Matching Methods

CEM: Software for Coarsened Exact Matching
Stefano M Iacus, Gary King, and Giuseppe Porro. 2009. “CEM: Software for Coarsened Exact Matching.” Journal of Statistical Software, 30. Publisher's VersionAbstract

This program is designed to improve causal inference via a method of matching that is widely applicable in observational data and easy to understand and use (if you understand how to draw a histogram, you will understand this method). The program implements the coarsened exact matching (CEM) algorithm, described below. CEM may be used alone or in combination with any existing matching method. This algorithm, and its statistical properties, are described in Iacus, King, and Porro (2008).

Multivariate Matching Methods That are Monotonic Imbalance Bounding
A technical paper that describes a new class of matching methods, of which coarsened exact matching is an example: Stefano M Iacus, Gary King, and Giuseppe Porro. 2011. “Multivariate Matching Methods That are Monotonic Imbalance Bounding.” Journal of the American Statistical Association, 106, 493, Pp. 345-361.Abstract

We introduce a new "Monotonic Imbalance Bounding" (MIB) class of matching methods for causal inference with a surprisingly large number of attractive statistical properties. MIB generalizes and extends in several new directions the only existing class, "Equal Percent Bias Reducing" (EPBR), which is designed to satisfy weaker properties and only in expectation. We also offer strategies to obtain specific members of the MIB class, and analyze in more detail a member of this class, called Coarsened Exact Matching, whose properties we analyze from this new perspective. We offer a variety of analytical results and numerical simulations that demonstrate how members of the MIB class can dramatically improve inferences relative to EPBR-based matching methods.

Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference
A unified approach to matching methods as a way to reduce model dependence by preprocessing data and then using any model you would have without matching: Daniel Ho, Kosuke Imai, Gary King, and Elizabeth Stuart. 2007. “Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference.” Political Analysis, 15, Pp. 199–236.Abstract

Although published works rarely include causal estimates from more than a few model specifications, authors usually choose the presented estimates from numerous trial runs readers never see. Given the often large variation in estimates across choices of control variables, functional forms, and other modeling assumptions, how can researchers ensure that the few estimates presented are accurate or representative? How do readers know that publications are not merely demonstrations that it is possible to find a specification that fits the author’s favorite hypothesis? And how do we evaluate or even define statistical properties like unbiasedness or mean squared error when no unique model or estimator even exists? Matching methods, which offer the promise of causal inference with fewer assumptions, constitute one possible way forward, but crucial results in this fast-growing methodological literature are often grossly misinterpreted. We explain how to avoid these misinterpretations and propose a unified approach that makes it possible for researchers to preprocess data with matching (such as with the easy-to-use software we offer) and then to apply the best parametric techniques they would have used anyway. This procedure makes parametric models produce more accurate and considerably less model-dependent causal inferences.

CEM: Coarsened Exact Matching in Stata
Matthew Blackwell, Stefano Iacus, Gary King, and Giuseppe Porro. 2009. “CEM: Coarsened Exact Matching in Stata.” The Stata Journal, 9, Pp. 524–546.Abstract
In this article, we introduce a Stata implementation of coarsened exact matching, a new method for improving the estimation of causal effects by reducing imbalance in covariates between treated and control groups. Coarsened exact matching is faster, is easier to use and understand, requires fewer assumptions, is more easily automated, and possesses more attractive statistical properties for many applications than do existing matching methods. In coarsened exact matching, users temporarily coarsen their data, exact match on these coarsened data, and then run their analysis on the uncoarsened, matched data. Coarsened exact matching bounds the degree of model dependence and causal effect estimation error by ex ante user choice, is monotonic imbalance bounding (so that reducing the maximum imbalance on one variable has no effect on others), does not require a separate procedure to restrict data to common support, meets the congruence principle, is approximately invariant to measurement error, balances all nonlinearities and interactions in sample (i.e., not merely in expectation), and works with multiply imputed datasets. Other matching methods inherit many of the coarsened exact matching method’s properties when applied to further match data preprocessed by coarsened exact matching. The cem command implements the coarsened exact matching algorithm in Stata.
A Theory of Statistical Inference for Matching Methods in Causal Research
Stefano M. Iacus, Gary King, and Giuseppe Porro. 2019. “A Theory of Statistical Inference for Matching Methods in Causal Research.” Political Analysis, 27, 1, Pp. 46-68.Abstract

Researchers who generate data often optimize efficiency and robustness by choosing stratified over simple random sampling designs. Yet, all theories of inference proposed to justify matching methods are based on simple random sampling. This is all the more troubling because, although these theories require exact matching, most matching applications resort to some form of ex post stratification (on a propensity score, distance metric, or the covariates) to find approximate matches, thus nullifying the statistical properties these theories are designed to ensure. Fortunately, the type of sampling used in a theory of inference is an axiom, rather than an assumption vulnerable to being proven wrong, and so we can replace simple with stratified sampling, so long as we can show, as we do here, that the implications of the theory are coherent and remain true. Properties of estimators based on this theory are much easier to understand and can be satisfied without the unattractive properties of existing theories, such as assumptions hidden in data analyses rather than stated up front, asymptotics, unfamiliar estimators, and complex variance calculations. Our theory of inference makes it possible for researchers to treat matching as a simple form of preprocessing to reduce model dependence, after which all the familiar inferential techniques and uncertainty calculations can be applied. This theory also allows binary, multicategory, and continuous treatment variables from the outset and straightforward extensions for imperfect treatment assignment and different versions of treatments.

