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The average treatment effect in a target population is $\mathbb{E}[Y(1)-Y(0)]$, the average additive contrast between two potential outcomes. It does not require every individual effect to be the same. If half the population has effect two and half effect minus one, the average treatment effect is $1/2$. Under consistency in causal inference, conditional exchangeability given pretreatment covariates $X$, and positivity, it is identified by ${\mathbb{E}}_X[\mathbb{E}(Y\mid A=1,X)-\mathbb{E}(Y\mid A=0,X)]$. The outer average uses the specified target population, not whichever group happens to be observed most often.

The average treatment effect on the treated is $\mathbb{E}[Y(1)-Y(0)\mid A=1]$, averaging over those who actually receive treatment. Their untreated potential outcomes are counterfactual. This can differ from the average treatment effect when effects vary and treatment uptake changes the population mix. Under conditional exchangeability, consistency in causal inference and adequate untreated overlap, it equals $\mathbb{E}[Y\mid A=1]-\mathbb{E}[\mathbb{E}(Y\mid A=0,X)\mid A=1]$. The second term standardizes untreated outcomes to the treated covariate distribution rather than using the crude untreated mean.

A causal effect compares potential outcomes under different specified interventions. For a binary treatment, a unit-level additive effect is $Y(1)-Y(0)$. Only one of those outcomes is observed for a given unit, so an observed treated-versus-untreated difference is not automatically the effect. For example, patients receiving treatment can differ in baseline severity from untreated patients. The average treatment effect averages the contrast over a target population; identifying it from data requires assumptions linking treatment assignment and the unobserved potential outcomes.

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]

The conditional average treatment effect at pretreatment covariates $X=x$ is $\tau (x)=\mathbb{E}[Y(1)-Y(0)\mid X=x]$. It describes average effect variation between strata rather than identifying a particular individual effect. If $\tau (x)=2$ in one equally common stratum and $-1$ in another, the average treatment effect is $1/2$. Under conditional exchangeability, consistency in causal inference and positivity, conditional outcome means identify $\tau (x)$. A predictor measured after treatment cannot simply be substituted for a pretreatment covariate without changing the causal question.

Donald Rubin's work connects Causal inference, missing data and Bayesian statistics. His Harvard profile lists Causal inference in experiments and observational studies, nonresponse in sample surveys, and Bayesian and empirical Bayesian methods as research interests.^[ref]

The potential outcomes framework makes a causal comparison explicit by distinguishing the outcomes that would occur under alternative treatments. Only the outcome under the received treatment is observed for each individual, so identifying an average treatment effect requires design or modelling assumptions. In missing-data analysis, multiple imputation propagates uncertainty by analyzing several completed datasets and pooling their results with Rubin's rules. These ideas explain why filling in one convenient value and then treating it as observed generally understates uncertainty.

Homepage: statistics.fas.harvard.edu/people/donald-b-rubin

Doubly robust estimation combines two nuisance functions so that either one can be correctly estimated while the other is misspecified. For example, let $A\in \{0,1\}$ indicate treatment, $X$ covariates, $m_a(X)=\mathbb{E}[Y\mid A=a,X]$, and $e_a(X)=\mathbb{P}(A=a\mid X)$. Under conditional exchangeability, positivity and the usual consistency of observed and potential outcomes, ${\mu }_a=\mathbb{E}[m_a(X)]$ is the mean potential outcome under treatment $a$.

For candidate functions ${\tilde{m}}_a,{\tilde{e}}_a$ with denominators bounded away from zero, the augmented signal isConditioning on $X$ shows thatThe bias vanishes if either nuisance function is correct. This algebra explains Double robustness and the second-order product bias. It does not protect against both models being wrong, failed Causal identification, or nearly zero denominators. cross-fitting helps when flexible functions are estimated from the same sample; valid confidence intervals require further rate and moment assumptions. The average treatment effect is ${\mu }_1-{\mu }_0$.^[ref]

For binary treatment and propensity score $e(X)=P(A=1\mid X)$, the observed-unit overlap weights areEach unit receives the probability of assignment to the opposite group. These weights are bounded between zero and one.^[ref]

The common target-population tilting function is instead $h(X)=e(X)\{1-e(X)\}$. If the original covariate density is $f(X)$, then the unnormalized weighted densities in the two arms are equal:Thus $h$ describes the target population, while $w_1,w_0$ are the arm-specific observation weights. Multiplying every observed unit by $h(X)$ would not give this balance identity.

