Whenever we report predicted values, we should also report some measure of the uncertainty of these estimates. In the linear case, this is relatively simple, and the answer well-known, but with nonlinear models the answer may not be apparent. This short article shows how to make these calculations. I first present this for the familiar linear case, also reviewing the two forms of uncertainty in these estimates, and then show how to calculate these for any arbitrary function. An example appears last.
This Note addresses the long-standing discrepancy between scholarly support for the effect of constituency service on incumbency advantage and a large body of contradictory empirical evidence. I show first that many of the methodological problems noticed in past research reduce to a single methodological problem that is readily resolved. The core of this Note then provides among the first systematic empirical evidence for the constituency service hypothesis. Specifically, an extra $10,000 added to the budget of the average state legislator gives this incumbent an additional 1.54 percentage points in the next election (with a 95% confidence interval of 1.14 to 1.94 percentage points).
"Politimetrics" (Gurr 1972), "polimetrics" (Alker 1975), "politometrics" (Hilton 1976), "political arithmetic" (Petty  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." 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 and we are developing persistent organizations, a growing body of scholarly literature, and an emerging consensus about important problems that need to be solved. I make one main point in this article: If political methodology is to play an important role in the future of political science, scholars will need to find ways of representing more interesting political contexts in quantitative analyses. This does not mean that scholars should just build more and more complicated statistical models. Instead, we need to represent more of the essence of political phenomena in our models. The advantage of formal and quantitative approaches is that they are abstract representations of the political world and are, thus, much clearer. We need methods that enable us to abstract the right parts of the phenomenon we are studying and exclude everything superfluous. Despite the fragmented history of quantitative political analysis, a version of this goal has been voiced frequently by both quantitative researchers and their critics (Sec. 2). However, while recognizing this shortcoming, earlier scholars were not in the position to rectify it, lacking the mathematical and statistical tools and, early on, the data. Since political methodologists have made great progress in these and other areas in recent years, I argue that we are now capable of realizing this goal. In section 3, I suggest specific approaches to this problem. Finally, in section 4, I provide two modern examples, ecological inference and models of spatial autocorrelation, to illustrate these points.
In an interesting and provocative article, Michael Lewis-Beck and Andrew Skalaban make an important contribution by emphasizing several philosophical issues in political methodology that have received too little attention from methodologists and quantitative researchers. These issues involve the role of systematic, and especially stochastic, variation in statistical models. After briefly discussing a few points of disagreement, hoping to reduce them to points of clarification, I turn to the philosophical issues. Examples with real data follow.
The dramatic increase in the electoral advantage of incumbency has sparked widespread interest among congressional researchers over the last 15 years. Although many scholars have studied the advantages of incumbency for incumbents, few have analyzed its effects on the underlying electoral system. We examine the influence of the incumbency advantage on two features of the electoral system in the U.S. House elections: electoral responsiveness and partisan bias. Using a district-level seats-votes model of House elections, we are able to distinguish systematic changes from unique, election-specific variations. Our results confirm the significant drop in responsiveness, and even steeper decline outside the South, over the past 40 years. Contrary to expectations, we find that increased incumbency advantage explains less than a third of this trend, indicating that some other unknown factor is responsible. Moreover, our analysis also reveals another dramatic pattern, largely overlooked in the congressional literature: in the 1940’s and 1950’s the electoral system was severely biased in favor of the Republican party. The system shifted incrementally from this severe Republican bias over the next several decades to a moderate Democratic bias by the mid-1980’s. Interestingly, changes in incumbency advantage explain virtually all of this trend in partisan bias since the 1940’s. By removing incumbency advantage and the existing configuration of incumbents and challengers analytically, our analysis reveals an underlying electoral system that remains consistently biased in favor of the Republican party. Thus, our results indicate that incumbency advantage affects the underlying electoral system, but contrary to conventional wisdom, this changes the trend in partisan bias more than electoral responsiveness.
