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Multilevel Analysis

Multilevel Analysis
An Introduction to Basic and Advanced Multilevel Modeling

Second Edition
  • Tom A B Snijders - University of Groningen, Netherlands, University of Groningen, University of Oxford, UK; University of Groningen, The Netherlands
  • Roel J Bosker - University of Groningen, Netherlands

December 2011 | 368 pages | SAGE Publications Ltd

The Second Edition of this classic text introduces the main methods, techniques, and issues involved in carrying out multilevel modeling and analysis.

Snijders and Boskers' book is an applied, authoritative, and accessible introduction to the topic, providing readers with a clear conceptual and practical understanding of all the main issues involved in designing multilevel studies and conducting multilevel analysis.

This book provides step-by-step coverage of:

  • Multilevel theories
  • Multi-stage sampling
  • The hierarchical linear model
  • Testing and model specification
  • Heteroscedasticity
  • Study designs
  • Longitudinal data
  • Multivariate multilevel models
  • Discrete dependent variables

There are also new chapters on:

  • Missing data
  • Multilevel Modeling for Surveys
  • Bayesian and MCMC estimation and latent-class models.

This book has been comprehensively revised and updated since the last edition, and now includes guides to modeling using HLM, MLwiN, SAS, Stata including GLLAMM, R, SPSS, Mplus, WinBugs, Latent Gold, and Mix.

This is a must-have text for any student, teacher, or researcher with an interest in conducting or understanding multilevel analysis.

Preface second edition
Preface to first edition
Multilevel analysis
Probability models

This book


Multilevel Theories, Multi-Stage Sampling and Multilevel Models
Dependence as a nuisance
Dependence as an interesting phenomenon
Macro-level, micro-level, and cross-level relations
Statistical Treatment of Clustered Data
The intraclass correlation
Within-group and between group variance

Testing for group differences

Design effects in two-stage samples
Reliability of aggregated variables
Within-and between group relations


Estimation of within-and between-group correlations

Combination of within-group evidence
The Random Intercept Model
Terminology and notation
A regression model: fixed effects only
Variable intercepts: fixed or random parameters?
When to use random coefficient models

Definition of the random intercept model
More explanatory variables
Within-and between-group regressions
Parameter estimation
'Estimating' random group effects: posterior means
Posterior confidence intervals

Three-level random intercept models
The Hierarchical Linear Model
Random slopes

Do not force ?01 to be 0!

Interpretation of random slope variances

Explanation of random intercepts and slopes
Cross-level interaction effects

A general formulation of fixed and random parts

Specification of random slope models
Centering variables with random slopes?

Three or more levels
Testing and Model Specification
Tests for fixed parameters
Multiparameter tests for fixed effects

Deviance tests
More powerful tests for variance parameters

Other tests for parameters in the random part
Confidence intervals for parameters in the random part

Model specification
Working upward from level one

Joint consideration of level-one and level-two variables

Concluding remarks on model specification

How Much Does the Model Explain?
Explained variance
Negative values of R2?

Definition of the proportion of explained variance in two-level models

Explained variance in three-level models

Explained variance in models with random slopes

Components of variance
Random intercept models

Random slope models

Heteroscedasticity at level one
Linear variance functions

Quadratic variance functions

Heteroscedasticity at level two
Missing Data
General issues for missing data
Implications for design

Missing values of the dependent variable
Full maximum likelihood
The imputation method

Putting together the multiple results

Multiple imputations by chained equations
Choice of the imputation model
Assumptions of the Hierarchical Linear Model
Assumptions of the hierarchical linear model
Following the logic of the hierarchical linear model
Include contextual effects

Check whether variables have random effects

Explained variance

Specification of the fixed part
Specification of the random part
Testing for heteroscedasticity

What to do in case of heteroscedasticity

Inspection of level-one residuals
Residuals at level two
Influence of level-two units
More general distributional assumptions
Designing Multilevel Studies
Some introductory notes on power
Estimating a population mean
Measurement of subjects
Estimating association between variables
Cross-level interaction effects

