Causal inference

Causal inference studies effects of interventions by comparing potential outcomes under different treatments.

Outcome standardization averages fitted conditional outcome risks over a specified target population's pretreatment covariate distribution. The same target distribution is used for each treatment strategy. Under conditional exchangeability, the positivity assumption and consistency, this average identifies the target mean of the corresponding potential outcomes. For a target consisting of treated subjects, average over their covariate distribution. A conditional odds ratio cannot generally be applied directly to an aggregate mean risk: the nonlinear odds-to-probability conversion must be performed before averaging. Outcome standardization supplies that individual-risk calculation.

Causal mediation analysis decomposes a total causal effect into effects transmitted through a designated mediator and effects operating through other pathways. Identification generally requires assumptions about exposure-outcome, exposure-mediator, and mediator-outcome confounding.

For binary exposure $A$, the total causal effect on an additive scale is $\mathbb{E}[Y(1)-Y(0)]$. With mediator potential outcome $M(a)$ and nested outcome $Y(a,m)$, consistency gives $Y(a)=Y(a,M(a))$.

One natural direct effect iswhich changes exposure while holding the mediator at its natural value under no exposure.

One natural indirect effect iswhich changes the mediator between its two natural values while holding exposure fixed.

Mediator-outcome confounding occurs when a common cause of the mediator and outcome is not adequately controlled. If exposure itself affects such a common cause, ordinary adjustment can fail to identify natural direct and indirect effects.

Confounding by indication occurs when treatment choice depends on disease severity or prognosis. The treated group can then have worse outcomes because it was at higher baseline risk, even if treatment is beneficial.

A causal estimand is identified when every causal model satisfying the stated assumptions and inducing the same observed-data distribution gives it the same value. An identifying formula expresses that value entirely through the observed-data distribution.

The g-formula identifies an interventional outcome distribution by averaging observed conditional outcome laws over the covariate distribution under suitable consistency, exchangeability, and positivity assumptions.

For treatment history ${\bar{A}}_t$, observed history $H_t$, and final potential outcome $Y(\bar{a})$, sequential exchangeability requiresat every treatment time. Together with consistency of potential outcomes and positivity in causal inference, it identifies longitudinal effects by G-computation.

An exposure is a condition, behavior, intervention, or environmental factor whose association or causal effect on an outcome is studied.

A treatment is an intervention assigned or received by a study unit. In causal inference it indexes the unit's potential outcomes.

A statistical association is a systematic dependence between variables. It need not represent a causal effect, because it may arise from confounding, selection, or reverse causation.

An observational study measures exposures and outcomes without assigning the exposure by an intervention. Its causal interpretation therefore requires assumptions that address confounding and selection bias.

A case–control study selects subjects according to an outcome: cases have the outcome and controls represent an appropriate comparison population. It compares their prior exposures or genetic variants to study association. Its retrospective sampling must be accounted for when interpreting risks; selection bias and confounding can distort an unadjusted association. In genetics, matched family pseudo-control genotypes can replace unrelated controls for a conditional transmission disequilibrium test.

A longitudinal study observes the same individuals repeatedly. Within-person changes distinguish temporal development from persistent differences between people, although calendar-period changes, selective dropout and measurement error still require attention. A random-intercept linear mixed model can represent persistent individual differences and correlated repeated measurements.

A cross-sectional study measures different individuals during one observation period. Differences between ages can reflect confounding or birth-cohort differences as well as changes within individuals; a longitudinal study is needed to measure the latter directly.

Reverse causality occurs when an apparent exposure-outcome association is partly or wholly generated by the outcome, or by an early stage of it, affecting the measured exposure.

An instrumental variable changes an exposure or treatment but affects the outcome only through that exposure. Together, instrument relevance, instrumental-variable independence, and the exclusion restriction identify causal effects under suitable structural assumptions.

