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Below results based on the criteria 'multilevel'
Total number of records returned: 19

The difference between ``significant'' and ``not significant'' is not itself statistically significant
Gelman, Andrew
Stern, Hal

Uploaded 12-23-2005
Keywords multilevel modeling
multiple comparisons
statistical significance
Abstract A common error in statistical analyses is to summarize comparisons by declarations of statistical significance or non-significance. There are a number of difficulties with this approach. First is the oft-cited dictum that statistical significance is not the same as practical significance. Another difficulty is that this dichotomization into significant and non-significant results encourages the dismissal of observed differences in favor of the usually less interesting null hypothesis of no difference. Here, we focus on a less commonly noted problem, namely that changes in statistical significance are not themselves significant. By this, we are not merely making the commonplace observation that any particular threshold is arbitrary---for example, only a small change is required to move an estimate from a 5.1% significance level to 4.9%, thus moving it into statistical significance. Rather, we are pointing out that even large changes in significance levels can correspond to small, non-significant changes in the underlying variables. We illustrate with a theoretical and an applied example.

Fitting Multilevel Models When Predictors and Group Effects Correlate
Bafumi, Joseph

Uploaded 04-27-2006
Keywords Multilevel models
random effects
fixed effects
unit effects
group effects
Abstract Random effects models (that is, regressions with varying intercepts that are modeled with error) are avoided by some social scientists because of potential issues with bias and uncertainty estimates. Particularly, when one or more predictors correlate with the group or unit effects, a key Gauss-Markov assumption is violated and estimates are compromised. However, this problem can easily be solved by including the average of each individual-level predictors in the group-level regression. We explain the solution, demonstrate its effectiveness using simulations, show how it can be applied in some commonly-used statistical software, and discuss its potential for substantive modeling.

A default prior distribution for logistic and other regression models
Gelman, Andrew
Jakulin, Aleks
Pittau, Maria Grazia
Su, Yu-Sung

Uploaded 08-03-2007
Keywords Bayesian inference
generalized linear model
least squares
hierarchical model
linear regression
logistic regression
multilevel model
noninformative prior distribution
Abstract We propose a new prior distribution for classical (non-hierarchical) logistic regression models, constructed by first scaling all nonbinary variables to have mean 0 and standard deviation 0.5, and then placing independent Student-$t$ prior distributions on the coefficients. As a default choice, we recommend the Cauchy distribution with center 0 and scale 2.5, which in the simplest setting is a longer-tailed version of the distribution attained by assuming one-half additional success and one-half additional failure in a logistic regression. We implement a procedure to fit generalized linear models in R with this prior distribution by incorporating an approximate EM algorithm into the usual iteratively weighted least squares. We illustrate with several examples, including a series of logistic regressions predicting voting preferences, an imputation model for a public health data set, and a hierarchical logistic regression in epidemiology. We recommend this default prior distribution for routine applied use. It has the advantage of always giving answers, even when there is complete separation in logistic regression (a common problem, even when the sample size is large and the number of predictors is small) and also automatically applying more shrinkage to higher-order interactions. This can be useful in routine data analysis as well as in automated procedures such as chained equations for missing-data imputation.

Causal Inference of Repeated Observations: A Synthesis of the Propensity Score Methods and Multilevel Modeling
Su, Yu-Sung

Uploaded 07-03-2008
Keywords causal inference
balancing score
multilevel modeling
propensity score
time-series-cross-sectional data
Abstract The fundamental problem of causal inference is that an individual cannot be simultaneously observed in both the treatment and control states (Holland 1986). The propensity score methods that compare the treatment and control groups by discarding the unmatched units are now widely used to deal with this problem. In some situations, however, it is possible to observe the same individual or unit of observation in the treatment and control states at different points in time. The data has the structure that is often refer to as time-series-cross-sectional (TSCS) data. While multilevel modeling is often applied to analyze TSCS data, this paper proposes that synthesizing the propensity score methods and multilevel modeling is preferable. The paper conducts a Monte Carlo simulation with 36 different scenarios to test the performance of the two combined methods. The result shows that synthesizing the propensity score matching with multilevel modeling performs better in that such method yields less biased and more efficient estimates. An empirical case study that reexamine the model of Przeworksi et al (2000) on democratization and development also shows the advantage of this synthesis.

