Dyadic (state-pair) data is completely inappropriate for analyzing multilateral events (such as large alliances and major wars). Scholars, particularly in international relations, often divide the actors in a multilateral event into a series of dyadic relations. Though this practice can dramatically increase the size of data sets, using dyadic data to analyze what are, in reality, k-adic events leads to model misspecification and, inevitably, statistical bias. In short, one cannot recover a k-adic data generating process using dy-adic data. In this paper, I accomplish three tasks. First, I use Monte Carlo simulations to confirm that analyzing k-adic events with dyadic data produces substantial bias. Second, I show that choice-based sampling, as popularized by King and Zeng (2001a, Explaining rare events in international relations. International Organization 55:693–715, and 2001b, Logistic regression in rare events data. Political Analysis 9:137–63), can be used to create feasibly sized k-adic data sets. Finally, I use the study of alliance formation by Gibler and Wolford (2006, Alliances, then democracy: An examination of the relationship between regime type and alliance formation. The Journal of Conflict Resolution 50:1–25) to illustrate how to apply this choice-based sampling solution and explain how to code independent variables in a k-adic context.

Since Beck, Katz, and Tucker (1998), the standard method for modeling time dependence in binary data has been to incorporate time dummies or splined time in logistic regressions. Although we agree with the need for modeling time dependence, we demonstrate that time dummies can induce estimation problems due to separation. Splines do not suffer from these problems. However, the complexity of splines has led substantive researchers (1) to use knot values that may be inappropriate for their data and (2) to ignore any substantive discussion concerning temporal dependence. We propose a relatively simple alternative: including t, t2, and t3 in the regression. This cubic polynomial approximation is trivial to implement—and, therefore, interpret—and it avoids problems such as quasi-complete separation. Monte Carlo analysis demonstrates that, for the types of hazards one often sees in substantive research, the polynomial approximation always outperforms time dummies and generally performs as well as splines or even more flexible autosmoothing procedures. Due to its simplicity, this method also accommodates nonproportional hazards in a straightforward way. We reanalyze Crowley and Skocpol (2001) using nonproportional hazards and find new empirical support for the historical-institutionalist perspective.

Political scientists lack methods to efficiently measure the priorities political actors emphasize in statements. To address this limitation, I introduce a statistical model that attends to the structure of political rhetoric when measuring expressed priorities: statements are naturally organized by author. The expressed agenda model exploits this structure to simultaneously estimate the topics in the texts, as well as the attention political actors allocate to the estimated topics. I apply the method to a collection of over 24,000 press releases from senators from 2007, which I demonstrate is an ideal medium to measure how senators explain their work in Washington to constituents. A set of examples validates the estimated priorities and demonstrates their usefulness for testing theories of how members of Congress communicate with constituents. The statistical model and its extensions will be made available in a forthcoming free software package for the R computing language.

Carter and Signorino (2010) (hereinafter “CS”) add another arrow, a simple cubic polynomial in time, to the quiver of the binary time series—cross-section data analyst; it is always good to have more arrows in one's quiver. Since comments are meant to be brief, I will discuss here only two important issues where I disagree: are cubic duration polynomials the best way to model duration dependence and whether we can substantively interpret duration dependence.

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.

Using a Bayesian latent variable approach, we synthesize a new measure of democracy, the Unified Democracy Scores (UDS), from 10 extant scales. Our measure eschews the difficult—and often arbitrary—decision to use one existing democracy scale over another in favor of a cumulative approach that allows us to simultaneously leverage the measurement efforts of numerous scholars. The result of this cumulative approach is a measure of democracy that, for every country-year, is at least as reliable as the most reliable component measure and is accompanied by quantitative estimates of uncertainty in the level of democracy. Moreover, for those who wish to continue using previously existing scales or to evaluate research performed using those scales, we extract information from the new measure to perform heretofore impossible direct comparisons between component scales. Specifically, we estimate the relative reliability of the constituent indicators, compare the specific ordinal levels of each of the existing measures in relationship to one another and assess overall levels of disagreement across raters. We make the UDS and associated parameter estimates freely available online and provide a detailed tutorial that demonstrates how to best use the UDS in applied work.

In this paper, we discuss an estimator for average treatment effects (ATEs) known as the augmented inverse propensity weighted (AIPW) estimator. This estimator has attractive theoretical properties and only requires practitioners to do two things they are already comfortable with: (1) specify a binary regression model for the propensity score, and (2) specify a regression model for the outcome variable. Perhaps the most interesting property of this estimator is its so-called “double robustness.” Put simply, the estimator remains consistent for the ATE if either the propensity score model or the outcome regression is misspecified but the other is properly specified. After explaining the AIPW estimator, we conduct a Monte Carlo experiment that compares the finite sample performance of the AIPW estimator to three common competitors: a regression estimator, an inverse propensity weighted (IPW) estimator, and a propensity score matching estimator. The Monte Carlo results show that the AIPW estimator has comparable or lower mean square error than the competing estimators when the propensity score and outcome models are both properly specified and, when one of the models is misspecified, the AIPW estimator is superior.