For a binary instrumental variable $Z$, exposure $X$, and outcome $Y$, the Wald estimator is the ratio of the instrument's effect on the outcome to its effect on the exposure:Under instrumental-variable assumptions it estimates a causal effect.

Instrument relevance requires the instrumental variable to change the conditional distribution of the exposure. A nearly irrelevant instrument is a weak instrument and can produce unstable estimates.

A weak instrument has only a small association with the endogenous exposure, making instrumental-variable estimators imprecise and potentially badly biased in finite samples.

Instrumental-variable independence requires the instrument to be independent of unmeasured causes of the outcome, usually conditionally on specified observed covariates.

The exclusion restriction requires an instrumental variable to affect the outcome only through the exposure whose causal effect is being studied.

Mendelian randomization uses inherited genetic variants associated with an exposure as instrumental variables. Its causal interpretation requires instrument relevance, instrumental-variable independence, and the exclusion restriction.

Within-family Mendelian randomization uses genetic contrasts within families, such as differences between siblings. Conditioning on shared parental background can reduce population stratification and family-level sources of confounding, although the exclusion restriction and sufficient instrument relevance remain necessary.

MR-Egger regression fits genetic outcome associations to genetic exposure associations with a free intercept. Its slope estimates a causal effect under the InSIDE assumption and appropriate instrument-strength assumptions; its intercept assesses average directional horizontal pleiotropy. A nonsignificant intercept, especially with few genetic variants, does not prove the exclusion restriction for every instrument.

InSIDE stands for instrument strength independent of direct effect. It requires the genetic associations with the exposure to be independent of the direct, pleiotropic effects of those genetic variants on the outcome. It is a key identifying assumption for MR-Egger regression.

This weighted median of genetic variant ratio estimates consistently estimates a common causal effect when more than half of its asymptotic weight comes from valid instrumental variables, subject to the usual sampling regularity conditions. It is a sensitivity analysis for horizontal pleiotropy, not an unconditional guarantee of validity.

Two-sample Mendelian randomization combines genetic variant associations with an exposure estimated in one sample and their associations with an outcome estimated in another. It requires the instrumental variable assumptions and compatible associations across the populations; sample overlap can change the behaviour of weak instruments.

Horizontal pleiotropy occurs when a genetic instrument affects an outcome through a pathway other than the exposure under study. It violates the exclusion restriction in Mendelian randomization.

Residual exposure stratification subtracts the genetically predicted component of an exposure before forming exposure strata. Under a suitable additive first-stage model, this avoids directly conditioning on the component caused by the instrument and can reduce the resulting collider bias.

A confounder-instrument interaction occurs when an instrument's effect on treatment varies with a variable that also affects the outcome. If the conditional first-stage effect is $\delta (U)=\mathbb{E}[A\mid Z=1,U]-\mathbb{E}[A\mid Z=0,U]$, absence of this interaction means that $\delta (U)$ is constant.

Instrumental-variable monotonicity requires changing the instrument in its encouraging direction never to move any unit's exposure in the opposite direction.

The local average treatment effect is the average causal effect among compliers, the units whose treatment status changes in the direction encouraged by the instrument.

Interference occurs when one unit's treatment changes another unit's potential outcome. It violates the usual assumption that each unit's potential outcomes depend only on that unit's own treatment.

The ecological fallacy is an invalid inference about individuals from associations measured only for groups or populations.

A positive association between a community's demographic composition and its incidence rate does not establish what proportion of incidents involved people from that demographic group. The same aggregate totals can be generated by different offender compositions and mechanisms. Individual incident records with age, sex and role would be needed to identify such proportions; aggregate regression coefficients concern community-level association.

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 probability distribution is causally minimal for a Directed acyclic graph when it satisfies the global Markov property for a directed acyclic graph for that graph and for no proper subgraph obtained by deleting edges on the same vertex set. It excludes redundant edges without excluding every extra conditional independence.

