## Overlap Weights for Causal Inference/Comparative Effectiveness Research

Basic Idea

To compare two treatments (say one treatment and one control), the overlap weight (OW) for each unit is the probability that unit is assigned to the opposite group, that is, 1-PS for the treated units, and PS for the control units, where PS is the propensity score of that unit. Overlap weights focus on the causal effects on the population with the most overlap in covariates between two treatments. The procedure involves three simple steps, no hidden caveats:
• 1. Estimate the propensity score from a model, e.g. a logistic regression model or fancier machine learning models (which, by my experience, rarely made a difference in the effect estimation)
• 2. Calculate the overlap weights based on the estimated propensity scores: treated units OW= 1-PS; control units OW=PS. Crucial to normalize the weights so that sum of the overlap weights equals 1 within each treatment group, i.e. dividing each unit's weight by the sum of all weights within that treatment group.
• 3. Estimate the average treatment effect for the overlap population by the difference of the OW-weighted average outcomes between the treatment groups.
Compared to the traditional inverse probability of treatment weights (IPW or IPTW) and associated trimming methods, overlap weights have several advantages:
• No extreme weights, minimum variance of the weighted estimator of causal effects among all balancing weights (including IPW)
• Exact mean balance of covariates when PS is estimated via a logistic regression -- the exact balance is finite-sample NOT asymptotic
• No need to choose an artificial cutoff point as trimming

Extensions to multiple treatments, binary outcomes, time-to-event outcomes, and dynamic treatment regime have also been established.

Some graphical illustration of the target population of the overlap weights (the tilting function h(x))

R package

PSweight [CRAN]: a R package that provides a comprehensive analysis platform for causal inference based on propensity score weighting methods, including overlap weighting, inverse probability weighting, trimming. It accommodates both binary and multiple treatments as well as different types of estimands (difference, odds ratio, risk ratio, relative risk). A detailed manual and illustration is given in the R vignette Zhou et al. (2022) [R vignette] and arxiv [arxiv].

SAS code

• An example SAS code that simulates a case with two treatments and a binary outcome [SAS code for binary outcome].
• An example SAS code that simulates a case with two treatments and a survival outcome. The analysis includes (1) a Cox proportional hazard model on OW-weighted sample; (2) a Breslow (Nelson-Aalen) estimator of the survival curves on the OW-weighted sample [SAS code for survival/time-to-event outcome].
• Tutorial

An accessible tutorial on overlap weights that includes binary, multiple treatments, subgroup analysis, code demo, and real examples is [OW Tutorial].

Papers and Slides

• Main theoretical paper: The theoretical foundation of overlap weights is established in Li, Morgan, Zaslavsky (2018, JASA) [DOI | slides]
• Methods brief in JAMA: A short and accessible introduction of overlap weighting for the general medical audience is given in Thomas, Li, Pencina (2020, JAMA) [DOI]
• Accessible version: Extensive simulation comparisons between OW, IPW, trimming, and closed-form variance estimator of the weighted causal effect estimator are given in Li, Thomas, Li (2019, AJE) [DOI | code | slides]
• Non-technical review: Simple high-level review with examples [Slides]
• Time-to-event/Survival outcomes: For survival outcomes, one can use a Cox proportional hazard model or estimate the Kaplan-Meier curve on an OW-weighted sample. Cheng et al. (2021, AJE) [arxiv] developed an OW-based estimator for the counterfactual survival function. Zeng et al. (2021, Sta Sinica) [arxiv] established the asymptotic variance of balancing weights (including OW, IPW) estimators of several estimands based on the pseudo observations.
• Multiple treatments: Extension to multiple (also called multi-valued) treatments is provided in Li and Li (2019, AOAS) [arxiv | supplement | code | slides]
• Subgroup analysis: Application to subgroup analysis is provided in Yang et al. (2021, SIM) [pdf | supplement |code]
• Covariate adjustment: OW can be used for covariate adjustment in randomized controlled trials, outperforming IPW and regression adjustment. See Zeng et al. (2021, SIM) [DOI|arxiv|slides].