Ideal point estimators are typically based on an assumption that all legislators are equally responsive to modeled dimensions of legislative disagreement; however, particularistic constituency interests and idiosyncrasies of individual legislators introduce variation in the degree to which legislators cast votes predictably. I introduce a Bayesian heteroskedastic ideal point estimator and demonstrate by Monte Carlo simulation that it outperforms standard homoskedastic estimators at recovering the relative positions of legislators. In addition to providing a refinement of ideal point estimates, the heteroskedastic estimator recovers legislator-specific error variance parameters that describe the extent to which each legislator's voting behavior is not conditioned on the primary axes of disagreement in the legislature. Through applications to the roll call histories of the U.S. Congress, the E.U. Parliament, and the U.N. General Assembly, I demonstrate how to use the heteroskedastic estimator to study substantive questions related to legislative incentives for low-dimensional voting behavior as well as diagnose unmodeled dimensions and nonconstant ideal points.

Many social processes are stable and smooth in general, with discrete jumps. We develop a sequential segmentation spline method that can identify both the location and the number of discontinuities in a series of observations with a time component, while fitting a smooth spline between jumps, using a modified Bayesian Information Criterion statistic as a stopping rule. We explore the method in a large-n, unbalanced panel setting with George W. Bush's approval data, a small-n time series with median DW-NOMINATE scores for each Congress over time, and a series of simulations. We compare the method to several extant smoothers, and the method performs favorably in terms of visual inspection, residual properties, and event detection. Finally, we discuss extensions of the method.

Researchers using latent class (LC) analysis often proceed using the following three steps: (1) an LC model is built for a set of response variables, (2) subjects are assigned to LCs based on their posterior class membership probabilities, and (3) the association between the assigned class membership and external variables is investigated using simple cross-tabulations or multinomial logistic regression analysis. Bolck, Croon, and Hagenaars (2004) demonstrated that such a three-step approach underestimates the associations between covariates and class membership. They proposed resolving this problem by means of a specific correction method that involves modifying the third step. In this article, I extend the correction method of Bolck, Croon, and Hagenaars by showing that it involves maximizing a weighted log-likelihood function for clustered data. This conceptualization makes it possible to apply the method not only with categorical but also with continuous explanatory variables, to obtain correct tests using complex sampling variance estimation methods, and to implement it in standard software for logistic regression analysis. In addition, a new maximum likelihood (ML)—based correction method is proposed, which is more direct in the sense that it does not require analyzing weighted data. This new three-step ML method can be easily implemented in software for LC analysis. The reported simulation study shows that both correction methods perform very well in the sense that their parameter estimates and their SEs can be trusted, except for situations with very poorly separated classes. The main advantage of the ML method compared with the Bolck, Croon, and Hagenaars approach is that it is much more efficient and almost as efficient as one-step ML estimation.

We thank Neal Beck for his comments. In general, there is little difference in our theoretical and practical positions. We all agree that in some applications, time dummies can be problematic; controlling for temporal dependence is important for less-biased estimation of the coefficients on our substantive regressors; a smoothed function of time (whether via a cubic polynomial or spline) is a useful way to model temporal dependence in grouped binary data; and viewing the graph of the hazard can be informative.

The increased use of models with limited-dependent variables has allowed researchers to test important relationships in political science. Often, however, researchers employing such models fail to acknowledge that the violation of some basic assumptions has in part difference consequences in nonlinear models than in linear ones. In this paper, I demonstrate this for binary probit models in which the dependent variable is systematically miscoded. Contrary to the linear model, such misclassifications affect not only the estimate of the intercept but also those of the other coefficients. In a Monte Carlo simulation, I demonstrate that a model proposed by Hausman, Abrevaya, and Scott-Morton (1998, Misclassification of the dependent variable in a discrete-response setting. Journal of Econometrics 87:239–69) allows for correcting these biases in binary probit models. Empirical examples based on reanalyses of models explaining the occurrence of rebellions and civil wars demonstrate the problem that comes from neglecting these misclassifications.

Randomized experiments commonly compare subjects receiving a treatment to subjects receiving a placebo. An alternative design, frequently used in field experimentation, compares subjects assigned to an untreated baseline group to subjects assigned to a treatment group, adjusting statistically for the fact that some members of the treatment group may fail to receive the treatment. This article shows the potential advantages of a three-group design (baseline, placebo, and treatment). We present a maximum likelihood estimator of the treatment effect for this three-group design and illustrate its use with a field experiment that gauges the effect of prerecorded phone calls on voter turnout. The three-group design offers efficiency advantages over two-group designs while at the same time guarding against unanticipated placebo effects (which would undermine the placebo-treatment comparison) and unexpectedly low rates of compliance with the treatment assignment (which would undermine the baseline-treatment comparison).

