= Overlap weight
{title2=$w_1=1-e(X),\quad w_0=e(X)$}
For binary <treatment> and <propensity score> $e(X)=P(A=1\mid X)$, the observed-unit <overlap weights> are
$$
w_1(X)=1-e(X)\quad(A=1),\qquad
w_0(X)=e(X)\quad(A=0).
$$
Each unit receives the <probability> of assignment to the opposite group. These weights are bounded between zero and one.<Balancing Covariates via Propensity Score Weighting by Fan Li, Kari Lock Morgan and Alan M. Zaslavsky>[Sections 2–3: balancing weights and overlap weights]{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:
$$
w_1(X)e(X)f(X)=w_0(X)\{1-e(X)\}f(X)=h(X)f(X).
$$
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 effect>
$$
\tau_{\rm ov}=\frac{\mathbb E[h(X)\{Y(1)-Y(0)\}]}{\mathbb E[h(X)]}.
$$
Units 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>.
Back to article page
