# A fast algorithm for maximal propensity score matching

## TL;DR

This paper introduces a fast algorithm for maximal propensity score matching that efficiently finds the largest set of matched pairs with caliper constraints, improving existing matching techniques in terms of speed and optimality.

## Contribution

The paper presents a novel algorithm for maximal propensity score matching that handles variable calipers and 1-to-n matching efficiently, advancing current matching methods.

## Key findings

- Matching with the new algorithm requires O(N) operations for ordered data.
- The algorithm can handle variable width calipers as Lipschitz functions.
- It improves upon greedy nearest neighbor matching in speed and optimality.

## Abstract

We present a new algorithm which detects the maximal possible number of matched disjoint pairs satisfying a given caliper when a bipartite matching is done with respect to a scalar index (e.g., propensity score), and constructs a corresponding matching. Variable width calipers are compatible with the technique, provided that the width of the caliper is a Lipschitz function of the index. If the observations are ordered with respect to the index then the matching needs $O(N)$ operations, where $N$ is the total number of subjects to be matched. The case of 1-to-$n$ matching is also considered. We offer also a new fast algorithm for optimal complete one-to-one matching on a scalar index when the treatment and control groups are of the same size. This allows us to improve greedy nearest neighbor matching on a scalar index. Keywords: propensity score matching, nearest neighbor matching, matching with caliper, variable width caliper.