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Alan M. Zaslavsky develops statistical methods for measuring health-care quality and disparities. His research includes missing data, hierarchical Bayesian models and Bayesian inference, bringing statistical adjustment to comparisons between health-care providers and populations.^[ref]

With Fan Li and Kari Lock Morgan, he developed a framework for balancing covariates through propensity score weighting. Their overlap weights emphasize people with a substantial chance of receiving either treatment, defining an overlap-weighted average treatment effect for that comparison population.^[ref]

Fan Li is the Duke Statistical Science researcher who coauthored Balancing Covariates via Propensity Score Weighting by Fan Li, Kari Lock Morgan and Alan M. Zaslavsky. Her author homepage and the paper's institutional byline identify this particular researcher.^[ref]

For propensity score $e(X)$, overlap weights assign weight $1-e(X)$ to a treated person and $e(X)$ to an untreated person. These bounded weights emphasize shared treatment eligibility and target a population with density proportional to $e(X)(1-e(X))$. Causal interpretation still requires the relevant no unmeasured confounding assumption and positivity in causal inference.^[ref]

Inverse probability weighting weights observations by reciprocals of the probabilities that generated their observed assignment or selection. For binary treatment with propensity score $e(X)$, the average-treatment-effect observation weight isIt uses the probability of the treatment actually received. Under conditional exchangeability, consistency in causal inference and positivity, the identification identity isTo prove it, condition on $X$: the assignment probability cancels its denominator, leaving $\mathbb{E}(Y\mid A=a,X)$. Average over $X$ and apply the causal identification assumptions. The same cancellation with a function $g(X)$ instead of $Y$ proves population balance of the measured covariates under correctly specified assignment probabilities.

If a stratum has treatment probability $0.2$, each treated observation there receives weight five: it represents units with the same measured baseline characteristics who usually receive the other treatment. Very small assignment probabilities give large weights and unstable estimates. Estimated assignment models, trimming and normalization can change finite-sample behavior and, for trimming, the target population. Overlap weights deliberately define a different target rather than estimating the same full-population contrast by another name.

Kari Lock Morgan studies the design of randomized experiments, especially how to improve baseline covariate balance. Her work on restricted randomization specifies acceptable balance before an experiment and redraws allocations that fail that criterion. Inference must account for the resulting assignment procedure.^[ref]

With Fan Li and Alan M. Zaslavsky, she studied propensity score weighting for observational studies. For a logistic propensity score fitted by maximum likelihood estimation with an intercept, their overlap weights exactly balance weighted means of the included covariates under the stated fitting conditions.^[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.

For fixed $\beta$, conditional least squares is minimized byTherefore the remaining objective iswhose normal equation givesLetbe the conditional average treatment effect. Since $A$ is binary, conditional exchangeability impliesand $\mathrm{Var}(A\mid X)=e(X)\{1-e(X)\}$. HenceThus ${\beta }_3$ is the overlap-weighted average treatment effect. It weights covariate strata by the overlap weight and generally differs from the ordinary ATE ${\beta }_1=\mathbb{E}[\tau (X)]$ when treatment effects are heterogeneous and overlap varies with $X$.

For binary $A$, conditional covariance satisfiesBy conditional exchangeability and consistency of potential outcomes, the difference in braces is $\mathbb{E}[Y(1)-Y(0)\mid X]$. Consequentlyso the required weight isThis overlap weight emphasizes covariate strata with treatment probabilities near one half and downweights strata near a violation of positivity in causal inference.