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  • AN INTRODUCTION TO MULTILEVEL MODELING TECHNIQUES

    Univariate and multivariate multilevel models are used to understand how to design studies and analyze data in this comprehensive text distinguished by its variety of applications from the educational, behavioral, and social sciences. Basic and advanced models are developed from the multilevel regression (MLM) and latent variable (SEM) traditions within one uni-fied analytic framework for investigating hierarchical data. The authors provide examples using each modeling approach and also explore situations where alternative approaches may be more appropriate, given the research goals. Numerous examples and exercises allow readers to test their understanding of the techniques presented.

    Changes to the new edition include:

    The use of Mplus 7.2 for running the analyses including the input and data files at www.routledge.com/9781848725522.

    Expanded discussion of MLM and SEM model building that outlines the steps taken in the process, the relevant Mplus syntax, and tips on how to evaluate the models.

    Expanded pedagogical program now with chapter objectives, boldfaced key terms, a glossary, and more tables and graphs to help students better understand key concepts and techniques.

    Numerous, varied examples developed throughout, which make this book appropriate for use in education, psychology, business, sociology, and the health sciences.

    Expanded coverage of missing data problems in MLM using ML estimation and mul-tiple imputation to provide currently accepted solutions (Ch. 10).

    New chapter on three-level univariate and multilevel multivariate MLM models pro-vides greater options for investigating more complex theoretical relationships (Ch. 4).

    New chapter on MLM and SEM models with categorical outcomes facilitates the specification of multilevel models with observed and latent outcomes (Ch. 8).

    New chapter on multilevel and longitudinal mixture models provides readers with options for identifying emergent groups in hierarchical data (Ch. 9).

    New chapter on the utilization of sample weights, power analysis, and missing data provides guidance on technical issues of increasing concern for research publication (Ch. 10).

    Ideal as a text for graduate courses on multilevel, longitudinal, and latent variable modeling; multivariate statistics; or advanced quantitative techniques taught in psychology, business, education, health, and sociology, this books practical approach also appeals to researchers. Recommended prerequisites are introductory univariate and multivariate statistics.

    Ronald H. Heck is professor of education at the University of Hawaii at Mnoa. His areas of interest include organizational theory, policy, and quantitative research methods.

    Scott L. Thomas is professor and Dean of the School of Educational Studies at Cla-remont Graduate University. His specialties include sociology of education, policy, and quantitative research methods.

  • Quantitative Methodology SeriesGeorge A. Marcoulides, Series Editor

    This series presents methodological techniques to investigators and students. The goal is to provide an understanding and working knowledge of each method with a mini-mum of mathematical derivations. Each volume focuses on a specific method (e.g. Factor Analysis, Multilevel Analysis, Structural Equation Modeling).

    Proposals are invited from interested authors. Each proposal should consist of: a brief description of the volumes focus and intended market; a table of contents with an outline of each chapter; and a curriculum vita. Materials may be sent to Dr. George A. Marcoulides, University of California Santa Barbara, gmarcoulides@education.ucsb.edu.

    Marcoulides Modern Methods for Business Research

    Marcoulides/Moustaki Latent Variable and Latent Structure Models

    Heck Studying Educational and Social Policy: Theoretical Concepts and Research Methods

    Van der Ark/Croon/Sijtsma New Developments in Categorical Data Analysis for the Social and Behavioral Sciences

    Duncan/Duncan/Strycker An Introduction to Latent Variable Growth Curve Modeling: Concepts, Issues, and Applications, Second Edition

    Cardinet/Johnson/Pini Applying Generalizability Theory Using EduG

    Creemers/Kyriakides/Sammons Methodological Advances in Educa-tional Effectiveness Research

    Hox Multilevel Analysis: Techniques and Applications, Second Edition

    Heck/Thomas/Tabata Multilevel Modeling of Categorical Outcomes Using IBM SPSS

    Heck/Thomas/Tabata Multilevel and Longitudinal Modeling with IBM SPSS, Second Edition

    McArdle/Ritschard Contemporary Issues in Exploratory Data Mining in the Behavioral Sciences

    Heck/Thomas An Introduction to Multilevel Modeling Techniques: MLM and SEM Approaches Using Mplus, Third Edition

  • AN INTRODUCTION TO MULTILEVEL MODELING TECHNIQUES

    MLM and SEM Approaches Using Mplus

    Third Edition

    Ronald H. Heck Scott L. Thomas

  • Third edition published 2015 by Routledge 711 Third Avenue, New York, NY 10017

    and by Routledge 27 Church Road, Hove, East Sussex BN3 2FA

    Routledge is an imprint of the Taylor & Francis Group, an informa business

    2015 Taylor & Francis

    The right of Ronald H. Heck and Scott L. Thomas to be identified as the authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.

