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The use of multilevel models—models in which lower-level (“micro”) units are nested within higher-level (“macro”) units—has blossomed recently in political science. Possible relationships in such models include macro variables influencing macro variables; micro variables influencing micro variables; macro variables influencing micro variables, and vice versa; and often most interestingly, micro-micro relationships varying interactively with macro variables. Most work in political science has drawn on the useful introductions of Raudenbush and Bryk (2002), Western (1998), and Steenbergen and Jones (2002). We refer readers to good general introductions/reviews of multi-level modeling in the articles in this issue by Bowers and Drake and by Franzese.
Nearly all hierarchical linear models presented to political science audiences are estimated using maximum likelihood under a repeated sampling interpretation of the results of hypothesis tests. Maximum likelihood estimators have excellent asymptotic properties but less than ideal small sample properties. Multilevel models common in political science have relatively large samples of units like individuals nested within relatively small samples of units like countries. Often these level-2 samples will be so small as to make inference about level-2 effects uninterpretable in the likelihood framework from which they were estimated. When analysts do not have enough data to make a compelling argument for repeated sampling based probabilistic inference, we show how visualization can be a useful way of allowing scientific progress to continue despite lack of fit between research design and asymptotic properties of maximum likelihood estimators.
Somewhere along the line in the teaching of statistics in the social sciences, the importance of good judgment got lost amid the minutiae of null hypothesis testing. It is all right, indeed essential, to argue flexibly and in detail for a particular case when you use statistics. Data analysis should not be pointlessly formal. It should make an interesting claim; it should tell a story that an informed audience will care about, and it should do so by intelligent interpretation of appropriate evidence from empirical measurements or observations.
—Abelson, 1995, p. 2
With neither prior mathematical theory nor intensive prior investigation of the data, throwing half a dozen or more exogenous variables into a regression, probit, or novel maximum-likelihood estimator is pointless. No one knows how they are interrelated, and the high-dimensional parameter space will generate a shimmering pseudo-fit like a bright coat of paint on a boat's rotting hull.
In recent years, large sets of national surveys with shared content have increasingly been used for cross-national opinion research. But scholars have not yet settled on the most flexible and efficient models for utilizing such data. We present a two-step strategy for such analysis that takes advantage of the fact that in such datasets each “cluster” (i.e., country sample) is large enough to sustain separate analysis of its internal variances and covariances. We illustrate the method by examining a puzzle of comparative electoral behavior—why does turnout decline rather than increase with the number of parties competing in an election (Blais and Dobryzynska 1998, for example)? This discussion demonstrates the ease with which a two-step strategy incorporates confounding variables operating at different levels of analysis. Technical appendices demonstrate that the two-step strategy does not lose efficiency of estimation as compared with a pooling strategy.
Researchers often use as dependent variables quantities estimated from auxiliary data sets. Estimated dependent variable (EDV) models arise, for example, in studies where counties or states are the units of analysis and the dependent variable is an estimated mean, proportion, or regression coefficient. Scholars fitting EDV models have generally recognized that variation in the sampling variance of the observations on the dependent variable will induce heteroscedasticity. We show that the most common approach to this problem, weighted least squares, will usually lead to inefficient estimates and underestimated standard errors. In many cases, OLS with White's or Efron heteroscedastic consistent standard errors yields better results. We also suggest two simple alternative FGLS approaches that are more efficient and yield consistent standard error estimates. Finally, we apply the various alternative estimators to a replication of Cohen's (2004) cross-national study of presidential approval.
This paper develops and tests arguments about how national-level social and institutional factors shape the propensity of individuals to form attachments to political parties. Our tests employ a two-step estimation procedure that has attractive properties when there is a binary dependent variable in the first stage and when the number of second-level units is relatively small. We find that voters are most likely to form party attachments when group identities are salient and complimentary. We also find that institutions that assist voters in retrospectively evaluating parties—specifically, strong party discipline and few parties in government—increase partisanship. These institutions matter most for those individuals with the fewest cognitive resources, measured here by education.
Voters use observed economic performance to infer the competence of incumbent politicians. These economic perceptions enter the voter's utility calculations modified by a weight that is minimized when the variance in exogenous shocks to the economy is very large relative to the variance in economic outcomes associated with the competence of politicians. Cross-national variations in the political and economic context systematically increase or undermine the voter's ability to ascertain the competency of incumbents. We test one hypothesis: As policy-making responsibility is shared more equally among parties, economic evaluations will be more important in the vote decision. We employ two multilevel modeling procedures for estimating the contextual variations in micro-level economic voting effects: a conventional pooled approach and a two-stage procedure. We compare the multivariate results of a pooled method with our two-stage estimation procedure and conclude that they are similar. Our empirical efforts use data from 163 national surveys from 18 countries over a 22-year period.
I analyze how the diffusion of power in parliaments affects voter choice. Using a two-step research design, I first estimate an individual-level model of voter choice in 14 parliamentary democracies, allowing voters to hold preferences both for the party most similar to them ideologically and for the party that pulls policy in their direction. While in systems in which power is concentrated the two motivations converge, in consensual systems they diverge: since votes will likely be watered down by bargaining in the parliament, outcome-oriented choice in consensual systems often leads voters to endorse parties whose positions differ from their own views. In the second step, I utilize institutional measures of power diffusion in the parliament to account for the degree to which voters in different polities pursue one motivation versus the other. I demonstrate that the more power diffusion and compromise built into the political system via institutional mechanisms, the more voters compensate for the watering down of their vote by endorsing parties whose positions differ from their own views.
Equivalent separate-subsample (two-step) and pooled-sample (one-step) strategies exist for any multilevel-modeling task, but their relative practicality and efficacy depend on dataset dimensions and properties and researchers' goals. Separate-subsample strategies have difficulties incorporating cross-subsample information, often crucial in time-series cross-section or panel contexts (subsamples small and/or cross-subsample information great) but less relevant in pools of independently random surveys (subsamples large; cross-sample information small). Separate-subsample estimation also complicates retrieval of macro-level-effect estimates, although they remain obtainable and may not be substantively central. Pooled-sample estimation, conversely, struggles with stochastic specifications that differ across levels (e.g., stochastic linear interactions in binary dependent-variable models). Moreover, pooled-sample estimation that models coefficient variation in a theoretically reduced manner rather than allowing each subsample coefficient vector to differ arbitrarily can suffer misspecification ills insofar as this reduced specification is lacking. Often, though, these ills are limited to inefficiencies and standard-error inaccuracies that familiar efficient (e.g., feasible generalized least squares) or consistent-standard-error estimation strategies can satisfactorily redress.
Two-step estimators for hierarchical models can be constructed even when neither stage is a conventional linear regression model. For example, the first stage might consist of probit models, or duration models, or event count models. The second stage might be a nonlinear regression specification. This note sketches some of the considerations that arise in ensuring that two-step estimators are consistent in such cases.
The articles in this special issue all use multilevel methods to study comparative political behavior. This is obviously a good thing, for both methodology and comparative politics. Clearly comparative politics means comparing things and not just studying nations other than the United States. This is equally true of micropolitical studies. These articles all do a very nice job of showing how one can do comparative micropolitics (and tie together micro and macro variables).
These articles demonstrate, in several different examples, the effectiveness of two-level regression: the procedure of fitting several separate regression models, and then fitting a second, higher-level, regression to the estimated coefficients (for example, fitting a separate regression model to survey data from each of several countries, then regressing the coefficient estimates on country-level predictors). For simplicity, we will refer to the first- and second-level units as “persons” and “countries,” respectively, but our points apply more generally.