Causal Inference Without Balance Checking: Coarsened Exact Matching
A simple and powerful method of matching: Stefano M. Iacus, Gary King, and Giuseppe Porro. 2012. “Causal Inference Without Balance Checking: Coarsened Exact Matching.” Political Analysis, 20, 1, Pp. 1--24. WebsiteAbstract

We discuss a method for improving causal inferences called "Coarsened Exact Matching'' (CEM), and the new "Monotonic Imbalance Bounding'' (MIB) class of matching methods from which CEM is derived. We summarize what is known about CEM and MIB, derive and illustrate several new desirable statistical properties of CEM, and then propose a variety of useful extensions. We show that CEM possesses a wide range of desirable statistical properties not available in most other matching methods, but is at the same time exceptionally easy to comprehend and use. We focus on the connection between theoretical properties and practical applications. We also make available easy-to-use open source software for R and Stata which implement all our suggestions.

See also:  An Explanation of CEM Weights

The Balance-Sample Size Frontier in Matching Methods for Causal Inference
Gary King, Christopher Lucas, and Richard Nielsen. 2017. “The Balance-Sample Size Frontier in Matching Methods for Causal Inference.” American Journal of Political Science, 61, 2, Pp. 473-489.Abstract

We propose a simplified approach to matching for causal inference that simultaneously optimizes balance (similarity between the treated and control groups) and matched sample size. Existing approaches either fix the matched sample size and maximize balance or fix balance and maximize sample size, leaving analysts to settle for suboptimal solutions or attempt manual optimization by iteratively tweaking their matching method and rechecking balance. To jointly maximize balance and sample size, we introduce the matching frontier, the set of matching solutions with maximum possible balance for each sample size. Rather than iterating, researchers can choose matching solutions from the frontier for analysis in one step. We derive fast algorithms that calculate the matching frontier for several commonly used balance metrics. We demonstrate with analyses of the effect of sex on judging and job training programs that show how the methods we introduce can extract new knowledge from existing data sets.

Easy to use, open source, software is available here to implement all methods in the paper.

MatchIt: Nonparametric Preprocessing for Parametric Causal Inference
Daniel E. Ho, Kosuke Imai, Gary King, and Elizabeth A. Stuart. 2011. “MatchIt: Nonparametric Preprocessing for Parametric Causal Inference.” Journal of Statistical Software, 42, 8, Pp. 1--28. Publisher's VersionAbstract
MatchIt implements the suggestions of Ho, Imai, King, and Stuart (2007) for improving parametric statistical models by preprocessing data with nonparametric matching methods. MatchIt implements a wide range of sophisticated matching methods, making it possible to greatly reduce the dependence of causal inferences on hard-to-justify, but commonly made, statistical modeling assumptions. The software also easily fits into existing research practices since, after preprocessing data with MatchIt, researchers can use whatever parametric model they would have used without MatchIt, but produce inferences with substantially more robustness and less sensitivity to modeling assumptions. MatchIt is an R program, and also works seamlessly with Zelig.
Comparative Effectiveness of Matching Methods for Causal Inference
Gary King, Richard Nielsen, Carter Coberley, James E. Pope, and Aaron Wells. 2011. “Comparative Effectiveness of Matching Methods for Causal Inference”.Abstract

Matching is an increasingly popular method of causal inference in observational data, but following methodological best practices has proven difficult for applied researchers. We address this problem by providing a simple graphical approach for choosing among the numerous possible matching solutions generated by three methods: the venerable ``Mahalanobis Distance Matching'' (MDM), the commonly used ``Propensity Score Matching'' (PSM), and a newer approach called ``Coarsened Exact Matching'' (CEM). In the process of using our approach, we also discover that PSM often approximates random matching, both in many real applications and in data simulated by the processes that fit PSM theory. Moreover, contrary to conventional wisdom, random matching is not benign: it (and thus PSM) can often degrade inferences relative to not matching at all. We find that MDM and CEM do not have this problem, and in practice CEM usually outperforms the other two approaches. However, with our comparative graphical approach and easy-to-follow procedures, focus can be on choosing a matching solution for a particular application, which is what may improve inferences, rather than the particular method used to generate it.

Please see our follow up paper on this topic: Why Propensity Scores Should Not Be Used for Matching.

Why Propensity Scores Should Not Be Used for Matching
Gary King and Richard Nielsen. 2019. “Why Propensity Scores Should Not Be Used for Matching.” Political Analysis, 27, 4, Pp. 435-454. Publisher's VersionAbstract

We show that propensity score matching (PSM), an enormously popular method of preprocessing data for causal inference, often accomplishes the opposite of its intended goal --- thus increasing imbalance, inefficiency, model dependence, and bias. The weakness of PSM comes from its attempts to approximate a completely randomized experiment, rather than, as with other matching methods, a more efficient fully blocked randomized experiment. PSM is thus uniquely blind to the often large portion of imbalance that can be eliminated by approximating full blocking with other matching methods. Moreover, in data balanced enough to approximate complete randomization, either to begin with or after pruning some observations, PSM approximates random matching which, we show, increases imbalance even relative to the original data. Although these results suggest researchers replace PSM with one of the other available matching methods, propensity scores have other productive uses.