Causal inference studies effects of interventions by comparing potential outcomes under different treatments.
A causal directed acyclic graph represents variables by vertices and direct causal relations by arrows. Its graphical separation rules encode conditional independences implied by the causal model.
A nonparametric structural equation model assigns each observed variable an arbitrary measurable function of its graphical parents and an exogenous variable. Independence or dependence among exogenous variables determines the graph's latent-confounding structure.
An acyclic directed mixed graph contains directed edges without directed cycles and bidirected edges representing latent dependence. It extends a directed acyclic graph to causal models with unobserved common causes.
A district is a maximal set of vertices connected by a path consisting entirely of bidirected edges.
A vertex is fixable when its district contains no proper directed descendant of that vertex. Equivalently, .
M-separation extends d-separation to mixed graphs: a path is open given a conditioning set when every noncollider is unconditioned and every collider has a conditioned descendant.
Front-door adjustment identifies an exposure effect through an observed mediator when the mediator intercepts every directed exposure-outcome path, the exposure-mediator relation has no unblocked backdoor path, and exposure blocks every backdoor path from mediator to outcome.
A backdoor path creates noncausal association by entering the exposure through an arrowhead rather than beginning with a causal arrow leaving it.
A backdoor path from exposure to outcome begins with an arrow entering . An adjustment set identifies the total effect when it blocks every such path without conditioning on descendants of .
In a linear structural equation model, each variable is a linear combination of its graphical parents and an exogenous error term. Edge coefficients quantify direct effects.
Wright's path tracing rule expresses a covariance in a standardized linear structural equation model as a sum of products of edge coefficients over admissible unblocked paths.
The potential outcomes framework represents causal effects by comparing outcomes assigned to the same unit under different hypothetical interventions.
A causal null hypothesis states that an intervention has no effect on a specified potential-outcome contrast. A sharp causal null sets every unit's treated and untreated potential outcomes equal; an average causal null sets their population mean difference to zero.
A heterogeneous treatment effect varies across units or covariate values rather than being constant throughout the population. The conditional average treatment effect describes one common form of this heterogeneity.
The average treatment effect on the treated is .
Confounding is distortion of an exposure-outcome association by common causes or associated background variables.
Ignorability is an assumption under which treatment assignment carries no information about relevant potential outcomes after conditioning on specified covariates.
Positivity requires every compared treatment to have positive probability at each target covariate value.
Inverse probability weighting reweights each observed outcome by the reciprocal probability of receiving its observed treatment, creating a weighted population in which treatment is balanced across measured covariates.
For binary treatment with propensity score , the overlap weight emphasizes covariate strata in which both treatments occur with appreciable probability.
Consistency equates the observed outcome with the potential outcome under the treatment received.
Selection bias occurs when inclusion in the analyzed sample depends on variables related to the exposure and outcome, making the observed association differ from its target-population counterpart.
Collider bias arises when analysis conditions on a common effect of two variables, thereby creating a statistical association between its causes.
Internal validity is the extent to which an estimate identifies the intended effect within the population and setting actually studied.
External validity is the extent to which a result transports or generalizes from the studied sample and setting to a target population.
A regression discontinuity design compares units immediately on either side of a treatment-assignment cutoff. Its causal interpretation requires potential outcomes to vary continuously through that cutoff in the absence of treatment.
Difference-in-differences compares outcome changes across exposed and unexposed groups. Its standard identifying condition is that their untreated outcomes would have followed parallel trends.
Matching compares treated and control units with similar pretreatment covariates.
A negative control outcome cannot plausibly be caused by treatment, so an estimated effect can reveal bias.
A negative control group is chosen so that the proposed causal mechanism should not operate in it. Reproducing the target association in that group is evidence for residual bias or an alternative mechanism.
This sensitivity analysis quantifies how strong an omitted association must be to alter a causal conclusion.
Random treatment assignment makes treatment independent of baseline potential outcomes in expectation.
Complete randomization chooses uniformly among all treatment assignments having prespecified treatment-arm sizes.
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Causal inference is a field of study that focuses on drawing conclusions about causal relationships between variables. Unlike correlation, which merely indicates that two variables change together, causal inference seeks to determine whether and how one variable (the cause) directly affects another variable (the effect). This is crucial in various fields such as epidemiology, economics, social sciences, and machine learning, as it informs decisions and policy-making based on understanding the underlying mechanisms of observed data.