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Causal Inference

Methods for detecting and reducing model dependence (i.e., when minor model changes produce substantively different inferences) in inferring casual effects and other counterfactuals. Matching methods; "politically robust" and cluster-randomized experimental designs; causal bias decompositions.

Methods for Observational Data

Evaluating Model Dependence

Evaluating whether counterfactual questions (predictions, what-if questions, and causal effects) can be reasonably answered from given data, or whether inferences will instead be highly model-dependent; also, a new decomposition of bias in causal inference. These articles overlap (and each as been the subject of a journal symposium):
For complete mathematical proofs, general notation, and other technical material, see: King, Gary, and Langche Zeng. 2006. The Dangers of Extreme Counterfactuals. Political Analysis 14: 131–159.Abstract
Article
We address the problem that occurs when inferences about counterfactuals – predictions, "what if" questions, and causal effects – are attempted far from the available data. The danger of these extreme counterfactuals is that substantive conclusions drawn from statistical models that fit the data well turn out to be based largely on speculation hidden in convenient modeling assumptions that few would be willing to defend. Yet existing statistical strategies provide few reliable means of identifying extreme counterfactuals. We offer a proof that inferences farther from the data are more model-dependent, and then develop easy-to-apply methods to evaluate how model-dependent our answers would be to specified counterfactuals. These methods require neither sensitivity testing over specified classes of models nor evaluating any specific modeling assumptions. If an analysis fails the simple tests we offer, then we know that substantive results are sensitive to at least some modeling choices that are not based on empirical evidence.
For more intuitive, but less general, notation, but with additional examples and more pedagogically oriented material, see: King, Gary, and Langche Zeng. 2007. When Can History Be Our Guide? The Pitfalls of Counterfactual Inference. International Studies Quarterly: 183-210.Abstract
Article
Inferences about counterfactuals are essential for prediction, answering "what if" questions, and estimating causal effects. However, when the counterfactuals posed are too far from the data at hand, conclusions drawn from well-specified statistical analyses become based on speculation and convenient but indefensible model assumptions rather than empirical evidence. Unfortunately, standard statistical approaches assume the veracity of the model rather than revealing the degree of model-dependence, and so this problem can be hard to detect. We develop easy-to-apply methods to evaluate counterfactuals that do not require sensitivity testing over specified classes of models. If an analysis fails the tests we offer, then we know that substantive results are sensitive to at least some modeling choices that are not based on empirical evidence. We use these methods to evaluate the extensive scholarly literatures on the effects of changes in the degree of democracy in a country (on any dependent variable) and separate analyses of the effects of UN peacebuilding efforts. We find evidence that many scholars are inadvertently drawing conclusions based more on modeling hypotheses than on their data. For some research questions, history contains insufficient information to be our guide.

Matching Methods

Iacus, Stefano M, and Gary King. 2012. How Coarsening Simplifies Matching-Based Causal Inference Theory.Abstract
Paper
The simplicity and power of matching methods have made them an increasingly popular approach to causal inference in observational data. Existing theories that justify these techniques are well developed but either require exact matching, which is usually infeasible in practice, or sacrifice some simplicity via asymptotic theory, specialized bias corrections, and novel variance estimators; and extensions to approximate matching with multicategory treatments have not yet appeared. As an alternative, we show how conceptualizing continuous variables as having logical breakpoints (such as phase transitions when measuring temperature or high school or college degrees in years of education) is both natural substantively and can be used to simplify causal inference theory. The result is a finite sample theory that is widely applicable, simple to understand, and easy to implement by using matching to preprocess the data, after which one can use whatever method would have been applied without matching. The theoretical simplicity also allows for binary, multicategory, and continuous treatment variables from the start and for extensions to valid inference under imperfect treatment assignment.
A technical paper that describes a new class of matching methods, of which coarsened exact matching is an example: Iacus, Stefano M, Gary King, and Giuseppe Porro. 2011. Multivariate Matching Methods That are Monotonic Imbalance Bounding. Journal of the American Statistical Association 106, no. 493: 345-361.Abstract
Article
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.
A simple and powerful method of matching: Iacus, Stefano M, Gary King, and Giuseppe Porro. 2011. Causal Inference Without Balance Checking: Coarsened Exact Matching. Political Analysis.Abstract
Article
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. Political Analysis version An Explanation of CEM Weights
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: Ho, Daniel, Kosuke Imai, Gary King, and Elizabeth Stuart. 2007. Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference. Political Analysis 15: 199–236.Abstract
Article
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.
Blackwell, Matthew, Stefano Iacus, Gary King, and Giuseppe Porro. 2009. CEM: Coarsened Exact Matching in Stata. The Stata Journal 9: 524–546.Abstract
Article
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.
King, Gary, Richard Nielsen, Carter Coberley, James E Pope, and Aaron Wells. 2011. Comparative Effectiveness of Matching Methods for Causal Inference.Abstract
Paper
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.

