We clarify the theoretical foundations of partisan fairness standards for district-based democratic electoral systems, including essential assumptions and definitions not previously recognized, formalized, or in some cases even discussed. We also offer extensive empirical evidence for assumptions with observable implications. We cover partisan symmetry, the most commonly accepted fairness standard, and other perspectives. Throughout, we follow a fundamental principle of statistical inference too often ignored in this literature—defining the quantity of interest separately so its measures can be proven wrong, evaluated, and improved. This enables us to prove which of the many newly proposed fairness measures are statistically appropriate and which are biased, limited, or not measures of the theoretical quantity they seek to estimate at all. Because real-world redistricting and gerrymandering involve complicated politics with numerous participants and conflicting goals, measures biased for partisan fairness sometimes still provide useful descriptions of other aspects of electoral systems.

The mission of the social sciences is to understand and ameliorate society’s greatest challenges. The data held by private companies, collected for different purposes, hold vast potential to further this mission. Yet, because of consumer privacy, trade secrets, proprietary content, and political sensitivities, these datasets are often inaccessible to scholars. We propose a novel organizational model to address these problems. We also report on the first partnership under this model, to study the incendiary issues surrounding the impact of social media on elections and democracy: Facebook provides (privacy-preserving) data access; eight ideologically and substantively diverse charitable foundations provide initial funding; an organization of academics we created, Social Science One, leads the project; and the Institute for Quantitative Social Science at Harvard and the Social Science Research Council provide logistical help.

Ecological inference (EI) is the process of learning about individual behavior from aggregate data. We relax assumptions by allowing for “linear contextual effects,” which previous works have regarded as plausible but avoided due to nonidentification, a problem we sidestep by deriving bounds instead of point estimates. In this way, we offer a conceptual framework to improve on the Duncan–Davis bound, derived more than 65 years ago. To study the effectiveness of our approach, we collect and analyze 8,430 $2\times 2$ EI datasets with known ground truth from several sources—thus bringing considerably more data to bear on the problem than the existing dozen or so datasets available in the literature for evaluating EI estimators. For the 88% of real data sets in our collection that fit a proposed rule, our approach reduces the width of the Duncan–Davis bound, on average, by about 44%, while still capturing the true district-level parameter about 99% of the time. The remaining 12% revert to the Duncan–Davis bound.

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.

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.

The Chinese government has long been suspected of hiring as many as 2 million people to surreptitiously insert huge numbers of pseudonymous and other deceptive writings into the stream of real social media posts, as if they were the genuine opinions of ordinary people. Many academics, and most journalists and activists, claim that these so-called 50c party posts vociferously argue for the government’s side in political and policy debates. As we show, this is also true of most posts openly accused on social media of being 50c. Yet almost no systematic empirical evidence exists for this claim or, more importantly, for the Chinese regime’s strategic objective in pursuing this activity. In the first large-scale empirical analysis of this operation, we show how to identify the secretive authors of these posts, the posts written by them, and their content. We estimate that the government fabricates and posts about 448 million social media comments a year. In contrast to prior claims, we show that the Chinese regime’s strategy is to avoid arguing with skeptics of the party and the government, and to not even discuss controversial issues. We show that the goal of this massive secretive operation is instead to distract the public and change the subject, as most of these posts involve cheerleading for China, the revolutionary history of the Communist Party, or other symbols of the regime. We discuss how these results fit with what is known about the Chinese censorship program and suggest how they may change our broader theoretical understanding of “common knowledge” and information control in authoritarian regimes.

“Politimetrics” (Gurr 1972), “polimetrics,” (Alker 1975), “politometrics” (Hilton 1976), “political arithmetic” (Petty [1672] 1971), “quantitative Political Science (QPS),” “governmetrics,” “posopolitics” (Papayanopoulos 1973), “political science statistics” (Rai and Blydenburgh 1973), “political statistics” (Rice 1926). These are some of the names that scholars have used to describe the field we now call “political methodology.”1 The history of political methodology has been quite fragmented until recently, as reflected by this patchwork of names. The field has begun to coalesce during the past decade; we are developing persistent organizations, a growing body of scholarly literature, and an emerging consensus about important problems that need to be solved.

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.

In response to the data-based measures of model dependence proposed in King and Zeng (2006), Sambanis and Michaelides (2008) propose alternative measures that rely upon assumptions untestable in observational data. If these assumptions are correct, then their measures are appropriate and ours, based solely on the empirical data, may be too conservative. If instead, and as is usually the case, the researcher is not certain of the precise functional form of the data generating process, the distribution from which the data are drawn, and the applicability of these modeling assumptions to new counterfactuals, then the data-based measures proposed in King and Zeng (2006) are much preferred. After all, the point of model dependence checks is to verify empirically, rather than to stipulate by assumption, the effects of modeling assumptions on counterfactual inferences.

We study rare events data, binary dependent variables with dozens to thousands of times fewer ones (events, such as wars, vetoes, cases of political activism, or epidemiological infections) than zeros (“nonevents”). In many literatures, these variables have proven difficult to explain and predict, a problem that seems to have at least two sources. First, popular statistical procedures, such as logistic regression, can sharply underestimate the probability of rare events. We recommend corrections that outperform existing methods and change the estimates of absolute and relative risks by as much as some estimated effects reported in the literature. Second, commonly used data collection strategies are grossly inefficient for rare events data. The fear of collecting data with too few events has led to data collections with huge numbers of observations but relatively few, and poorly measured, explanatory variables, such as in international conflict data with more than a quarter-million dyads, only a few of which are at war. As it turns out, more efficient sampling designs exist for making valid inferences, such as sampling all available events (e.g., wars) and a tiny fraction of nonevents (peace). This enables scholars to save as much as 99% of their (nonfixed) data collection costs or to collect much more meaningful explanatory variables. We provide methods that link these two results, enabling both types of corrections to work simultaneously, and software that implements the methods developed.

