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# Multilevel Analysis : An Introduction to Basic and Advanced Multilevel Modeling

**by**Tom A B Snijders

2nd

### 9781849202015

184920201X

Paperback

12/6/2011

SAGE Publications Ltd

## Questions About This Book?

What version or edition is this?

This is the 2nd edition with a publication date of 12/6/2011.

What is included with this book?

- The
**New**copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any CDs, lab manuals, study guides, etc. - The
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## Summary

The Second Edition of this classic text outlines 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.

## Table of Contents

Preface second edition | |

Preface to first edition | |

Introduction | |

Multilevel analysis | |

Probability models | |

This book | |

Prerequisites | |

Notation | |

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 | |

Glommary | |

Statistical Treatment of Clustered Data | |

Aggregation | |

Disaggregation | |

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 | |

Regressions | |

Correlations | |

Estimation of within-and between-group correlations | |

Combination of within-group evidence | |

Glommary | |

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 | |

Glommary | |

The Hierarchical Linear Model | |

Random slopes | |

Heteroscedasticity | |

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? | |

Estimation | |

Three or more levels | |

Glommary | |

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 | |

Glommary | |

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 | |

Glommary | |

Heteroscedasticity | |

Heteroscedasticity at level one | |

Linear variance functions | |

Quadratic variance functions | |

Heteroscedasticity at level two | |

Glommary | |

Missing Data | |

General issues for missing data | |

Implications for design | |

Missing values of the dependent variable | |

Full maximum likelihood | |

Imputation | |

The imputation method | |

Putting together the multiple results | |

Multiple imputations by chained equations | |

Choice of the imputation model | |

Glommary | |

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 | |

Glommary | |

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 | |

Glommary | |

Other Methods and Models | |

Bayesian inference | |

Sandwich estimators for standard errors | |

Latent class models | |

Glommary | |

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 | |

Glommary | |

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 | |

Glommary | |

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 | |

Glommary | |

Multivariate Multilevel Models | |

Why analyze multiple dependent variables simultaneously? | |

The multivariate random intercept model | |

Multivariate random slope models | |

Glommary | |

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 | |

Estimation | |

Aggregation | |

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 | |

Glommary | |

Software | |

Special software for multilevel modeling | |

HLM | |

MLwiN | |

The MIXOR suite and SuperMix | |

Modules in general-purpose software packages | |

SAS procedures VARCOMP, MIXED, GLIMMIX, and NLMIXED | |

R | |

Stata | |

SPSS, commands VARCOMP and MIXED | |

Other multilevel software | |

PinT | |

Optimal Design | |

MLPowSim | |

Mplus | |

Latent Gold | |

REALCOM | |

WinBUGS | |

References | |

Index | |

Table of Contents provided by Publisher. All Rights Reserved. |