Public Opinion and Senate Confirmation of Supreme Court Nominees
Kastellec, Jonathan
Lax, Jeffrey
Phillips, Justin

Uploaded 08-22-2008
Keywords Supreme Court
public opinion
multilevel models

Abstract We study the relationship between state-level public opinion and the roll call votes of senators on Supreme Court nominees. Applying recent advances in multilevel modeling, we use national polls on nine recent Supreme Court nominees to produce state-of-the-art estimates of public support for the confirmation of each nominee in all 50 states. We show that greater public support strongly increases the probability that a senator will vote to approve a nominee, even after controlling for standard predictors of roll call voting. We also find that the impact of opinion varies with context: it has a greater effect on opposition party senators, on ideologically opposed senators, and for generally weak nominees. These results establish a systematic and powerful link between constituency opinion and voting on Supreme Court nominees.

Cosponsorship in the U.S. Senate: A Multilevel Approach to Detecting Subtle Social Predictors of Legilslative Support
Gross, Justin

Uploaded 09-14-2008
Keywords Congress
social network analysis
multilevel models
mixed effects
Abstract Why do members of Congress choose to cosponsor legislation proposed by their colleagues and what can we learn from their patterns of cosponsorship? To answer these questions properly requires models that respect the relational nature of the relevant data and the resulting interdependence among observations. We show how the inclusion of carefully selected random effects can capture network-type dependence, allowing us to more confidently investigate senators' propensity to support colleagues' proposals. To illustrate, we examine whether certain social factors such as demographic similarities, opportunities for interaction, and institutional roles are associated with varying odds of cosponsorship during the 2003-04 (108th) Senate.

Beyond "Fixed Versus Random Effects": A Framework for Improving Substantive and Statistical Analysis of Panel, TSCS, and Multilevel Data
Bartels, Brandon

Uploaded 09-30-2008
Keywords random effects
fixed effects
time-series cross-sectional data
panel data
multilevel modeling
Abstract Researchers analyzing panel, time-series cross-sectional, and multilevel data often choose between a random effects, fixed effects, or complete pooling modeling approach. While pros and cons exist for each approach, I contend that some core issues concerning clustered data continue to be ignored. I present a unified and simple modeling framework for analyzing clustered data that solves many of the substantive and statistical problems inherent in extant approaches. The approach: (1) solves the substantive interpretation problems associated with cluster confounding, which occurs when one assumes that within- and between-cluster effects are equal; (2) accounts for cluster-level unobserved heterogeneity via a random intercept model; (3) satisfies the controversial statistical assumption that level-1 variables be uncorrelated with the random effects term; (4) allows for the inclusion of level-2 variables; and (5) allows for statistical tests of cluster confounding. I illustrate this approach using three substantive examples: global human rights abuse, oil production for OPEC countries, and Senate voting on Supreme Court nominations. Reexaminations of these data produce refined interpretations of some of the core substantive conclusions.

Prior Distributions for Variance Parameters in Hierarchical Models
Gelman, Andrew

Uploaded 03-28-2004
Keywords Bayesian inference
hierarchical model
multilevel model
noninformative prior distribution
weakly informative prior distribution
Abstract Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new folded-noncentral-$t$ family of conditionally conjugate priors for hierarchical standard deviation parameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inverse-gamma family of "noninformative" prior distributions. We suggest instead to use a uniform prior on the hierarchical standard deviation, using the half-$t$ family when the number of groups is small and in other settings where a weakly informative prior is desired.

Language Access and Initiative Outcomes: Did the Voting Rights Act Influence Support for Bilingual Education?

