The average treatment effect is a population functional, but its meaning changes with the target population, treatment versions, and effect scale. This page derives its principal representations and distinguishes it from nearby causal contrasts.
Population and subgroup effects
Definition. For a target distribution of \(X\), \[ \tau_{\mathrm{ATE}}=\mathbb E\{Y(1)-Y(0)\}, \quad \tau_{\mathrm{ATT}}=\mathbb E\{Y(1)-Y(0)\mid A=1\}, \] and \[ \tau(x)=\mathbb E\{Y(1)-Y(0)\mid X=x\}. \] More generally, a weighted average treatment effect is \[ \tau_h= \frac{\mathbb E[h(X)\{Y(1)-Y(0)\}]} {\mathbb Eh(X)} \] for a nonnegative target weight \(h\).
Proposition. \[ \tau_{\mathrm{ATE}}=\mathbb E\{\tau(X)\}. \]
Proof. Apply iterated expectation to \(Y(1)-Y(0)\) conditional on \(X\).
Observed-data identifying functionals
Theorem. Under consistency, conditional exchangeability given \(X\), and positivity, the following are equal: \[ \begin{aligned} \tau_{\mathrm{ATE}} &=\mathbb E\{m_1(X)-m_0(X)\}\\ &=\mathbb E\left[ \frac{AY}{e(X)}-\frac{(1-A)Y}{1-e(X)} \right]\\ &=\mathbb E\left[ m_1(X)-m_0(X)+ \frac{A\{Y-m_1(X)\}}{e(X)} -\frac{(1-A)\{Y-m_0(X)\}}{1-e(X)} \right]. \end{aligned} \]
Proof. The first equality is standardization. The inverse-weighted equality follows by conditioning each weighted outcome on \(X\). In the third line, each residual term has conditional expectation zero when its corresponding outcome regression is correct, so its expectation equals the first line.
The ATT weighting identity
Theorem. Under mean exchangeability for \(Y(0)\) and positivity, \[ \tau_{\mathrm{ATT}} = \mathbb E(Y\mid A=1) -\frac{1}{P(A=1)} \mathbb E\left[ \frac{(1-A)e(X)}{1-e(X)}Y \right]. \]
Proof. Consistency identifies \(\mathbb EY(1)\mid A=1\) as \(\mathbb E(Y\mid A=1)\). For the missing treated-group control mean, \[ \mathbb E\left[ \frac{(1-A)e(X)}{1-e(X)}Y\right] =\mathbb E[e(X)m_0(X)] =P(A=1)\mathbb E\{m_0(X)\mid A=1\}. \] Exchangeability identifies the final conditional mean with \(\mathbb E\{Y(0)\mid A=1\}\).
Risk ratios and odds ratios
Definition. For binary outcomes, \[ \operatorname{RD}=\mu_1-\mu_0,\qquad \operatorname{RR}=\frac{\mu_1}{\mu_0},\qquad \operatorname{OR}= \frac{\mu_1/(1-\mu_1)}{\mu_0/(1-\mu_0)}, \] where \(\mu_a=\mathbb EY(a)\).
These are marginal causal effects. A conditional logistic-regression coefficient is generally not the marginal causal log-odds ratio, even without confounding, because odds ratios are noncollapsible.
Example. Suppose treatment has no effect and \(P(Y=1\mid A,X)=\operatorname{expit}(\alpha+\gamma X)\), with \(A\perp X\). The conditional treatment odds ratio is one, and so is the marginal one. If a nonzero constant conditional log-odds effect is introduced, averaging over heterogeneous baseline risks generally yields a marginal odds ratio different from the conditional coefficient, despite randomization.
Treatment-effect heterogeneity
Proposition. For any partition \(G\in\{1,\ldots,K\}\), \[ \tau_{\mathrm{ATE}}=\sum_{g=1}^K P(G=g)\tau_g, \qquad \tau_g=\mathbb E\{Y(1)-Y(0)\mid G=g\}. \]
A zero ATE can conceal large effects of opposite signs. Subgroup discovery, however, is a model-selection problem: estimating many \(\tau_g\) on the same data without honest splitting creates adaptive bias. Causal forests and meta-learners require complexity control and overlap within subgroups.
Policy value is not the ATE
Definition. A deterministic treatment rule \(d:\mathcal X\to\{0,1\}\) has value \[ V(d)=\mathbb E\{Y(d(X))\} =\mathbb E[m_{d(X)}(X)] \] under identification. Relative to treating nobody, \[ V(d)-V(0)=\mathbb E[d(X)\tau(X)]. \]
Proof. Pointwise, \(Y(d(X))=d(X)Y(1)+\{1-d(X)\}Y(0)\). Take conditional expectations given \(X\), then subtract \(\mathbb Em_0(X)\).
The unconstrained optimal rule treats when \(\tau(X)>0\). Resource constraints, treatment costs, and risk aversion change the decision criterion. Estimating the ATE alone is insufficient for individualized policy learning.
Finite-sample estimators under randomization
Theorem. Under Bernoulli randomization with known probability \(p\), the Horvitz-Thompson estimator \[ \widehat\tau_{\mathrm{HT}}= \frac1n\sum_{i=1}^n \left\{\frac{A_iY_i}{p}- \frac{(1-A_i)Y_i}{1-p}\right\} \] is unbiased for the finite-sample average effect conditional on all potential outcomes.
Proof. Conditional on potential outcomes, \[ \mathbb E_A\left(\frac{A_iY_i}{p}\right)=Y_i(1), \qquad \mathbb E_A\left(\frac{(1-A_i)Y_i}{1-p}\right)=Y_i(0). \] Average over units.
Estimand discipline
A reported effect should state the target population, treatment intervention, comparison intervention, outcome horizon, summary scale, and handling of competing events. Changing any of these changes the estimand before it changes the estimator.