• Introduction
• General Overviews
• Classic Works
• Textbooks
• Edited Volumes
• Statistical Software
• Longitudinal Models
• Three-Level and Cross-Classified Models
• Hierarchical Generalized Linear Models (HGLMs)
• Centering
• Meta-Analysis
• Cross-National Applications
• Labor Market Applications
• Education Applications

# Multilevel ModelsbyAndrew S. FullertonLAST REVIEWED: 05 September 2014LAST MODIFIED: 21 November 2012DOI: 10.1093/obo/9780199756384-0088

## Introduction

Multilevel models are a set of statistical techniques for analyzing quantitative data measured at two or more levels of analysis. In the basic two-level linear model, level 1 units (e.g., students) are nested within level 2 units (e.g., schools) and variables are included in the model from both levels of analysis. Although one can specify separate equations for each level, they are linked through a random intercept and random slopes. Prior to the development of multilevel models, scholars used methods such as Ordinary Least Squares (OLS) regression to examine data from multiple levels of analysis. However, the clustering of micro-level units within meso- and macro-level units violates several key assumptions of the OLS regression model. This led statisticians to develop a set of models that explicitly incorporates the multilevel nature of the data into the statistical models. Although the earliest formulations and applications of the multilevel model focused on two levels of analysis and a continuous outcome, the basic model has been extended in recent years to include three or more levels of analysis and a wide variety of different outcome types. The prevalence of categorical variables in quantitative sociological research makes hierarchical generalized linear models essential for multilevel scholarship in sociology. Longitudinal data analysis represents another major application of multilevel modeling dating back to some of the earliest multilevel research on the effects of schools on student achievement over time. The alternative “fixed effects” approach to longitudinal analysis captures (time-invariant) between-subject variation with a set of nuisance parameters. In other words, fixed effects models control for between-subject variation but do not allow one to model this variation based on one or more variables. However, researchers may use multilevel models to predict variation within and between subjects. Multilevel models are an extension of the “random effects” approach to longitudinal analysis, which allows one to predict between-subject variation based on subject-level characteristics. The intercept is the only random coefficient in a random effects model, but multilevel models for longitudinal data may have random slopes as well. There is now a much broader range of multilevel models available for the analysis of longitudinal data. This article reviews the literature on multilevel models from the classic works to recent methodological advances and applications in an attempt to trace the development of multilevel models since the 1970s and provide an overview of the potential of this diverse set of methods for analyzing multilevel data structures.

## General Overviews

Overview articles on multilevel models are an important complement to textbooks and edited collections. There are review articles on multilevel modeling in several social science disciplines that serve as essential references for students and researchers in these fields. Raudenbush 1988 reviewed early applications in educational research and helped popularize the term hierarchical linear model (HLM). Blalock 1984 reviewed models for data at multiple levels of analysis, referred to as “contextual models,” before the hierarchical linear model was introduced and gained popularity in the late 1980s and early 1990s. DiPrete and Forristal 1994 builds on Blalock’s early work in its review of multilevel models and applications in sociology. Sociologists should also find review articles from related disciplines very useful, including the work of Duncan, et al. 1998 and Goldstein 2002 in health and medicine and Hofmann 1997, a work in organizational studies. More recent overviews focus either exclusively on multilevel models for categorical outcomes or devote a substantial portion of the review to these models. For example, Guo and Zhao 2000 provides a very comprehensive review of multilevel models for binary outcomes, and Fullerton, et al. 2010 illustrates the use of multilevel modeling with an example based on a binary outcome (opposition to welfare spending). In addition to providing concise introductions to multilevel modeling, these overviews also point readers to recent applications of multilevel models in several different disciplines and substantive areas.

• Blalock, Hubert M. 1984. Contextual-effects models: Theoretical and methodological issues. Annual Review of Sociology 10:353–372.

A review article on contextual effects models before the introduction of the hierarchical linear model for studying multilevel data. Covers several features of multilevel models, such as cross-nesting and slope variation, that have been incorporated in the recent advances in multilevel modeling (e.g., random coefficient models and cross-nested random effects).

• DiPrete, Thomas A., and Jerry D. Forristal. 1994. Multilevel models: Methods and substance. Annual Review of Sociology 20:331–357.

One of the earliest and most influential annual review articles in sociology on multilevel models. Presents the basic setup for the two-level model. Considers extensions of the basic model, including three-level, cross-classified, and nonlinear models. Authors also review substantive applications.

• Duncan, Craig, Kelvyn Jones, and Graham Moon. 1998. Context, composition, and heterogeneity: Using multilevel models in health research. Social Science &amp; Medicine 46:97–117.

Highly cited review of multilevel modeling in health research. Covers the basic two-level model, cross-level interactions, and other types of multilevel models, including three-level, repeated cross-section, longitudinal, multivariate, and cross-classified models.

• Fullerton, Andrew S., Michael Wallace, and Michael J. Stern. 2010. Multilevel models. In Handbook of politics: State and society in global perspective. Edited by Kevin T. Leicht and J. Craig Jenkins, 589–604. New York: Springer.

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Overview of multilevel models and their applications in political sociology. Reviews the logic of multilevel modeling, basic characteristics, and examples in political research. Focus is on two-level linear and binary models with an example on opposition to welfare spending.

• Goldstein, Harvey, William Browne, and Jon Rasbash. 2002. Multilevel modelling of medical data. Statistics in Medicine 21:3291–3315.

Reviews the logic and method of multilevel modeling and applications in medical research. Presents the basic two-level linear model and several extensions, including growth models, crossed random effects, multiple memberships, and Bayesian estimation. Includes examples using the statistical software package MLwiN.

• Guo, Guang, and Hongxin Zhao. 2000. Multilevel modeling for binary data. Annual Review of Sociology 26:441–462.

One of the first annual review articles focused on multilevel models for binary outcomes. Notes the relatively more complex estimation required in binary models compared to linear multilevel models. Compares several different estimation procedures and their relative advantages, disadvantages, and potential biases.

• Hofmann, David A. 1997. An overview of the logic and rationale of hierarchical linear models. Journal of Management 23:723–744.

Overview of multilevel modeling with a focus on applications in organizational research. Focuses on the logic of multilevel models, basic estimation issues, extensions of the two-level linear model, and other important considerations such as centering, assumptions, and sample size requirements.

• Raudenbush, Stephen W. 1988. Educational applications of hierarchical linear models: A review. Journal of Educational Statistics 13:85–116.

An early review of multilevel modeling in educational research. One of the first uses of the term hierarchical linear model (HLM). Reviews the history and development of multilevel models, estimation theory, properties of HLM estimators, and educational applications.