An Introduction to Multilevel Modeling Techniques: MLM and SEM Approaches

405 Pages·2020·36.136 MB·English

by  Ronald H. Heck, Scott L. Thomas

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An Introduction to Multilevel Modeling

Techniques
Multilevel modeling is a data analysis method that is frequently used to inves-
tigate hierarchical data structures in educational, behavioral, health, and social
science disciplines. Multilevel data analysis exploits data structures that cannot
be adequately investigated using single-l evel analytic methods such as multiple
regression, path analysis, and structural modeling. This text offers a comprehen-
sive treatment of multilevel models for univariate and multivariate outcomes. It
explores their similarities and differences and demonstrates why one model may
be more appropriate than another, given the research objectives.
New to this edition:
• An expanded focus on the nature of different types of multilevel data
structures (e.g., cross- sectional, longitudinal, cross- classified, etc.) for
addressing specific research goals;
• Varied modeling methods for examining longitudinal data including
random- effect and fixed- effect approaches;
• Expanded coverage illustrating different model-b uilding sequences and how
to use results to identify possible model improvements;
• An expanded set of applied examples used throughout the text;
• Use of four different software packages (i.e., Mplus, R, SPSS, Stata), with
selected examples of model- building input files included in the chapter
appendices and a more complete set of files available online.
This is an ideal text for graduate courses on multilevel, longitudinal, latent
variable modeling, multivariate statistics, or advanced quantitative techniques
taught in psychology, business, education, health, and sociology. Recommended
prerequisites are introductory univariate and multivariate statistics.
Ronald H. Heck is Professor of Education at the University of Hawai’i at
Mānoa. His areas of interest include organizational theory, policy, and quanti-
tative research methods.
Scott L. Thomas is Professor and Dean of the College of Education and Social
Services, University of Vermont. His specialties include sociology of education,
policy, and quantitative research methods.
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Quantitative Methodology Series

George 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 minimum of mathematical derivations. Each volume focuses on a spe-
cific 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 volume’s 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,
[email protected].
Creemers/ Kyriakides/ Sammons • Methodological Advances in Educational
Effectiveness Research
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
Hox/ Moerbeek/ van de Schoot • Multilevel Analysis: Techniques and Applications,
Third Edition
Lamprianou • Applying the Rasch Model in Social Sciences Using R and BlueSky
Statistics
Heck/ Thomas • An Introduction to Multilevel Modeling Techniques: MLM and
SEM Approaches, Fourth Edition
iii
An Introduction to Multilevel

Modeling Techniques
MLM and SEM Approaches
Fourth Edition
Ronald H. Heck
Scott L. Thomas
iv
Fourth Edition published 2020
by Routledge
52 Vanderbilt Avenue, New York, NY 10017
and by Routledge
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2020 Taylor & Francis
The right of Ronald H. Heck and Scott L. Thomas to be identified as 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 utilised
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
Third Edition published by Routledge, March 2015
Reprinted IBM SPSS output (Figures 2.1– 2.2, 3.1– 3.8, 5.2– 5.3, 9.1– 9.3) 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-i n- Publication Data
A catalog record for this title has been requested
ISBN: 978- 0- 367- 18242- 7 (hbk)
ISBN: 978- 0- 367- 18244- 1 (pbk)
ISBN: 978- 0- 429- 06027- 4 (ebk)
Typeset in Times New Roman
by Newgen Publishing UK
Visit the eResource: www.routledge.com/ 9780429060274
v
Contents