Uploaded 12-17-2009
Keywords regression discontinuity design
multilevel modeling
immigrant political incorporation
language access
Voting Rights Act
Abstract This paper investigates one tool designed to enfranchise immigrants: foreign-language election materials. Specifically, it estimates the impact of Spanish-language assistance provided under Section 203 of the Voting Rights Act. Focusing on a California initiative on bilingual education, it tests how Spanish-language materials influenced turnout and election outcomes in Latino neighborhoods. It also considers the possibility of an anti-Spanish backlash in non-Hispanic white neighborhoods. Empirically, the analysis couples a regression discontinuity design with multilevel modeling to isolate the impact of Section 203. The analysis finds that Spanish-language assistance increased turnout and reduced support for ending bilingual education in Latino neighborhoods with many Spanish speakers. It finds hints of backlash among non-Hispanic white precincts, but not with the same certainty. The turnout finding gains additional support from multilevel regression discontinuity analyses of 2004 Latino voter turnout nationwide. For Latino citizens who speak little English, the availability of Spanish ballots increases turnout and influences election outcomes as well.

Bayesian Measures of Explained Variance and Pooling in Multilevel (Hierarchical) Models
Gelman, Andrew
Pardoe, Iain

Uploaded 04-16-2004
Keywords adjusted R-squared
Bayesian inference
hierarchical model
multilevel regression
partial pooling
Abstract Explained variance (R2) is a familiar summary of the fit of a linear regression and has been generalized in various ways to multilevel (hierarchical) models. The multilevel models we consider in this paper are characterized by hierarchical data structures in which individuals are grouped into units (which themselves might be further grouped into larger units), and there are variables measured on individuals and each grouping unit. The models are based on regression relationships at different levels, with the first level corresponding to the individual data, and subsequent levels corresponding to between-group regressions of individual predictor effects on grouping unit variables. We present an approach to defining R2 at each level of the multilevel model, rather than attempting to create a single summary measure of fit. Our method is based on comparing variances in a single fitted model rather than comparing to a null model. In simple regression, our measure generalizes the classical adjusted R2. We also discuss a related variance comparison to summarize the degree to which estimates at each level of the model are pooled together based on the level-specific regression relationship, rather than estimated separately. This pooling factor is related to the concept of shrinkage in simple hierarchical models. We illustrate the methods on a dataset of radon in houses within counties using a series of models ranging from a simple linear regression model to a multilevel varying-intercept, varying-slope model.

Modeling Electoral Coordination: Voters, Parties and Legislative Lists in Uruguay
Levin, Ines
Katz, Gabriel

Uploaded 04-20-2011
Keywords electoral coordination
number of parties
Bayesian estimation
multilevel modeling
strategic voting
Abstract During each electoral period, the strategic interaction between voters and political elites determines the number of viable candidates in a district. In this paper, we implement a hierarchical seemingly unrelated regression model to explain electoral coordination at the district level in Uruguay as a function of district magnitude, previous electoral outcomes and electoral regime. Elections in this country are particularly useful to test for institutional effects on the coordination process due to the large variations in district magnitude, to the simultaneity of presidential and legislative races held under different rules, and to the reforms implemented during the period under consideration. We find that district magnitude and electoral history heuristics have substantial effects on the number of competing and voted-for parties and lists. Our modeling approach uncovers important interaction-effects between the demand and supply side of the political market that were often overlooked in previous research.

Multiculturalism, Diversity, and Prejudice
Branton, Regina P.
Jones, Bradford S.