We demonstrate that the expected value and variance commonly given for a well-known probability distribution are incorrect. We also provide corrected versions and report changes in a computer program to account for the known practical uses of this distribution.

The probability distribution in question, named the continuous parameter binomial (CPB) by King (1989a), has been known for at least six decades. The publications that reported the moments incorrectly, or were at least unclear about them, include a dated article (Guldberg 1931), a dissertation and several resulting published articles (Katz 1945, 1965), a popular reference book (Johnson and Kotz 1969),' work in political science on event count regression models (King 1989a), and extensions of these event count models in econometrics (Winkelmann and Zimmermann 1991). Event count regression models have become increasingly common in empirical political science research; some recent examples include Wang et al. (1993) and Krause (1994).

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 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, Stata, and SPSS that implement all our suggestions.

When respondents use the ordinal response categories of standard survey questions in different ways, the validity of analyses based on the resulting data can be biased. Anchoring vignettes is a survey design technique, introduced by King et al. (2004, Enhancing the validity and cross-cultural comparability of measurement in survey research. American Political Science Review 94 [February]: 191–205), intended to correct for some of these problems. We develop new methods both for evaluating and choosing anchoring vignettes and for analyzing the resulting data. With surveys on a diverse range of topics in a range of countries, we illustrate how our proposed methods can improve the ability of anchoring vignettes to extract information from survey data, as well as saving in survey administration costs.

“Robust standard errors” are used in a vast array of scholarship to correct standard errors for model misspecification. However, when misspecification is bad enough to make classical and robust standard errors diverge, assuming that it is nevertheless not so bad as to bias everything else requires considerable optimism. And even if the optimism is warranted, settling for a misspecified model, with or without robust standard errors, will still bias estimators of all but a few quantities of interest. The resulting cavernous gap between theory and practice suggests that considerable gains in applied statistics may be possible. We seek to help researchers realize these gains via a more productive way to understand and use robust standard errors; a new general and easier-to-use “generalized information matrix test” statistic that can formally assess misspecification (based on differences between robust and classical variance estimates); and practical illustrations via simulations and real examples from published research. How robust standard errors are used needs to change, but instead of jettisoning this popular tool we show how to use it to provide effective clues about model misspecification, likely biases, and a guide to considerably more reliable, and defensible, inferences. Accompanying this article is software that implements the methods we describe.

We take this opportunity to comment on Herron and Shotts (2003; hereinafter HS) because of its interesting and productive ideas and because of the potential to affect the way a considerable body of practical research is conducted. This article, and the literature referenced therein, is based on the suggestions in three paragraphs in King (1997, pp. 289–290). Because these paragraphs were not summarized in HS, we thought they might be a useful place to start.

Binary, count, and duration data all code discrete events occurring at points in time. Although a single data generation process can produce all of these three data types, the statistical literature is not very helpful in providing methods to estimate parameters of the same process from each. In fact, only a single theoretical process exists for which known statistical methods can estimate the same parameters—and it is generally used only for count and duration data. The result is that seemingly trivial decisions about which level of data to use can have important consequences for substantive interpretations. We describe the theoretical event process for which results exist, based on time independence. We also derive a set of models for a time-dependent process and compare their predictions to those of a commonly used model. Any hope of understanding and avoiding the more serious problems of aggregation bias in events data is contingent on first deriving a much wider arsenal of statistical models and theoretical processes that are not constrained by the particular forms of data that happen to be available. We discuss these issues and suggest an agenda for political methodologists interested in this very large class of aggregation problems.

Katz and King have previously developed a model for predicting or explaining aggregate electoral results in multiparty democracies. Their model is, in principle, analogous to what least-squares regression provides American political researchers in that two-party system. Katz and King applied their model to three-party elections in England and revealed a variety of new features of incumbency advantage and sources of party support. Although the mathematics of their statistical model covers any number of political parties, it is computationally demanding, and hence slow and numerically imprecise, with more than three parties. In this paper we produce an approximate method that works in practice with many parties without making too many theoretical compromises. Our approach is to treat the problem as one of missing data. This allows us to use a modification of the fast EMis algorithm of King, Honaker, Joseph, and Scheve and to provide easy-to-use software, while retaining the attractive features of the Katz and King model, such as the t distribution and explicit models for uncontested seats.

The directional and proximity models offer dramatically different theories for how voters make decisions and fundamentally divergent views of the supposed microfoundations on which vast bodies of literature in theoretical rational choice and empirical political behavior have been built. We demonstrate here that the empirical tests in the large and growing body of literature on this subject amount to theoretical debates about which statistical assumption is right. The key statistical assumptions have not been empirically tested and, indeed, turn out to be effectively untestable with existing methods and data. Unfortunately, these assumptions are also crucial since changing them leads to different conclusions about voter decision processes.

Few better ways of checking and improving statistical methods exist than having other researchers go over your results, and so I especially appreciate the efforts in Anselin and Cho (2002), hereinafter AC. In this note, I make two main points.