Acknowledgments x
Preface xi
1 Introduction 1
Providing a Conceptual Overview 3
Contrasting Linear Models 7
Univariate Analysis 11
Multiple Regression 12
Analysis of Variance 12
Multivariate Analysis 13
Multivariate Analysis of Variance 13
Structural Equation Modeling 15
Multilevel Data Structures 16
Multilevel Multivariate Model 18
Multilevel Structural Model 19
Summary 21
References 23
2 Getting Started with Multilevel Analysis 26
Introduction 26
The Big Picture 27
From Single- level to Multilevel Analysis 28
Summarizing Some Differences between Single- level and Multilevel
Analyses 32
Developing a General Multilevel Modeling Strategy 35
Step 1: Partitioning the Variance in an Outcome 36
Step 2: Adding Level- 1 Predictors to Explain Variability in Intercepts 40
Step 3: Adding Level- 2 Predictors to Explain Variability in Intercepts 41
Step 4: Examining Possible Variation in Level- 1 Slopes 42
Step 5: Adding Predictors to Explain Variation in Slopes 44
Specifying the Basic Two- level Model with Path Diagrams 46
Model Estimation 48
Maximum Likelihood Estimation 49
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vi Contents
Model Convergence and Fit 51
Considerations for ML Estimation 52
An ML Illustration 54
Bayesian Estimation 54
Bayesian and ML Estimation with a Limited Number of Groups 56
Summary 59
References 61
3 Multilevel Regression Models 66
Introduction 66
Building a Model to Explain Employee Morale 67
Model 1: One- way ANOVA or Null Model 70
Model 2: Random- Intercept with Level- 1 Predictors 74
Model 3: Specifying a Level- 1 Random Slope 81
Model 4: Explaining Variation in the Level- 2 Intercept
and Slope 83
Model 4 Output 84
Examining Residuals 87
Centering Predictors 92
Centering Predictors in Models with Random Slopes 96
Summary 98
References 100
4 Extending the Two- level Regression Model 101
Introduction 101
Three- level Univariate Model 101
Developing a Three- level Univariate Model 103
Research Questions 103
Data 104
Model 1: Null (No Predictors) Model 105
Model 2: Defining Predictors at Level 1 106
Model 3: Defining Predictors at Level 2 107
Model 4: Examining an Interaction at Level 2 110
Model 5: Examining a Randomly Varying Slope at Level 3 111
Model 6: Adding Level- 3 Predictors 112
Accounting for Variance 114
Cross- classified Data Structures 114
Students Cross- classified in High Schools and Postsecondary
Institutions 117
Research Questions 118
Model 1: Developing a Null Model 118
Model 2: Adding Within- cell (Level-1) Predictors 121
Model 3: Adding Between- cell Predictors 122
Model 4: Adding a Randomly Varying Level- 1 Slope 123
Model 5: Explaining Variability in Random Slopes 124
Summary 126
References 130
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Contents vii
5 Methods for Examining Individual and Organizational Change 131
Introduction 132
Analyzing Longitudinal Data 132
Repeated Measures ANOVA 133
Growth Modeling and Other Approaches 133
Random- coefficients Growth Modeling 134
Defining the Level- 1 Model 135
Defining the Level- 2 Model 138
Extending the Model to Examine Changes Between Groups 139
Examining Changes in Students’ Math Scores 139
Model 1: Unconditional Means Model 140
Model 2: Unconditional Growth Model 141
Model 3: Unconditional Growth Model with Random Time Parameter 144
Investigating Subsets of Individuals’ Trajectories 144
Model 4: Adding a Quadratic Polynomial Term 144
Model 5: Adding a Between- subjects Predictor 148
Model 6: Deciding on the Level- 1 Covariance Structure 149
Building a Two- level Growth Model Using Age 152
Model 1: Unconditional Growth Model with Random Age Variable 153
Model 2: Final Growth Model with Between- subjects Predictor 154
Examining Changes in Institutions’ Graduation Rates 155
Model 1: Unconditional Means Model 157
Model 2: Unconditional Growth Model with Random Time Slope 157
Model 3: Adding a Time- Varying Covariate 159
Model 4: Examining Whether Instructional Support Influences Growth in
Graduation 160
Model 5: Testing a Random Effect for Instructional Support 161
Model 6: Adding Between- Institution Covariates 161
Developing Piecewise Growth Models 163
Examining Student Growth in Literacy 163
Changes Due to a Policy 165
Fixed- effects Regression Models 167
Graduation Growth in One Higher Education System 170
Summary 173
References 178
6 Multilevel Models with Categorical Variables 180
Introduction 180
Estimating the Models 182
Specifying Models for Binary, Ordinal, and Nominal Outcomes 185
Binary Outcome 185
Logit Link Function 186
Probit Link Function 189
Estimating the Intraclass Correlation 193
Ordinal Logit Outcome 193
Ordinal Probit Outcome 198
MPLUS Latent Response Formulation 200
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viii Contents
Unordered Categorical (Nominal) Outcome 201
Explaining Student Persistence 202
Binary Outcome 202
Null Model 203
Ordinal Outcome 205
Estimating Probabilities from Logit and Probit Coefficients 206
Dichotomous Outcome: Adding Level- 1 and Level- 2 Predictors 207
Ordinal Outcomes: Adding Level- 1 and Level- 2 Predictors 209
Examining a Cross- level Interaction with Continuous by Categorical
Predictors 211
Examining a Cross- level Interaction with Two Continuous Variables 213
Count Data 217
Building a Level- 1 and Level- 2 Model 220
Summary 223
References 226
7 Multilevel Structural Equation Models 228
Multilevel Models with Latent Variables 228
Multilevel Measurement Models 231
Multilevel Factor Variance Components 234
Types of Multilevel Factors 236
Model 1: Individual-level Factor 236
Model 2: Within- cluster Factor 236
Model 3: Shared Cluster- level Factor 237
Model 4: Configural Factor Structure 238
Model 5: Shared and Configural Factors 239
Estimating MCFA Models 239
Developing a Two- level Model 242
Model 1 and Model 2 Results 243
Model 3 and Model 4 Results 246
Extending the CFA Model to Three Levels 249
Model 5: Defining a Configural Factor Model at Levels 1, 2, and 3 251
Model 6: Restricting Errors to Zero at Level 2 252
Multilevel CFA with Ordinal Observed Indicators 255
Developing a CFA Model 256
Multilevel Models with Latent Variables and Covariates 261
Model 1: Specifying a Two- level Latent Factor Model with Covariates 263
Model 2: Specifying a Random Level- 1 Slope 268
Model 3: Adding a Latent Factor Between Groups 270
Model 4: Testing an Indirect (or Mediating) Effect 275
Summary 277
References 281
8 Multilevel Latent Growth and Mixture Models 285
Introduction 286
Latent Growth Models 286
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Contents ix
Intercept and Slope (IS) and Level and Shape (LS) Models 287
Defining the Latent Growth Model 288
Measurement Model 289
Structural Model 291
Multilevel Analysis of Growth 293
Examining Variables that Influence Student Growth in Science 294
Piecewise Latent Growth Model 298
Latent Variable Mixture Modeling 300
Defining Latent Profiles and Classes 302
An Example Latent Profile Analysis 306
Two- level Models 313
Examining Heterogeneity in Intercepts 313
Investigating Latent Classes for Random Slopes at Level 2 320
Alternative Model Specification 324
Growth Mixture Models 324
Examining Latent Classes in Students’ Growth in Science 326
Two- level GMM 330
Summary 335
References 339
9 Data Considerations in Examining Multilevel Models 343
Complex Samples, Design Effects, and Sample Weights 343
An Example Using Multilevel Weights 347
Parameter Bias and Statistical Power 350
Parameter Bias 350
Power 351
An Example 352
Anticipated Effect Size and Power 355
Mplus Monte Carlo Study 358
Design Complexity 362
Missing Data 363
Missing Data at Level 2 367
Imputed Data Sets 368
Concluding Thoughts 372
References 375
Index 380