Uploaded 03-27-1999
Keywords random coefficients
multilevel analysis
racial politics
Abstract In this paper, we consider the relationship between racial and ethnic diversity and individuals' assessments of racial and ethnic groups measured on several public opinion items. To examine these issues, we merge 1992 National Election Study data with U.S. Census Bureau demographic data measured at the congressional district level. We then develop an index of diversity that is based on the distribution of racial and ethnic groups in the congressional district. To examine the relationship between diversity and individual-level attitudes toward racial and ethnic groups, we estimate a series of models treating the response variable as a function of both individual-level attributes and district-level attributes. This approach allows us to assess, among other things, if diversity is associated with more positive or negative evaluations of racial and ethnic groups. The models herein are all estimated as mixed effects models to account for the clustering of observations within congressional districts. We find that diversity is associated with group affect and individuals' placement on policy issues; however, in contrast to some of the extant literature, we find that racial and ethnic diversity is nonlinearly associated with group affect: extremely low and extremely high levels of racial and ethnic diversity are associated with lower racial and ethnic group evaluations, while districts that are moderately diverse are associated with higher evaluations. This result also holds for some of the policy items examined. Specifically, we find that support for government assistance to blacks, and to a lesser extent, support for affirmative action, exhibits this nonlinearity with regard to racial and ethnic diversity. We also find this pattern for individuals' assessment of welfare recipients.

Should I Use Fixed or Random Effects?
Clark, Tom
Linzer, Drew

Uploaded 03-26-2012
Keywords Fixed effects
Random effects
Panel data
Abstract Empirical analyses in political science very commonly confront data that are grouped---multiple votes by individual legislators, multiple years in individual states, multiple conflicts during individual years, and so forth. Modeling these data presents a series of potential challenges, of which accounting for differences across the groups is perhaps the most well-known. Two widely-used methods are the use of either "fixed" or "random" effects models. However, how best to choose between these approaches remains unclear in the applied literature. We employ a series of simulation experiments to evaluate the relative performance of fixed and random effects estimators for varying types of datasets. We further investigate the commonly-used Hausman test, and demonstrate that it is neither a necessary nor sufficient statistic for deciding between fixed and random effects. We summarize the results into a typology of datasets to offer practical guidance to the applied researcher.

Aggregate Economic Conditions and Indivdual Forecasts: A Mulilevel Model of EconomicExpectations
Jones, Bradford S.
Haller, H. Brandon

Uploaded 00-00-0000
Keywords random coefficient modeling
multilevel analysis
hierarchical linear models
Abstract To what extent are individual economic expectations related to actual economic conditions? This is the central question examined in this paper. Surprisingly, little research exists examining how economic expectations are formed. Moreover, even less research has been done examining the interaction between the state of the national economy and individual forecasts. Most research addressing expectation formation has resided at the aggregate level. In this paper, we utilize the methodology of random coefficient models to explore the linkage between individuals and the macroeconomic environment. We conceptualize individuals as being "nested" within time periods. Individual forecasts are treated as contextually conditioned by the state of the economy. We find evidence that aggregate economic indicators do influence the parameters predicting economic expectations. Furthermore, the relationship between the macroeconomy and individual expectations provides strong support for Katona's (1972, 1975) notion of "psychological economics." We find that individual forecasts of the future are "brighter" when aggregate economic conditions are "darkest." Additionally, we find that individuals tend to rely less on retrospective evaluations of the economy when the economy is faring poorly.

Modeling Multilevel Data Structures
Jones, Bradford S.
Steenbergen, Marco R.

Uploaded 07-17-1997
Keywords multilevel models
random coefficients
contextual analysis
Abstract Although integrating multiple levels of data into an analysis can often yield better inferences about the phenomenon under study, traditional methodologies used to combine multiple levels of data are problematic. In this paper, we discuss several methodologies under the rubric of multilevel analysis. Multilevel methods, we argue, provide researchers, particularly researchers using comparative data, substantial leverage in overcoming the typical problems associated with either ignoring multiple levels of data, or problems associated with combining lower-level and higher-level data (including overcoming implicit assumptions of fixed and constant effects). The paper discusses several variants of the multilevel model and provides an application of individual-level support for European integration using comparative political data from Western Europe.

Multilevel (hierarchical) modeling: what it can and can't do
Gelman, Andrew

Uploaded 01-26-2005
Keywords Bayesian inference
hierarchical model
multilevel regression
Abstract Multilevel (hierarchical) modeling is a generalization of linear and generalized linear modeling in which regression coefficients are themselves given a model, whose parameters are also estimated from data. We illustrate the strengths and limitations of multilevel modeling through an example of the prediction of home radon levels in U.S. counties. The multilevel model is highly effective for predictions at both levels of the model but could easily be misinterpreted for causal inference.