    All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.

    Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

    First Edition published by Taylor and Francis, November 1999 Second Edition published by Routledge, September 2008

    Reprinted IBM SPSS output (Figures 2.12.2, 3.1, 8.18.2, 10.110.3) is courtesy of International Business Machines Corporation, SPSS, Inc, an IBM Companya

    a. SPSS was acquired by IBM in October, 2009.

    Library of Congress Cataloging-in-Publication Data Heck, Ronald H. An introduction to multilevel modeling techniques : MLM and SEM approaches using Mplus / by Ronald H. Heck and Scott L. Thomas. Third edition. pages cm. (Quantitative methodology series) Includes bibliographical references and index. 1. Social sciencesMathematical models. 2. Social sciencesResearchMathematical models. I. Thomas, Scott Loring. II. Title. H61.25.H43 2015 001.422dc23 2014038512

    ISBN: 978-1-84872-551-5 (hbk) ISBN: 978-1-84872-552-2 (pbk) ISBN: 978-1-31574649-4 (ebk)

    Typeset in Bembo by Apex CoVantage, LLC

  • Preface xiii

    1 Introduction 1Chapter Objectives 1Introduction 1Providing a Conceptual Overview 2Analysis of Multilevel Data Structures 5Contrasting Linear Models 6Univariate Analysis 9Multiple Regression 10Analysis of Variance 10

    Multivariate Analysis 11Multivariate Analysis of Variance 11Structural Equation Modeling 13

    Multilevel Data Structures 15Multilevel Multivariate Model 17Multilevel Structural Model 18

    Summary 20References 21

    2 Getting Started With Multilevel Analysis 23Chapter Objectives 23Introduction 23From Single-Level to Multilevel Analysis 25Summarizing Some Differences 29

    Developing a General Multilevel Modeling Strategy 31

    CONTENTS

  • vi Contents

    Step 1: Partitioning the Variance in an Outcome 33Step 2: Adding Level-1 Predictors to Explain Intercept Variability 37Step 3: Specifying Level-2 Predictors to Explain Intercept Variability 38Step 4: Examining Possible Variation in Slopes 40Step 5: Adding Predictors to Explain Variation in Slopes 41

    Specifying Random Effects at Level 2 43Methods for Estimating Model Parameters 44Maximum Likelihood Estimation 45Full Information ML 48Model Convergence 51Considerations for ML Estimation 52Other Model Estimation Approaches in Mplus 54

    WLS Estimation 55Bayesian Estimation 56

    A Comparison of Estimation Approaches With Small Numbers of Level-2 Units 57Summary 60References 62

    3 Multilevel Regression Models 67Chapter Objectives 67Introduction 67Overview of Multilevel Regression Models 69Building a Model to Explain Employee Morale 70Model 1: One-Way ANOVA model 74

    Model 1 Statements 75Model 1 Output 77

    Model 2: Level-1 Random-Intercept Model 79Model 2 Statements 81Model 2 Output 82

    Model 3: Specifying a Level-1 Random Slope 83Model 3 Statements 83Model 3 Output 84

    Model 4: Explaining Variation in the Level-2 Intercept and Slope 85Model 4 Statements 85Model 4 Output 86

    Centering Predictors 87Centering Predictors in Models With Random Slopes 91

    Summary 93References 94

    4 Extending the Two-Level Regression Model 97Chapter Objectives 97Introduction 97

  • Contents vii

    Three-Level Univariate Model 98Developing a Three-Level Univariate Model 99Research Questions 100Data 100Model 1: Null (No Predictors) Model 101

    Model 1 Statements 101Model 1 Output 102

    Model 2: Defining Predictors at Each Level 103Grand-Mean Centering 103Model 2 Statements 105Model 2: Grand-Mean Centered Output 105Group-Mean Centering 107Model 2 Statements 107Model 2: Group-Mean Centered Output 108

    Model 3: Does the Slope Vary Randomly Across Schools? 109Model 3 Statements 110Model 3 Output 111

    Model 4: Developing a Model to Explain Variability in Slopes 111Model 4 Statements 112Model 4 Output 112

    Defining Path Models 113Single-Level Path Model 114Multilevel Path Model 115Model 1: Two-Level Model With Multivariate Outcomes 1

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