## Overlap Weights for 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. Compared to the traditional inverse probability of treatment weights (IPW or IPTW) and associated trimming methods, overlap weights have several advantages:
• 1. No extreme weights, minimum variance of the weighted estimator of causal effects among all balancing weights (including IPW)
• 2. Exact mean balance of covariates when PS is estimated via a logistic regression
• 3. 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)) Fig.1 - Overlap weights for two normally distributed groups with different means. In the upper panel, the red, blue, black dashed, and blue lines represent the density of the covariate in the control, treated, combined (h(x) = 1), and overlap weighted populations (h(x) = e(x)(1 − e(x))), respectively (e(x) is the propensity score). In the lower panel, the red and green lines represent the overlap weights for treated and control units, and the tilting function h(x) = e(x)(1 − e(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 paper Zhou et al. (2020) [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]
• Time-to-event/Survival outcomes: For survival outcomes, one can easily use a Cox proportional hazard model or estimate the Kaplan-Meier curve on an OW-weighted sample. A more flexible, model-free method that is applicable to a range of estimands is via the pseudo observations. The paper is Zeng et al. (2021) [arxiv]
• Multiple treatments: Extension to multiple (also called multi-valued) treatments is provided in Li and Li (2019, AOAS) [arxiv | supplement | code | slides]
• Covariate adjustment: OW can be used for covariate adjustment in randomized controlled trials, outperforming IPW and regression adjustment. See Zeng et al. (2020) [DOI|arxiv|slides].