Allocating treatment to groups or individuals
Exploring the variance structure
The intraclass correlation

Variance parameters

Other Methods and Models
Bayesian inference
Sandwich estimators for standard errors
Latent class models
Imperfect Hierarchies
A two-level model with a crossed random factor
Crossed random effects in three-level models
Multiple membership models
Multiple membership multiple classification models
Survey Weights
Model-based and design-based inference
Descriptive and analytic use of surveys

Two kinds of weights
Choosing between model-based and design-based analysis
Inclusion probabilities and two-level weights

Exploring the informativeness of the sampling design

Example: Metacognitive strategies as measured in the PISA study
Sampling design

Model-based analysis of data divided into parts

Inclusion of weights in the model

How to assign weights in multilevel models
Appendix. Matrix expressions for the single-level estimators
Longitudinal Data
Fixed occasions
The compound symmetry models

Random slopes

The fully multivariate model

Multivariate regression analysis

Explained variance

Variable occasion designs
Populations of curves

Random functions

Explaining the functions 27415.2.4

Changing covariates

Autocorrelated residuals

Multivariate Multilevel Models
Why analyze multiple dependent variables simultaneously?
The multivariate random intercept model
Multivariate random slope models
Discrete Dependent Variables
Hierarchical generalized linear models
Introduction to multilevel logistic regression
Heterogeneous proportions

The logit function: Log-odds

The empty model

The random intercept model



Further topics on multilevel logistic regression
Random slope model

Representation as a threshold model

Residual intraclass correlation coefficient

Explained variance

Consequences of adding effects to the model

Ordered categorical variables
Multilevel event history analysis
Multilevel Poisson regression
Special software for multilevel modeling


The MIXOR suite and SuperMix

Modules in general-purpose software packages



SPSS, commands VARCOMP and MIXED

Other multilevel software

Optimal Design



Latent Gold




'Overall, Snijders and Bosker provide an accessible and readable text on the subject of multilevel analysis. It is as much tailored to the needs of advanced quantitative researchers, as to relative beginners in the field. As a reader with limited experience in quantitative research, but an interest in advancing my knowledge and understanding of multilevel analysis, I found this text an ideal introduction to the area' -
Emma Smith
Methodspace Book Reviews Club

This new edition is still intended for students, teachers, or researchers with an interest in conducting or understanding multilevel analysis.  The book is definitely an excellent reference book for researchers working in social sciences, education, environmental sciences, and economic, biological, medical, and health disciplines.

Yuehua Wu
York University, Toronto
Zentralblatt Math

I found the book useful for researchers who want to know more about multi level statistics.

Dr Muhammet Mustafa Alpaslan
Education, Mugla Sitki Kocman Universitesi
July 10, 2017

This is one the most comprehensive books about multilevel analysis. It's like a "must have" book for those who want to have a deep understanding of different aspects of multilevel analysis.

Dr Amin Mousavi
Educational Psychology , University Of Saskatchewan
September 7, 2016

Good book on multilevel modeling

Professor Jost Sieweke
Business Administration , Heinrich Heine University
January 18, 2016

A must for scholars engaged in multilevel research. High on analytical treatment.

Professor Israr Qureshi
Business Administration , IE Business School
November 29, 2015

The content of the textbook is great. However, the text is very difficult to follow when students do not have a very strong background on statistics.

Dr Bruno Schivinski
Nottingham Business School, Nottingham Trent University
December 1, 2015

This is a thorough and approachable treatment of the topic, and the associated web resources are great.

Professor Corey Sparks
Demography, UTSA
June 29, 2015

Very specialiced book. Only recommended for details.

Professor Bernd Kaltenhaeuser
Technical Management, Duale Hochschule BW VS
March 31, 2015

This is the book!, however, the technical level is not appropriate to be adopted as a course material in my particular module.

Dr Guillermo Perez Algorta
Division of Health and Research, Lancaster University
January 16, 2015

Sample Materials & Chapters

Chapter Two