Heterogeneity in Supreme Court Decision-Making: How Case-Level Factors Alter Preference-Based Behavior
Bartels, Brandon

Uploaded 07-19-2005
Keywords Supreme Court decision-making
multilevel modeling
Abstract Many theoretical perspectives of Supreme Court decision-making, most notably the attitudinal model, assume that justicesí policy preferences exhibit a uniform impact on their decisions across a wide variety of situations. I argue that there exists meaningful heterogeneity in the impact of policy preferences that can be explained theoretically and tested empirically. Adopting social psychological insights from theories of the attitude-behavior relationship, I develop a theoretical framework specifying the mechanisms--attitude strength and accountability--that explain variation in the preference-behavior relationship for justices. Case-level factors associated with each mechanism are hypothesized to moderate the impact of preferences. To test the hypotheses, I use a multilevel (hierarchical) modeling framework and conceive of Supreme Court voting data from the 1994-2002 terms as a two-level hierarchy: justicesí choices nested within cases. Estimates from a random coefficient model indicate that case-level variables associated with both attitude strength and accountability systematically explain variation in the preference-behavior relationship. Using an average partial effects post-estimation procedure, I present in-depth substantive interpretations of the results that highlight the compelling ways in which these case-level factors alter the nature of preference-based behavior. In addition to providing important substantive conclusions about Supreme Court decision-making, the paper also illustrates how a multilevel modeling framework is well-qualified to test heterogeneity-related hypotheses in social and behaviorial processes.

Struggles with survey weighting and regression modeling
Gelman, Andrew

Uploaded 10-12-2005
Keywords multilevel modeling
poststrati cation
sampling weights
Abstract The general principles of Bayesian data analysis imply that models for survey responses should be constructed conditional on all variables that affect the probability of inclusion and nonresponse, which are also the variables used in survey weighting and clustering. However, such models can quickly become very complicated, with potentially thousands of post-stratification cells. It is then a challenge to develop general families of multilevel probability models that yield reasonable Bayesian inferences. We discuss in the context of several ongoing public health and social surveys. This work is currently open-ended, and we conclude with thoughts on how research could proceed to solve these problems.

Rich state, poor state, red state, blue state:What's the matter with Connecticut?
Gelman, Andrew
Shor, Boris
Bafumi, Joseph
Park, David

Uploaded 11-29-2005
Keywords availability heuristic
ecological fallacy
hierarchical model
income and voting
multilevel model
presidential elections
public opinion
secret weapon
varying-slope model
Abstract We find that income matters more in ``red America'' than in ``blue America.'' In poor states, rich people are much more likely than poor people to vote for the Republican presidential candidate, but in rich states (such as Connecticut), income has a very low correlation with vote preference. In addition to finding this pattern and studying its changes over time, we use the concepts of typicality and availability from cognitive psychology to explain how these patterns can be commonly misunderstood. Our results can be viewed either as a debunking of the journalistic image of rich ``latte'' Democrats and poor ``Nascar'' Republicans, or as support for the journalistic images of political and cultural differences between red and blue states---differences which are not explained by differences in individuals' incomes. For decades, the Democrats have been viewed as the party of the poor, with the Republicans representing the rich. Recent presidential elections, however, have shown a reverse pattern, with Democrats performing well in the richer ``blue'' states in the northeast and west coast, and Republicans dominating in the ``red'' states in the middle of the country. Through multilevel modeling of individual-level survey data and county- and state-level demographic and electoral data, we reconcile these patterns. Key methods used in this research are: (1) plots of repeated cross-sectional analyses, (2) varying-intercept, varying-slope multilevel models, and (3) a graph that simultaneously shows within-group and between-group patterns in a multilevel model. These statistical tools help us understand patterns of variation within and between states in a way that would not be possible from classical regressions or by looking at tables of coefficient estimates.

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