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.
Because the goals of local and national representation are inherently incompatible, there is an uncertain relationship between aggregates of citizen votes and the national allocation of legislative seats in almost all democracies. In particular electoral systems, this uncertainty leads to diverse configurations of electoral responsiveness and partisian bias, two fundamental concepts in empirical democratic theory. This paper unifies virtually all existing multiyear seats-votes models as special cases of a new general model. It also permits the first formalization of, and reliable method for empirically estimating, electoral responsiveness and partisian bias in electoral systems with any number of political parties. I apply this model to data from nine democratic countries, revealing clear patterns in responsiveness and bias across different types of electoral rules.
In this paper we prove theoretically and demonstrate empirically that all existing measures of incumbency advantage in the congressional elections literature are biased or inconsistent. We then provide an unbiased estimator based on a very simple linear regression model. We apply this new method to congressional elections since 1900, providing the first evidence of a positive incumbency advantage in the first half of the century.
We analyze the effects of redistricting as revealed in the votes received by the Democratic and Republican candidates for state legislature. We develop measures of partisan bias and the responsiveness of the composition of the legislature to changes in statewide votes. Our statistical model incorporates a mixed hierarchical Bayesian and non-Bayesian estimation, requiring simulation along the lines of Tanner and Wong (1987). This model provides reliable estimates of partisan bias and responsiveness along with measures of their variabilities from only a single year of electoral data. This allows one to distinguish systematic changes in the underlying electoral system from typical election-to-election variability.
In this paper, we formalize existing normative criteria used to judge presidential selection contests by modeling the translation of citizen votes in primaries and caucuses into delegates to the national party conventions. We use a statistical model that enables us to separate the form of electoral responsiveness in presidential selection systems, as well as the degree of bias toward each of the candidates. We find that (1) the Republican nomination system is more responsive to changes in citizen votes than the Democratic system and (2) non-PR primaries are always more responsive than PR primaries and (3) surprisingly, caucuses are more proportional than even primaries held under PR rules and (4) significant bias in favor of a candidate was a good prediction of the winner of the nomination contest. We also (5) evaluate the claims of Ronald Reagan in 1976 and Jesse Jackson in 1988 that the selection systems were substantially biased against their candidates. We find no evidence to support Reagan’s claim, but substantial evidence that Jackson was correct.
The literature on cabinet duration is split between two apparently irreconcilable positions. The attributes theorists seek to explain cabinet duration as a fixed function of measured explanatory variables, while the events process theorists model cabinet durations as a product of purely stochastic processes. In this paper we build a unified statistical model that combines the insights of these previously distinct approaches. We also generalize this unified model, and all previous models, by including (1) a stochastic component that takes into account the censoring that occurs as a result of governments lasting to the vicinity of the maximum constitutional interelection period, (2) a systematic component that precludes the possibility of negative duration predictions, and (3) a much more objective and parsimonious list of explanatory variables, the explanatory power of which would not be improved by including a list of indicator variables for individual countries.
International relations theorists tend to think in terms of continuous processes. Yet we observe only discrete events, such as wars or alliances, and summarize them in terms of the frequency of occurrence. As such, most empirical analyses in international relations are based on event count variables. Unfortunately, analysts have generally relied on statistical techniques that were designed for continuous data. This mismatch between theory and method has caused bias, inefficiency, and numerous inconsistencies in both theoretical arguments and empirical findings throughout the literature. This article develops a much more powerful approach to modeling and statistical analysis based explicity on estimating continuous processes from observed event counts. To demonstrate this class of models, I present several new statistical techniques developed for and applied to different areas of international relations. These include the influence of international alliances on the outbreak of war, the contagious process of multilateral economic sanctions, and reciprocity in superpower conflict. I also show how one can extract considerably more information from existing data and relate substantive theory to empirical analyses more explicitly with this approach.