Additional Approaches

A method to estimate base probabilities or any quantity of interest from case-control data, even with no (or partial) auxiliary information. Discusses problems with odds-ratios. King, Gary, and Langche Zeng. 2002. Estimating Risk and Rate Levels, Ratios, and Differences in Case-Control Studies. Statistics in Medicine 21: 1409–1427.Abstract
Article
Classic (or "cumulative") case-control sampling designs do not admit inferences about quantities of interest other than risk ratios, and then only by making the rare events assumption. Probabilities, risk differences, and other quantities cannot be computed without knowledge of the population incidence fraction. Similarly, density (or "risk set") case-control sampling designs do not allow inferences about quantities other than the rate ratio. Rates, rate differences, cumulative rates, risks, and other quantities cannot be estimated unless auxiliary information about the underlying cohort such as the number of controls in each full risk set is available. Most scholars who have considered the issue recommend reporting more than just the relative risks and rates, but auxiliary population information needed to do this is not usually available. We address this problem by developing methods that allow valid inferences about all relevant quantities of interest from either type of case-control study when completely ignorant of or only partially knowledgeable about relevant auxiliary population information.
King, Gary. 1991. 'Truth' is Stranger than Prediction, More Questionable Than Causal Inference. American Journal of Political Science 35: 1047–1053.Abstract
Article
Robert Luskin’s article in this issue provides a useful service by appropriately qualifying several points I made in my 1986 American Journal of Political Science article. Whereas I focused on how to avoid common mistakes in quantitative political sciences, Luskin clarifies ways to extract some useful information from usually problematic statistics: correlation coefficients, standardized coefficients, and especially R2. Since these three statistics are very closely related (and indeed deterministic functions of one another in some cases), I focus in this discussion primarily on R2, the most widely used and abused. Luskin also widens the discussion to various kinds of specification tests, a general issue I also address. In fact, as Beck (1991) reports, a large number of formal specification tests are just functions of R2, with differences among them primarily due to how much each statistic penalizes one for including extra parameters and fewer observations. Quantitative political scientists often worry about model selection and specification, asking questions about parameter identification, autocorrelated or heteroscedastic disturbances, parameter constancy, variable choice, measurement error, endogeneity, functional forms, stochastic assumptions, and selection bias, among numerous others. These model specification questions are all important, but we may have forgotten why we pose them. Political scientists commonly give three reasons: (1) finding the "true" model, or the "full" explanation and (2) prediction and and (3) estimating specific causal effects. I argue here that (1) is used the most but useful the least and (2) is very useful but not usually in political science where forecasting is not often a central concern and and (3) correctly represents the goals of political scientists and should form the basis of most of our quantitative empirical work.

Experimental Design

King, Gary, Richard Nielsen, Carter Coberley, James E Pope, and Aaron Wells. 2011. Avoiding Randomization Failure in Program Evaluation. Population Health Management 14, no. 1: S11-S22.Abstract An evaluation of the Mexican Seguro Popular program (designed to extend health insurance and regular and preventive medical care, pharmaceuticals, and health facilities to 50 million uninsured Mexicans), one of the world's largest health policy reforms of the last two decades. The evaluation features the largest randomized health policy experiment in history, a new design for field experiments that is more robust to the political interventions that have ruined many similar previous efforts, and new statistical methods that produce more reliable and efficient results using substantially fewer resources, assumptions, and data.
Article
We highlight common problems in the application of random treatment assignment in large scale program evaluation. Random assignment is the defining feature of modern experimental design. Yet, errors in design, implementation, and analysis often result in real world applications not benefiting from the advantages of randomization. The errors we highlight cover the control of variability, levels of randomization, size of treatment arms, and power to detect causal effects, as well as the many problems that commonly lead to post-treatment bias. We illustrate with an application to the Medicare Health Support evaluation, including recommendations for improving the design and analysis of this and other large scale randomized experiments.
(Articles on the Seguro Popular Evaluation: Website)
Clarifying serious misunderstandings in the advantages and uses of the most common research designs for making causal inferences. Imai, Kosuke, Gary King, and Elizabeth Stuart. 2008. Misunderstandings Among Experimentalists and Observationalists about Causal Inference. Journal of the Royal Statistical Society, Series A 171, part 2: 481–502.Abstract
Article
We attempt to clarify, and suggest how to avoid, several serious misunderstandings about and fallacies of causal inference in experimental and observational research. These issues concern some of the most basic advantages and disadvantages of each basic research design. Problems include improper use of hypothesis tests for covariate balance between the treated and control groups, and the consequences of using randomization, blocking before randomization, and matching after treatment assignment to achieve covariate balance. Applied researchers in a wide range of scientific disciplines seem to fall prey to one or more of these fallacies, and as a result make suboptimal design or analysis choices. To clarify these points, we derive a new four-part decomposition of the key estimation errors in making causal inferences. We then show how this decomposition can help scholars from different experimental and observational research traditions better understand each other’s inferential problems and attempted solutions.