Let $E_1,E_2,E_3$ be independent random variables with the standard normal distribution and set $Z_1=E_1$, $Z_2=Z_1+E_2$, $Z_3=Z_2-Z_1+E_3$. The complete acyclic graph with edges $1\to 2$, $1\to 3$, $2\to 3$ is causally minimal: deleting each edge imposes a false marginal or conditional independence. Yet $Z_1\perp Z_3$ because $Z_3=E_2+E_3$, so faithfulness of a directed acyclic graph fails. Nonzero structural coefficients alone do not ensure faithfulness.

The PC algorithm starts with a candidate undirected graph, removes edges using conditional independence tests over increasing conditioning-set sizes, records separating sets, then orients unshielded colliders and propagates orientations. Under oracle tests and faithfulness of a directed acyclic graph, it estimates the graph's Markov equivalence class.

With an exact conditional independence oracle and faithfulness of a directed acyclic graph, the skeleton phase of the PC algorithm deletes precisely the nonedges. A true edge cannot be D-separated by a set excluding its endpoints. For nonadjacent vertices, choose the later vertex $v$ in a topological ordering; its parents separate it from the earlier nonparent. True parent edges survive throughout, so this separator remains available among the algorithm's candidate neighbour sets. For an unshielded triple $u-v-w$, any recorded separator of $u,w$ excludes $v$ exactly when the triple is an unshielded collider. The skeleton and collider characterization of Markov equivalence therefore identifies the correct equivalence class. Directions compelled throughout that class give its completed partially directed acyclic graph.

Initialize the PC skeleton search with an estimated conditional independence graph to reduce candidate pairs and conditioning sets. An exact graph contains the true Directed acyclic graph skeleton because it equals its moral graph under faithfulness. For initially absent pairs record the full complement as a separating set; an empty default can create false collider orientations. Estimation errors can affect this screening step.

A distribution factorizes according to a Directed acyclic graph when its joint density is the product of the conditional density of each variable given its graphical parents. This factorization implies the graph's global Markov property of a directed acyclic graph.

A distribution has the global Markov property for a Directed acyclic graph when every D-separation of vertex sets by a conditioning set implies the corresponding conditional independence of their random variables.

A distribution is faithful to a Directed acyclic graph when, for all disjoint vertex sets $A,B,S$, $Z_A\perp Z_B\mid Z_S$ holds if and only if $A$ and $B$ are D-separated by $S$. This includes the graph's global Markov property of a directed acyclic graph and excludes additional independences caused by parameter cancellations. If faithfulness is used to name only the reverse implication, the Markov property must be assumed separately.

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 Verma constraint is an equality restriction on an observed distribution implied by a latent-variable causal graph that need not be a conditional independence. Such constraints can therefore distinguish latent causal models whose ordinary observed conditional-independence relations coincide.

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, $\mathrm{dis}(v)\cap \mathrm{de}(v)=\{v\}$.

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.

Conditional front-door adjustment applies the front-door argument within strata of observed covariates and then averages over those covariates. The covariates must remove exposure-mediator confounding, while the exposure together with the covariates must remove mediator-outcome confounding.

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 $X$ to outcome $Y$ begins with an arrow entering $X$. An adjustment set identifies the total effect when it blocks every such path without conditioning on descendants of $X$.

A backdoor adjustment set contains no descendant of the exposure and blocks every backdoor path from the exposure to the outcome. Averaging the conditional outcome distribution over such a set identifies the total causal effect.

A sufficient adjustment set is a collection of pretreatment variables for which treatment and every relevant potential outcome are conditionally independent. Adjustment for that set identifies the corresponding causal contrast under consistency and positivity.

A minimal sufficient adjustment set is a sufficient adjustment set none of whose proper subsets is sufficient. Minimality is by set inclusion and does not imply minimum cardinality or uniqueness.

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.

Principal stratification groups units by the joint values of their potential values of an intermediate variable under every treatment assignment. With a binary instrument and binary treatment, the principal strata are always-takers, never-takers, compliers, and defiers.

A principal stratum is one group in a principal stratification.