This paper builds a stochastic model of the processes that give rise to observed patterns of representation and bias in congressional and state legislative elections. The analysis demonstrates that partisan swing and incumbency voting, concepts from the congressional elections literature, have determinate effects on representation and bias, concepts from the redistricting literature. The model shows precisely how incumbency and increased variability of partisan swing reduce the responsiveness of the electoral system and how partisan swing affects whether the system is biased toward one party or the other. Incumbency, and other causes of unresponsive representation, also reduce the effect of partisan swing on current levels of partisan bias. By relaxing the restrictive portions of the widely applied "uniform partisan swing" assumption, the theoretical analysis leads directly to an empirical model enabling one more reliably to estimate responsiveness and bias from a single year of electoral data. Applying this to data from seven elections in each of six states, the paper demonstrates that redistricting has effects in predicted directions in the short run: partisan gerrymandering biases the system in favor of the party in control and, by freeing up seats held by opposition party incumbents, increases the system’s responsiveness. Bipartisan-controlled redistricting appears to reduce bias somewhat and dramatically to reduce responsiveness. Nonpartisan redistricting processes substantially increase responsiveness but do not have as clear an effect on bias. However, after only two elections, prima facie evidence for redistricting effects evaporate in most states. Finally, across every state and type of redistricting process, responsiveness declined significantly over the course of the decade. This is clear evidence that the phenomenon of "vanishing marginals," recognized first in the U.S. Congress literature, also applies to these different types of state legislative assemblies. It also strongly suggests that redistricting could not account for this pattern.
This article introduces a new estimator for the analysis of two contemporaneously correlated endogenous event count variables. This seemingly unrelated Poisson regression model (SUPREME) estimator combines the efficiencies created by single equation Poisson regression model estimators and insights from "seemingly unrelated" linear regression models.
This paper discusses the problem of variance specification in models for event count data. Event counts are dependent variables that can take on only nonnegative integer values, such as the number of wars or coups d’etat in a year. I discuss several generalizations of the Poisson regression model, presented in King (1988), to allow for substantively interesting stochastic processes that do not fit into the Poisson framework. Individual models that cope with, and help analyze, heterogeneity, contagion, and negative contagion are each shown to lead to specific statistical models for event count data. In addition, I derive a new generalized event count (GEC) model that enables researchers to extract significant amounts of new information from existing data by estimating features of these unobserved substantive processes. Applications of this model to congressional challenges of presidential vetoes and superpower conflict demonstrate the dramatic advantages of this approach.
This paper presents analytical, Monte Carlo, and empirical evidence on models for event count data. Event counts are dependent variables that measure the number of times some event occurs. Counts of international events are probably the most common, but numerous examples exist in every empirical field of the discipline. The results of the analysis below strongly suggest that the way event counts have been analyzed in hundreds of important political science studies have produced statistically and substantively unreliable results. Misspecification, inefficiency, bias, inconsistency, insufficiency, and other problems result from the unknowing application of two common methods that are without theoretical justification or empirical unity in this type of data. I show that the exponential Poisson regression (EPR) model provides analytically, in large samples, and empirically, in small, finite samples, a far superior model and optimal estimator. I also demonstrate the advantage of this methodology in an application to nineteenth-century party switching in the U.S. Congress. Its use by political scientists is strongly encouraged.
The translation of citizen votes into legislative seats is of central importance in democratic electoral systems. It has been a longstanding concern among scholars in political science and in numerous other disciplines. Through this literature, two fundamental tenets of democratic theory, partisan bias and democratic representation, have often been confused. We develop a general statistical model of the relationship between votes and seats and separate these two important concepts theoretically and empirically. In so doing, we also solve several methodological problems with the study of seats, votes and the cube law. An application to U.S. congressional districts provides estimates of bias and representation for each state and deomonstrates the model’s utility. Results of this application show distinct types of representation coexisting in U.S. states. Although most states have small partisan biases, there are some with a substantial degree of bias.