Software

Blackwell, Matthew, Stefano Iacus, Gary King, and Giuseppe Porro. 2009. CEM: Coarsened Exact Matching in Stata. The Stata Journal 9: 524–546.Abstract
Article
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.
Tomz, Michael, Jason Wittenberg, and Gary King. 2003. CLARIFY: Software for Interpreting and Presenting Statistical Results. Journal of Statistical Software.Abstract
This is a set of easy-to-use Stata macros that implement the techniques described in Gary King, Michael Tomz, and Jason Wittenberg's "Making the Most of Statistical Analyses: Improving Interpretation and Presentation". To install Clarify, type "net from http://gking.harvard.edu/clarify" at the Stata command line. The documentation [ HTML | PDF ] explains how to do this. We also provide a zip archive for users who want to install Clarify on a computer that is not connected to the internet. Winner of the Okidata Best Research Software Award. Also try -ssc install qsim- to install a wrapper, donated by Fred Wolfe, to automate Clarify's simulation of dummy variables.

Applications

Epstein, Lee, Daniel E Ho, Gary King, Jeffrey A Segal, Samuel Kernell, and Steven S Smith. 2006. The Effect of War on the Supreme Court. In Principles and Practice in American Politics: Classic and Contemporary Readings. 3rd ed. Washington, D.C.: Congressional Quarterly Press.Abstract
Article
Does the U.S. Supreme Court curtail rights and liberties when the nation’s security is under threat? In hundreds of articles and books, and with renewed fervor since September 11, 2001, members of the legal community have warred over this question. Yet, not a single large-scale, quantitative study exists on the subject. Using the best data available on the causes and outcomes of every civil rights and liberties case decided by the Supreme Court over the past six decades and employing methods chosen and tuned especially for this problem, our analyses demonstrate that when crises threaten the nation’s security, the justices are substantially more likely to curtail rights and liberties than when peace prevails. Yet paradoxically, and in contradiction to virtually every theory of crisis jurisprudence, war appears to affect only cases that are unrelated to the war. For these cases, the effect of war and other international crises is so substantial, persistent, and consistent that it may surprise even those commentators who long have argued that the Court rallies around the flag in times of crisis. On the other hand, we find no evidence that cases most directly related to the war are affected. We attempt to explain this seemingly paradoxical evidence with one unifying conjecture: Instead of balancing rights and security in high stakes cases directly related to the war, the Justices retreat to ensuring the institutional checks of the democratic branches. Since rights-oriented and process-oriented dimensions seem to operate in different domains and at different times, and often suggest different outcomes, the predictive factors that work for cases unrelated to the war fail for cases related to the war. If this conjecture is correct, federal judges should consider giving less weight to legal principles outside of wartime but established during wartime, and attorneys should see it as their responsibility to distinguish cases along these lines.
A brief summary of the above article for an undergraduate audience: Epstein, Lee, Daniel E Ho, Gary King, and Jeffrey A Segal. 2005. The Supreme Court During Crisis: How War Affects only Non-War Cases. New York University Law Review 80: 1–116.Abstract
Article
Does the U.S. Supreme Court curtail rights and liberties when the nation’s security is under threat? In hundreds of articles and books, and with renewed fervor since September 11, 2001, members of the legal community have warred over this question. Yet, not a single large-scale, quantitative study exists on the subject. Using the best data available on the causes and outcomes of every civil rights and liberties case decided by the Supreme Court over the past six decades and employing methods chosen and tuned especially for this problem, our analyses demonstrate that when crises threaten the nation’s security, the justices are substantially more likely to curtail rights and liberties than when peace prevails. Yet paradoxically, and in contradiction to virtually every theory of crisis jurisprudence, war appears to affect only cases that are unrelated to the war. For these cases, the effect of war and other international crises is so substantial, persistent, and consistent that it may surprise even those commentators who long have argued that the Court rallies around the flag in times of crisis. On the other hand, we find no evidence that cases most directly related to the war are affected. We attempt to explain this seemingly paradoxical evidence with one unifying conjecture: Instead of balancing rights and security in high stakes cases directly related to the war, the Justices retreat to ensuring the institutional checks of the democratic branches. Since rights-oriented and process-oriented dimensions seem to operate in different domains and at different times, and often suggest different outcomes, the predictive factors that work for cases unrelated to the war fail for cases related to the war. If this conjecture is correct, federal judges should consider giving less weight to legal principles outside of wartime but established during wartime, and attorneys should see it as their responsibility to distinguish cases along these lines.