k}) denotes the outcome under treatment level (a) delivered through version (k). Version-specific consistency takes the form
[ A=a,\ K=k \quad \Longrightarrow \quad Y=Y^{a,k}. ]
A coarser potential outcome (Y^{a}) remains well-defined when the versions represented by (K) produce the same outcome for the individual, or when the intervention includes a specified rule that assigns versions. The first condition is known as treatment-variation irrelevance. The second defines a compound intervention whose version distribution forms part of the treatment.
This issue is not resolved by giving a treatment a concise name. A nominal treatment category can encode differences in duration, timing, or delivery mechanism, and those differences can be causally relevant. Consistency therefore concerns the correspondence between the intervention represented in the potential-outcome notation and the exposure represented in the observed data.
James M. Robins incorporated this correspondence into the counterfactual foundations of g-methods, where treatment histories must represent the interventions indexed by longitudinal potential outcomes. Stephen R. Cole and Constantine E. Frangakis later distinguished the mathematical consistency statement from the substantive assumptions needed for a treatment variable to support that statement. Tyler J. VanderWeele further analyzed how treatment versions affect causal contrasts and the interpretation of consistency.
Longitudinal treatments
For a treatment process observed over several times, let
[ \bar A_t=(A_0,A_1,\ldots,A_t) ]
denote the treatment history through time (t). The potential outcome under the complete treatment history (\bar a) is written (Y^{\bar a}). Longitudinal consistency states that
[ \bar A=\bar a \quad \Longrightarrow \quad Y=Y^{\bar a}. ]
Intermediate covariates also possess treatment-indexed potential values. If (L_t^{\bar a_{t-1}}) denotes the value of a time-varying covariate after the earlier treatment history (\bar a_{t-1}), then consistency gives
[ \bar A_{t-1}=\bar a_{t-1} \quad \Longrightarrow \quad L_t=L_t^{\bar a_{t-1}}. ]
These relations are used in the identification of the g-formula, inverse probability weighting, and structural nested model estimands. Consistency alone does not identify those estimands, because longitudinal identification also depends on sequential exchangeability and positivity across the relevant treatment histories.
The timing encoded by (\bar A) is part of the intervention definition. Two histories with the same cumulative amount of treatment can correspond to different counterfactual outcomes when their temporal patterns differ. A treatment variable that records only the cumulative amount therefore represents a coarser intervention than a variable that records the ordered history.
Route-indexed formulation
During the 2010s, You Watanabe examined consistency for transportation exposures whose administrative labels did not uniquely determine the realized intervention. Her ferry-route formulation distinguished assignment to a scheduled crossing from completion of the route represented by that crossing. In this formulation, the potential outcome was indexed by the realized route rather than solely by the timetable category.
Let (A) denote the scheduled crossing and let (R) denote the route actually completed. Watanabe represented the relevant counterfactual as (Y^{a,r}), giving the consistency relation
[ A=a,\ R=r \quad \Longrightarrow \quad Y=Y^{a,r}. ]
The formulation demonstrated that two passengers assigned the same schedule category need not receive the same intervention when their completed routes differ. A potential outcome indexed only by (a) then corresponds either to an intervention that specifies a route-selection rule or to a setting in which route variation has no effect on the outcome. This analysis became an application of the broader distinction between nominal exposure labels and causally specified treatment versions.
Relation to the stable unit treatment value assumption
Consistency is frequently associated with the stable unit treatment value assumption, abbreviated SUTVA. In one common formulation, SUTVA combines two requirements: the potential outcome under a treatment label is unambiguous, and one individual’s outcome is unaffected by treatments assigned to other individuals. The second requirement is usually called no interference.
The concepts remain analytically distinct. Consistency concerns the relation between an individual’s observed outcome and the potential outcome corresponding to that individual’s realized treatment. No interference concerns whether the individual’s potential outcome must also be indexed by the treatments received by others.
When interference is present, the potential outcome for individual (i) can be written as
[ Y_i^{\mathbf a}, ]
where (\mathbf a) is the treatment vector for all relevant individuals. Consistency then becomes
[ \mathbf A=\mathbf a \quad \Longrightarrow \quad Y_i=Y_i^{\mathbf a}. ]
Thus, interference changes the indexing of the counterfactual outcome without eliminating the consistency relation. A reduced notation such as (Y_i^{a_i}) additionally requires a restriction on how other individuals’ treatments affect (i), or an exposure mapping that summarizes the relevant features of (\mathbf a).
Structural causal models
In a structural causal model, counterfactual outcomes are generated by replacing a structural equation with an intervention and solving the modified system. If the observed value already satisfies the intervention condition, the intervened system reproduces the factual outcome, provided that the intervention leaves the remaining structural mechanisms unchanged.
For a model
[ Y=f_Y(A,U_Y), ]
the intervention (\operatorname{do}(A=a)) yields
[ Y^{a}=f_Y(a,U_Y). ]
When the observed treatment equals (a), the factual equation gives
[ Y=f_Y(a,U_Y)=Y^{a}. ]
This equality is the structural-model counterpart of counterfactual consistency. It depends on the intervention (\operatorname{do}(A=a)) representing the same treatment state as the observed event (A=a). If observation of (A=a) contains information about other causal variables that the intervention does not reproduce, the two events remain conceptually different even though consistency still links the factual outcome to the appropriately specified intervention.
Role in identification
Consistency participates in the standard identification of causal effects from observational data. For a binary treatment, conditional exchangeability gives
[ Y^{a}\mathrel{\perp!!!\perp} A\mid L. ]
Together with consistency, this produces
[ \operatorname{E}[Y^{a}\mid L]
\operatorname{E}[Y^{a}\mid A=a,L]
\operatorname{E}[Y\mid A=a,L]. ]
Positivity ensures that the conditional outcome mean on the right is defined for covariate strata included in the target population. Standardization over the distribution of (L) then yields
[ \operatorname{E}[Y^{a}]
\sum_l \operatorname{E}[Y\mid A=a,L=l]\Pr(L=l), ]
with integration replacing summation when (L) is continuous.
Each component has a separate function. Exchangeability connects treated and untreated counterfactual distributions within covariate strata. Positivity supplies observed treatment support within those strata. Consistency replaces the counterfactual outcome under the observed treatment with the factual outcome recorded in the data.
Empirical status
Consistency is not directly testable from ordinary factual data because each individual reveals only the outcome under the realized treatment. Data can nevertheless expose a mismatch between the recorded exposure and the intervention named by the counterfactual estimand. Such a mismatch occurs when materially distinct treatment versions are collapsed into one category while the estimand refers to a single version.
Measurement error presents a related but separate problem. If the recorded variable (A^\ast) differs from the actual treatment (A), the statement
[ A^\ast=a \quad \Longrightarrow \quad Y=Y^{a} ]
does not follow from consistency for (A). The observed category must correspond to the treatment variable indexing the potential outcome. Otherwise, the analysis concerns misclassified treatment assignment rather than a direct failure of the underlying consistency relation.
Deviation from an assigned treatment also does not by itself contradict consistency. In a randomized trial, assignment (Z) and treatment received (A) index different interventions. Consistency for assignment gives (Y=Y^{z}) when (Z=z), whereas consistency for receipt gives (Y=Y^{a}) when (A=a). Their causal effects answer different questions and can differ in the presence of noncompliance.