Under conditional exchangeability, consistency in causal inference and nonzero total overlap, the weighted contrast identifies the overlap-weighted average treatment effectUnits with deterministic treatment assignment have $h=0$ and contribute nothing to this target. The estimand therefore generally differs from the full-population average treatment effect.

Let $Y(1)$ and $Y(0)$ be the potential outcomes for the same individual under well-defined interventions setting the binary exposure to 1 and 0. The average treatment effect, also called the average causal effect on the additive scale, isThe average is over the target population. This compares hypothetical outcomes under two interventions, rather than the observed difference $\mathbb{E}[Y\mid X=1]-\mathbb{E}[Y\mid X=0]$, which can be distorted by confounding. For binary $Y$, the displayed contrast is a causal risk difference. Consistency in causal inference and a well-defined intervention, including an appropriate no-interference in causal inference assumption, give these potential outcomes their meaning.

Let $e(x)=\mathbb{P}(A=1\mid X=x)$ and let $\hat{e}$ be a consistent estimate. The inverse-probability-weighted estimator of the average treatment effect isUnder the exchangeability, consistency, and positivity conditions in part ii, and consistent estimation of the propensity score, its probability limit isso it consistently estimates the average treatment effect. It does not require the additive outcome-regression model used to interpret the ordinary-least-squares coefficient.

A sufficient causal condition is conditional exchangeabilitytogether with consistency of potential outcomes and positivity in causal inference. For the ordinary-least-squares coefficient itself to equal one common causal effect, also require the correctly specified additive conditional-mean modelThen $\beta$ is the homogeneous treatment effect, and the adjusted coefficient consistently estimates both conditional effects and the average treatment effect in the analyzed population.

The no unmeasured confounding assumption, or conditional exchangeability, isAlso assume consistency of potential outcomes, no interference in causal inference, positivity in causal inference, and finite expectations. For $a\in \{0,1\}$, the law of total expectation, exchangeability, and consistency givePositivity ensures that the observed conditional means exist on the covariate support being averaged. Subtracting the two cases identifies the average treatment effect as

WriteSince $X_1$ has no parents, its total effect has no backdoor path, and

For $X_2$, $X_3$ intercepts every directed path to $X_4$. Conditional on $X_1$, there is no unblocked backdoor path from $X_2$ to $X_3$, and $(X_1,X_2)$ blocks every backdoor path from $X_3$ to $X_4$. The conditional front-door adjustment therefore givesandPart ii shows that the inner sum, after also averaging $X_4$, does not actually depend on $x_1$.

For $X_3$, $(X_1,X_2)$ is a sufficient backdoor adjustment set. HenceThese three formulas identify the requested average treatment effects from the observed joint distribution.

LetandBecause $U$ contains all exposure-outcome confounding and $Z$ is independent of $U$, the exclusion restriction and the law of total expectation giveLikewise,The no confounder-instrument interaction assumption says $\delta (U)=\delta$ is constant. Relevance gives $\delta \ne 0$, so the Wald ratio isthe overall average treatment effect.

The average treatment effect is more relevant than the average treatment effect on the treated because the contemplated policy changes treatment status for schools that are presently non-UTC, not merely the 43 schools that selected into existing UTC status. More precisely, the ideal target would be the average effect among the candidate schools the minister might convert; if that population is represented by all eligible schools, it is the ATE. An ATT from existing UTCs answers a narrower historical question and may not transport to the policy targets.

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.