A potential outcome $Y(a)$ is the outcome that would be observed under intervention $a$.

A causal effect compares potential outcomes under different interventions.

A treatment effect is homogeneous when the relevant contrast of potential outcomes is constant across units. For an additive treatment, this takes the form $Y(a)-Y(0)=\beta a$.

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 structural zero is an outcome value fixed at zero by the data-generating mechanism under a specified intervention or condition, rather than a zero arising by chance.

The average treatment effect is $\mathbb{E}[Y(1)-Y(0)]$ in a target population.

The conditional average treatment effect at covariates $X=x$ is

The overlap-weighted average treatment effect averages the conditional average treatment effect with weight proportional to $e(X)\{1-e(X)\}$, where $e(X)$ is the propensity score. It emphasizes covariate strata in which both treatments are plausible.

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 $\mathbb{E}[Y(1)-Y(0)\mid A=1]$.

For binary potential outcomes, the causal risk ratio compares intervention risks:

Confounding is distortion of an exposure-outcome association by common causes or associated background variables.

Case mix describes the distribution of patient characteristics affecting prognosis, such as age, severity, comorbidity and urgency. Comparing provider outcomes without accounting for case mix may confuse differences in patients with differences in care. Adjustment requires comparable, reliably measured baseline variables and does not remove all possible confounding.

A risk-adjusted provider comparison contrasts observed outcomes with outcomes expected from a prespecified patient-risk model, or standardizes providers to a common patient population. For individual fitted risks $p_i$, the expected event count is ${\sum }_i{p}_i$. Patient selection, model calibration, omitted risk factors, small samples and dependence between outcomes can still affect interpretation. Funnel plots display precision differences that an ordered league table can obscure.

A comparison can have the same direction within every background stratum but reverse after pooling strata with different group compositions. For example, let unexposed event risks be $0.001$ in a low-risk stratum and $0.05$ in a high-risk stratum, and multiply the odds by $2.7$ for exposure within each stratum. The exposed risks are then approximately $0.002695$ and $0.124424$. If 80% of unexposed subjects but only 1% of exposed subjects belong to the high-risk stratum, the pooled unexposed risk is $0.0402$ while the pooled exposed risk is about $0.003913$. Every conditional odds ratio is above one, yet the pooled exposed risk is smaller. The differing background distributions create the reversal; this example does not identify a causal effect without assumptions on how exposure was assigned.

A confounder predicts both treatment assignment and outcome and can create a noncausal association.

Ignorability is an assumption under which treatment assignment carries no information about relevant potential outcomes after conditioning on specified covariates.

Conditional exchangeability requires $(Y(0),Y(1))\perp A\mid X$ for sufficient pretreatment covariates $X$.

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.

The propensity score is the conditional probability $\pi (X)=\mathbb{P}(A=1\mid X)$ of receiving a binary treatment given observed covariates. Under conditional exchangeability, conditioning on the propensity score balances the covariate distribution between treatment groups.

For binary treatment, outcomes $Y_i$, and estimated propensity score $\hat{e}(X_i)$, the inverse-probability-weighted estimator isIt is consistent under conditional exchangeability, consistency of potential outcomes, positivity in causal inference, and a consistent propensity-score estimator.

For binary treatment with propensity score $\pi (X)$, the overlap weight $\pi (X)\{1-\pi (X)\}$ emphasizes covariate strata in which both treatments occur with appreciable probability.

Consistency equates the observed outcome with the potential outcome under the treatment received.

An effect modifier is a variable across whose values the causal effect differs.

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.

Survivorship bias occurs when analysis conditions on remaining observable or event-free long enough to enter a sample, thereby excluding earlier failures and overrepresenting longer survivors.

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.

Covariate balance means treatment groups have similar pretreatment covariate distributions.

A standardized mean difference divides a group mean difference by a pooled standard deviation.

This gives a dimensionless effect size, so mean differences can be compared across measurement scales.