We are grateful to DeFord et al. for the continued attention to our work and the crucial issues of fair representation in democratic electoral systems. Our response (Katz, King, and Rosenblatt Forthcoming) was designed to help readers avoid being misled by mistaken claims in DeFord et al. (Forthcoming-a), and does not address other literature or uses of our prior work. As it happens, none of our corrections were addressed (or contradicted) in the most recent submission (DeFord et al. Forthcoming-b).

Katz, King, and Rosenblatt (2020, American Political Science Review 114, 164–178) introduces a theoretical framework for understanding redistricting and electoral systems, built on basic statistical and social science principles of inference. DeFord et al. (2021, Political Analysis, this issue) instead focuses solely on descriptive measures, which lead to the problems identified in our article. In this article, we illustrate the essential role of these basic principles and then offer statistical, mathematical, and substantive corrections required to apply DeFord et al.’s calculations to social science questions of interest, while also showing how to easily resolve all claimed paradoxes and problems. We are grateful to the authors for their interest in our work and for this opportunity to clarify these principles and our theoretical framework.

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.

Local abiotic and biotic conditions can alter the strength of exotic species impacts. To better understand the effects of exotic species on invaded ecosystems and to prioritize management efforts, it is important that exotic species impacts are put in local environmental context. We studied how differences in plant community composition, photosynthetically active radiation (PAR), and available soil N associated with Russian olive presence are conditioned by local environmental variation within a western U.S. riparian ecosystem. In four sites along the South Fork of the Republican River in Colorado, we established 200 pairs of plots (underneath and apart from Russian olive) to measure the effects of invasion across the ecosystem. We used a series of a priori mixed models to identify environmental variables that altered the effects of Russian olive. For all response variables, models that included the interaction of environmental characteristics, such as presence/absence of an existing cottonwood canopy, with the presence/absence of Russian olive canopy were stronger candidate models than those that just included Russian olive canopy presence as a factor. Compared with reference plots outside of Russian olive canopy, plots underneath Russian olive had higher relative exotic cover (exotic/total cover), lower perennial C4 grass cover, and higher perennial forb cover. These effects were reduced, however, in the presence of a cottonwood canopy. As expected, Russian olive was associated with reduced PAR and increased N, but these effects were reduced under cottonwood canopy. Our results demonstrate that local abiotic and biotic environmental factors condition the effects of Russian olive within a heterogeneous riparian ecosystem and suggest that management efforts should be focused in open areas where Russian olive impacts are strongest.

In a previous article we showed that ordinary least squares with panel corrected standard errors is superior to the Parks generalized least squares approach to the estimation of time-series-cross-section models. In this article we compare our proposed method with another leading technique, Kmenta's “cross-sectionally heteroskedastic and timewise autocorrelated” model. This estimator uses generalized least squares to correct for both panel heteroskedasticity and temporally correlated errors. We argue that it is best to model dynamics via a lagged dependent variable rather than via serially correlated errors. The lagged dependent variable approach makes it easier for researchers to examine dynamics and allows for natural generalizations in a manner that the serially correlated errors approach does not. We also show that the generalized least squares correction for panel heteroskedasticity is, in general, no improvement over ordinary least squares and is, in the presence of parameter heterogeneity, inferior to it. In the conclusion we present a unified method for analyzing time-series-cross-section data.

Ordinal variables—categorical variables with a defined order to the categories, but without equal spacing between them—are frequently used in social science applications. Although a good deal of research exists on the proper modeling of ordinal response variables, there is not a clear directive as to how to model ordinal treatment variables. The usual approaches found in the literature for using ordinal treatment variables are either to use fully unconstrained, though additive, ordinal group indicators or to use a numeric predictor constrained to be continuous. Generalized additive models are a useful exception to these assumptions. In contrast to the generalized additive modeling approach, we propose the use of a Bayesian shrinkage estimator to model ordinal treatment variables. The estimator we discuss in this paper allows the model to contain both individual group—level indicators and a continuous predictor. In contrast to traditionally used shrinkage models that pull the data toward a common mean, we use a linear model as the basis. Thus, each individual effect can be arbitrary, but the model “shrinks” the estimates toward a linear ordinal framework according to the data. We demonstrate the estimator on two political science examples: the impact of voter identification requirements on turnout and the impact of the frequency of religious service attendance on the liberality of abortion attitudes.

This article considers random coefficient models (RCMs) for time-series—cross-section data. These models allow for unit to unit variation in the model parameters. The heart of the article compares the finite sample properties of the fully pooled estimator, the unit by unit (unpooled) estimator, and the (maximum likelihood) RCM estimator. The maximum likelihood estimator RCM performs well, even where the data were generated so that the RCM would be problematic. In an appendix, we show that the most common feasible generalized least squares estimator of the RCM models is always inferior to the maximum likelihood estimator, and in smaller samples dramatically so.

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.