Causal inference in statistics: An overview

This paper addresses the fundamental gap between statistical association and causal interpretation. Traditional statistics excels at describing patterns in data—correlations, conditional probabilities, and regression coefficients—but these quantities alone cannot answer causal questions: Will raising the minimum wage increase unemployment? Did the drug cause the patient's recovery? What proportion of the effect of education on earnings is mediated through cognitive skills? The core problem is that observational data confound causal mechanisms with selection bias, measurement error, and confounding by common causes. Prior to the formalization of causal diagrams and the do-calculus, researchers lacked a systematic language to state causal assumptions precisely, derive testable implications, and identify causal effects from observational data. This paper synthesizes the Structural Causal Model (SCM) framework, providing a unified mathematical foundation for three classes of causal queries: (1) estimating effects of interventions (causal effects), (2) computing probabilities of counterfactuals (attribution), and (3) decomposing effects into direct and indirect pathways (mediation). It shows how these queries reduce to estimable quantities when the causal graph satisfies certain graphical criteria, and it clarifies the relationship between the structural and potential-outcome frameworks.

The paper builds on the Structural Causal Model (SCM), which consists of three components: a set of endogenous variables (those we model), a set of exogenous variables (unobserved background factors), and a set of structural equations that express each endogenous variable as a deterministic function of its direct causes and an exogenous error term. The causal graph is a directed acyclic graph (DAG) where nodes represent variables and directed edges represent direct causal relationships.

The core idea: Causal inference is fundamentally about reasoning under intervention. The do-operator, denoted do(X=x), represents an intervention that sets variable X to value x, breaking all incoming edges to X in the causal graph. The interventional distribution P(Y | do(X=x)) describes the distribution of Y when we force X to x, which generally differs from the conditional distribution P(Y | X=x) due to confounding.

Three key identification strategies emerge from the graph:

1. Back-door criterion: A set of variables Z satisfies the back-door criterion relative to (X, Y) if (a) no node in Z is a descendant of X, and (b) Z blocks every path between X and Y that contains an arrow into X (back-door paths). When Z satisfies this criterion, the causal effect is identified as: P(Y | do(X=x)) = Σ_z P(Y | X=x, Z=z) P(Z=z) This is the adjustment formula—essentially, condition on all common causes of X and Y.

2. Front-door criterion: When there is an unobserved confounder between X and Y, but we have a mediator M that is not directly confounded with Y, the front-door criterion can identify the effect. It requires: (a) M is not affected by the confounder, (b) X affects Y only through M, and (c) there is no unobserved confounder of M and Y. The effect is then: P(Y | do(X=x)) = Σ_m P(M=m | X=x) Σ_x' P(Y | X=x', M=m) P(X=x') This works by using the mediator to "bootstrap" around the hidden confounder.

3. do-calculus: A complete set of three rules that allow transformation of expressions involving the do-operator into ordinary conditional probabilities. Rule 1 allows insertion/deletion of observations given independence in the interventional graph. Rule 2 allows swapping actions and observations given back-door conditions. Rule 3 allows insertion/deletion of actions given independence conditions. The do-calculus is complete: if a causal effect is identifiable, repeated application of these rules will derive the identification formula.

For counterfactuals: The SCM defines counterfactuals through the structural equations. For a unit with observed covariates U=u, the counterfactual Y_{X=x}(u) is computed by replacing the equation for X with X=x and solving the remaining equations. This yields a three-step procedure: (1) abduction—update the distribution of exogenous variables given observed evidence, (2) action—perform the intervention do(X=x), (3) prediction—compute the resulting distribution of Y. This allows answering queries like "What is the probability that the patient would have recovered had they not taken the drug, given that they actually took the drug and recovered?" (probability of necessity).

For mediation: The paper decomposes the total effect into natural direct effect (NDE) and natural indirect effect (NIE). NDE measures the change in Y when X is set to x while holding the mediator at the level it would take under the reference treatment. NIE measures the change in Y when the mediator is set to the level it would take under treatment x while holding X at the reference. These are identified under sequential ignorability assumptions (no unobserved confounding of X-Y, X-M, and M-Y relationships) using the mediation formula: NDE = Σ_m [E(Y | X=1, M=m) - E(Y | X=0, M=m)] P(M=m | X=0) NIE = Σ_m E(Y | X=0, M=m) [P(M=m | X=1) - P(M=m | X=0)]

Prefer the SCM/do-calculus framework over traditional regression when:

• You have a clear causal question but only observational data
• The causal structure involves multiple pathways, mediators, or colliders
• You need to reason about counterfactuals (e.g., "what if this specific patient had received different treatment?")
• You want to test the logical consistency of your causal assumptions using d-separation and conditional independence tests
• The treatment is not randomly assigned and you suspect complex confounding patterns

Prefer potential-outcomes (Rubin Causal Model) over SCM when:

• You are working in a randomized experiment setting where the assignment mechanism is known
• You need to handle non-compliance, instrumental variables, or principal stratification
• The focus is on finite-sample inference and exact randomization-based tests
• You are uncomfortable with the deterministic structural equations and prefer the inherently probabilistic potential-outcomes formulation

Prefer instrumental variables over SCM when:

• You have a valid instrument (randomly assigned encouragement, natural experiment) and are willing to estimate local average treatment effects (LATE)
• The treatment is endogenous and you cannot measure all confounders
• You are willing to accept that the effect is identified only for compliers

Prefer difference-in-differences or fixed effects over SCM when:

• You have panel data and are willing to assume parallel trends
• The causal structure is simple (single treatment, single outcome, time-invariant confounding)
• You need robustness to time-invariant unobserved confounders without specifying the full causal graph

Prefer the SCM framework over all others when:

• The causal graph is complex with multiple mediators, time-varying treatments, or feedback loops (though DAGs require acyclicity)
• You need to derive identification from first principles using graphical criteria
• You want to combine multiple data sources (observational + experimental) for identification
• You need to reason about transportability (whether causal effects from one population apply to another)

1. Graph specification is subjective and consequential: The entire analysis depends on the correctness of the causal graph. If a key confounder is omitted or an edge direction is reversed, all subsequent inferences can be wrong. In practice, researchers often draw graphs based on domain knowledge, which may be incomplete or incorrect.

2. Hidden confounding remains the central challenge: Even with perfect graph specification, if there are unobserved confounders that are common causes of treatment and outcome, the back-door criterion fails. The front-door criterion offers an alternative but requires a mediator that is not confounded with the outcome—a strong assumption.

3. Deterministic structural equations are unrealistic: The SCM assumes that each variable is a deterministic function of its parents and an exogenous error. In practice, this is a modeling convenience. The framework extends to probabilistic SCMs, but the deterministic formulation can be misleading.

4. Acyclicity is restrictive: The DAG assumption rules out feedback loops, reciprocal causation, and dynamic systems where variables influence each other over time. While cyclic graphs can be handled in some extensions, the standard SCM framework is limited to acyclic systems.

5. No interference is often violated: In social networks, infectious disease, or market settings, one unit's treatment affects another's outcome. The SCM framework can be extended to network settings, but the standard tools (back-door, front-door, do-calculus) assume independence across units.

6. Positivity violations are common in high dimensions: When adjusting for many covariates, it is easy to find strata where no treated or control units exist. This makes nonparametric identification impossible without extrapolation.

7. **Mediation analysis