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Causal inference uses data and explicit assumptions to learn effects of specified interventions. For binary treatment $A$, a target average treatment effect is $\mathbb{E}[Y(1)-Y(0)]$, comparing potential outcomes for the same population. Observing only $Y=Y(A)$ leaves one of each unit's potential outcomes unobserved.

An observed treated-minus-control difference can include confounding. For a concrete model, suppose half a population is in a low-baseline stratum with $(Y(0),Y(1))=(0,1)$ and half in a high-baseline stratum with $(2,3)$. Let treatment probabilities be $1/4$ and $3/4$ in the two strata. Every unit's causal effect is one. The treated mean is $\frac{1}{4}(1)+\frac{3}{4}(3)=2.5$, while the untreated mean is $\frac{3}{4}(0)+\frac{1}{4}(2)=0.5$. Their observed difference is two because their stratum compositions differ.

Under conditional exchangeability, positivity and consistency in causal inference, outcome standardization identifies the effect byIn the example, using the same half-and-half target weights for both strategies recovers one. The substantive assumptions, rather than the arithmetic alone, justify interpreting the standardized contrast as an intervention effect.^[ref]

G-computation calculates the outcome distribution under a specified intervention from conditional laws identified under a causal model. For a single binary treatment, pretreatment covariates $X$, and outcome $Y$, the G-formula isIt follows by averaging over $X$, using conditional exchangeability to replace the conditional potential outcome mean by the mean among recipients of $a$, and using consistency in causal inference to replace $Y(a)$ by observed $Y$. Positivity ensures the required conditional outcome laws are represented in the target strata.^[ref]

For an illustrative binary outcome, suppose target weights are $0.8$ and $0.2$ in two baseline strata. Untreated risks $0.1,0.4$ give intervention risk $0.8(0.1)+0.2(0.4)=0.16$. Treated risks $0.05,0.2$ give $0.08$, so the standardized causal risk ratio is $0.5$ and the risk difference is $-0.08$. Both strategies use the same target weights.

With repeated treatments and time-varying covariates, the longitudinal G-formula integrates products of the covariate transition laws while setting each treatment according to the intervention strategy. This requires sequential exchangeability, longitudinal positivity and consistency in causal inference. Simply conditioning on every observed intermediate variable can block part of the intended effect; the sequential intervention calculation specifies which laws are averaged.

James M. Robins develops statistical methods for causal inference in epidemiology. His joint exposition with Miguel A. Hernán explains how inverse probability weighting can estimate population treatment effects from observational studies under appropriate identifying assumptions.^[ref]

Weighting gives more influence to people whose observed treatment was relatively unlikely given their measured baseline covariates. In the resulting weighted comparison, measured treatment-selection differences are removed under the assumed model. Validity still depends on conditional exchangeability, consistency of potential outcomes and positivity in causal inference; extreme weights also expose a practical difficulty when some treatment choices are rare.^[ref]

Miguel A. Hernán develops causal inference methods for epidemiological data. With James M. Robins, he explains how observational studies can estimate treatment effects when the treatment groups satisfy appropriate conditional exchangeability assumptions.^[ref]

One method is outcome standardization: estimate outcome risks within baseline covariate strata under each treatment and average both sets of risks over the same target population. The comparison depends on identifying assumptions, including adequate measurement of confounders, consistency of potential outcomes and positivity in causal inference.^[ref]

A randomized controlled trial assigns experimental units to competing treatments by a specified random mechanism and compares their outcomes. Under complete randomization of $N$ units, exactly $n_1$ receive assignment one and $n_0=N-n_1$ assignment zero. With fixed potential outcomes, each unit has assignment probability $n_1/N$. ThereforeThis proves design-unbiasedness of the mean difference for the finite-population average treatment effect, assuming consistency in causal inference and no interference in causal inference.

Randomization controls the assignment distribution; it does not force every realized baseline covariate difference to be zero. Equal assignment probabilities yield unconditional exchangeability in the design, whereas covariate-dependent randomization can require conditional exchangeability and an analysis respecting that design.^[ref]

When assigned and received treatments differ, an intention-to-treat analysis estimates an assignment-policy effect. A treatment-receipt effect needs additional assumptions. Allocation concealment, blinded outcome assessment and handling of missing follow-up address other sources of distortion beyond random assignment itself.