This paper proposes a Bayesian generalized linear multilevel model with a pth-order autoregressive error process to analyze unbalanced binary time-series cross-sectional (TSCS) data. The model specification is motivated by the generic TSCS data structure and is intended to handle the associated inefficiency and endogeneity problems. It accommodates heterogeneity across units and between time periods in the form of random intercepts and random-effect coefficients. At the same time, its pth-order autoregressive error process, employed either by itself or in concert with other dynamic methods, adequately corrects serial correlation and improves statistical inference and forecasting. With a stationarity restriction on the error process, the model can also be used as a residual-based cointegration test on discrete TSCS data. This is especially valuable because cointegration testing on discrete TSCS data is methodologically challenging and rarely conducted in practice. To handle the estimation difficulties, I developed an efficient Markov chain Monte Carlo (MCMC) algorithm by orthogonalizing the error term with the Cholesky decomposition and adding an auxiliary variable. The parameter expansion method, that is, partial group move—multigrid Monte Carlo updating (PGM-MGMC), is employed to further improve MCMC mixing and speed up convergence. The paper also provides a computational scheme to approximate the Bayes's factor for the purposes of serial correlation diagnostics, lag order determination, and variable selection. Simulated and empirical examples are used to assess the model and techniques.

Although sample selection bias is a frequent problem of applied research, there has been no generalization of sample selection models with binary dependent variables of interest to data with temporal error correlations. We suggest a generalized estimating equation approach to panel data selection models, considering binary responses in both equations. We demonstrate the utility of this model by a simulation study and by analyzing highly unbalanced annual panel data taken from the German Socio-Economic Panel Study covering two decades of Green party support.

The issue of how qualitative and quantitative information can be used together is critical. Brady, Collier, and Seawright (BCS) have argued that “causal process observations” can be adjoined to “data set observations.” This implies that qualitative methods can be used to add information to quantitative data sets. In a symposium in Political Analysis, I argued that such qualitative information cannot be adjoined in any meaningful way to quantitative data sets. In that symposium, the original authors offered several defenses, but, in the end, BCS can be seen as recommending good, but hopefully standard, research design practices that are normally thought of as central in the quantitative arena. It is good that BCS remind us that no amount of fancy statistics can save a bad research design.

There has been no attention to circular (purely cyclical) data in political science research. We show that such data exist and are mishandled by models that do not take into account the inherently recycling nature of some phenomenon. Clock and calendar effects are the obvious cases, but directional data are observed as well. We describe a standard maximum likelihood regression modeling framework based on the von Mises distribution, then develop a general Bayesian regression procedure for the first time, providing an easy-to-use Metropolis-Hastings sampler for this approach. Applications include a chronographic analysis of U.S. domestic terrorism and directional party preferences in a two-dimensional ideological space for German Bundestag elections. The results demonstrate the importance of circular models to handle periodic and directional data in political science.

The Cox proportional hazards model is widely used to model durations in the social sciences. Although this model allows analysts to forgo choices about the form of the hazard, it demands careful attention to the proportional hazards assumption. To this end, a standard diagnostic method has been developed to test this assumption. I argue that the standard test for nonproportional hazards has been misunderstood in current practice. This test detects a variety of specification errors, and these specification errors must be corrected before one can correctly diagnose nonproportionality. In particular, unmodeled nonlinearity can appear as a violation of the proportional hazard assumption for the Cox model. Using both simulation and empirical examples, I demonstrate how an analyst might be led astray by incorrectly applying the nonproportionality test.

Many claims about political behavior are based on implicit assumptions about how people think. One such assumption, that political actors use identical conjectures when assessing others' strategies, is nested within applications of widely used game-theoretic equilibrium concepts. When empirical findings call this assumption into question, the self-confirming equilibrium (SCE) concept provides an alternate criterion for theoretical claims. We examine applications of SCE to political science. Our main example focuses on the claim of Feddersen and Pesendorfer that unanimity rule can lead juries to convict innocent defendants (1998. Convicting the innocent: The inferiority of unanimous jury verdicts under strategic voting. American Political Science Review 92:23–35). We show that the claim depends on the assumption that jurors have identical beliefs about one another's types and identical conjectures about one another's strategies. When jurors' beliefs and conjectures vary in ways documented by empirical jury research, fewer false convictions can occur in equilibrium. The SCE concept can confer inferential advantages when actors have different beliefs and conjectures about one another.