Matching is a conceptually straightforward method to make groups of units comparable on observed characteristics. The method is, however, limited to settings where the study design is simple and the sample is moderately sized. We illustrate these limitations by asking what the causal effects would have been if a large-scale voter mobilization experiment that took place in Michigan for the 2006 election were scaled up to the full population of registered voters. Matching could help us answer this question, but no existing matching method can accommodate the six treatment arms and the 6,762,701 observations involved in the study. To offer a solution for this and similar empirical problems, we introduce a generalization of the full matching method that can be used with any number of treatment conditions and complex compositional constraints. The associated algorithm produces near-optimal matchings; the worst-case maximum within-group dissimilarity is guaranteed to be no more than four times greater than the optimal solution, and simulation results indicate that it comes considerably closer to the optimal solution on average. The algorithm’s ability to balance the treatment groups does not sacrifice speed, and it uses little memory, terminating in linearithmic time using linear space. This enables investigators to construct well-performing matchings within minutes even in complex studies with samples of several million units.

The popularity of online surveys has increased the prominence of using sampling weights to enhance claims of representativeness. Yet, much uncertainty remains regarding how these weights should be employed in survey experiment analysis: should they be used? If so, which estimators are preferred? We offer practical advice, rooted in the Neyman–Rubin model, for researchers working with survey experimental data. We examine simple, efficient estimators, and give formulas for their biases and variances. We provide simulations that examine these estimators as well as real examples from experiments administered online through YouGov. We find that for examining the existence of population treatment effects using high-quality, broadly representative samples recruited by top online survey firms, sample quantities, which do not rely on weights, are often sufficient. We found that sample average treatment effect (SATE) estimates did not appear to differ substantially from their weighted counterparts, and they avoided the substantial loss of statistical power that accompanies weighting. When precise estimates of population average treatment effects (PATE) are essential, we analytically show poststratifying on survey weights and/or covariates highly correlated with outcomes to be a conservative choice. While we show substantial gains in simulations, we find limited evidence of them in practice.

There is increasing interest in experiments where outcomes are measured by surveys and treatments are delivered by a separate mechanism in the real world, such as by mailers, door-to-door canvasses, phone calls, or online ads. However, common designs for such experiments are often prohibitively expensive, vulnerable to bias, and raise ethical concerns. We show how four methodological practices currently uncommon in such experiments have previously undocumented complementarities that can dramatically relax these constraints when at least two are used in combination: (1) online surveys recruited from a defined sampling frame (2) with at least one baseline wave prior to treatment (3) with multiple items combined into an index to measure outcomes and, (4) when possible, a placebo control. We provide a general and extensible framework that allows researchers to determine the most efficient mix of these practices in diverse applications. Two studies then examine how these practices perform empirically. First, we examine the representativeness of online panel respondents recruited from a defined sampling frame and find that their representativeness compares favorably to phone panel respondents. Second, an original experiment successfully implements all four practices in the context of a door-to-door canvassing experiment. We conclude discussing potential extensions.

Following David Lee's pioneering work, numerous scholars have applied the regression discontinuity (RD) design to popular elections. Contrary to the assumptions of RD, however, we show that bare winners and bare losers in U.S. House elections (1942–2008) differ markedly on pretreatment covariates. Bare winners possess large ex ante financial, experience, and incumbency advantages over their opponents and are usually the candidates predicted to win by Congressional Quarterly's pre-election ratings. Covariate imbalance actually worsens in the closest House elections. National partisan tides help explain these patterns. Previous works have missed this imbalance because they rely excessively on model-based extrapolation. We present evidence suggesting that sorting in close House elections is due mainly to activities on or before Election Day rather than postelection recounts or other manipulation. The sorting is so strong that it is impossible to achieve covariate balance between matched treated and control observations, making covariate adjustment a dubious enterprise. Although RD is problematic for postwar House elections, this example does highlight the design's advantages over alternatives: RD's assumptions are clear and weaker than model-based alternatives, and their implications are empirically testable.

Legislative redistricting alters the political and electoral context for some voters but not others, thus offering a potentially promising research design to study many questions of interest in political science. We apply this design to study the effect that descriptive representation has on co-ethnic political engagement, focusing on Hispanic participation following California's 2000 redistricting cycle. We show that when redistrictors draw legislative boundaries in California's 1990, 2000, and 2010 apportionment cycles, they systematically sort higher-participating Hispanic voters into majority-Hispanic (MH) jurisdictions represented by co-ethnic candidates, biasing subsequent comparisons of Hispanic participation across districts. Similar sorting occurs during redistricting in Florida and Texas, though here the pattern is reversed, with less-participating Hispanic voters redistricted to MH districts. Our study highlights important heterogeneity in redistricting largely unknown or underappreciated in previous research. Ignoring this selection problem could significantly bias estimates of the effect of Hispanic representation, either positively or negatively. After we correct for these biases using a hierarchical genetic matching algorithm, we find that, in California, being moved to a district with an Hispanic incumbent has little impact on Hispanic participation in our data.

We describe a new computer program that combines evolutionary algorithm methods with a derivative-based, quasi-Newton method to solve difficult unconstrained optimization problems. The program, called GENOUD (GENetic Optimization Using Derivatives), effectively solves problems that are nonlinear or perhaps even discontinuous in the parameters of the function to be optimized. When a statistical model's estimating function (for example, a log-likelihood) is nonlinear in the model's parameters, the function to be optimized will usually not be globally concave and may contain irregularities such as saddlepoints or discontinuous jumps. Optimization methods that rely on derivatives of the objective function may be unable to find any optimum at all. Or multiple local optima may exist, so that there is no guarantee that a derivative-based method will converge to the global optimum. We discuss the theoretical basis for expecting GENOUD to have a high probability of finding global optima. We conduct Monte Carlo experiments using scalar Normal mixture densities to illustrate this capability. We also use a system of four simultaneous nonlinear equations that has many parameters and multiple local optima to compare the performance of GENOUD to that of the Gauss-Newton algorithm in SAS's PROC MODEL.

We look at conventional methods for removing endogeneity bias in regression models, including the linear model and the probit model. It is known that the usual Heckman two-step procedure should not be used in the probit model: from a theoretical perspective, it is unsatisfactory, and likelihood methods are superior. However, serious numerical problems occur when standard software packages try to maximize the biprobit likelihood function, even if the number of covariates is small. We draw conclusions for statistical practice. Finally, we prove the conditions under which parameters in the model are identifiable. The conditions for identification are delicate; we believe these results are new.

Natural experiments help to overcome some of the obstacles researchers face when making causal inferences in the social sciences. However, even when natural interventions are randomly assigned, some of the treatment–control comparisons made available by natural experiments may not be valid. We offer a framework for clarifying the issues involved, which are subtle and often overlooked. We illustrate our framework by examining four different natural experiments used in the literature. In each case, random assignment of the intervention is not sufficient to provide an unbiased estimate of the causal effect. Additional assumptions are required that are problematic. For some examples, we propose alternative research designs that avoid these conceptual difficulties.