Longitudinal Structural Equation Modeling

423 Pages·2015·1.735 MB·english

by  Jason T. Newsom

Longitudinal
Structural Equation Modeling
A Comprehensive Introduction
Jason T. Newsom
First 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
Library of Congress Cataloging in Publication data
Newsom, Jason T.
Longitudinal structural equation modeling : a comprehensive introduction /
Jason T. Newsom.
pages cm. – (Multivariate applications series)
1. Longitudinal method–Mathematical models. 2. Social sciences–Statistical
methods. 3. Structural equation modeling. I. Title.
H61.25.N49 2015
519.5′3–dc23
2014046412
ISBN: 978-1-84872-696-3 (hbk)
ISBN: 978-1-84872-697-0 (pbk)
ISBN: 978-1-315-87131-8 (ebk)
Typeset in Sabon
by Out of House Publishing
Contents
List of Figures xii
List of Tables xv
Preface xvii
Acknowledgments xx
Example Data Sets xxi
About the Author xxiii
1 Review of some Key Latent Variable Principles 1
2 Longitudinal Measurement Invariance 27
3 Structural Models for Comparing Dependent Means and Proportions 53
4 Fundamental Concepts of Stability and Change 91
5 Cross-Lagged Panel Models 122
6 Latent State-Trait Models 152
7 Linear Latent Growth Curve Models 171
8 Nonlinear Latent Growth Curve Models 216
9 Latent Difference Score Models 248
10 Latent Transition and Growth Mixture Models 264
11 Time Series Models 292
12 Survival Analysis Models 322
13 Missing Data and Attrition 344
Appendix A: Notation 383
Appendix B: A Primer on the Calculus of Change 390
Glossary 394
Index 402
Figures
1.1 Single Factor Measurement Model, no Mean Structure 3
1.2 Four-Indicator Measurement Model with Mean Structure 6
1.3 The Relation between the Unobserved Distribution of y* and the
Observed Values of y 13
2.1 Graphic Depiction of Meredith’s Factorial Invariance Definitions 32
2.2 Two-Wave Four-Indicator Measurement Model 36
3.1 Four Approaches to Testing for the Difference between Two
Observed Means 55
3.2 Three Approaches to Testing for the Difference Between Three
Observed Means 66
3.3 Three Approaches to Testing for the Difference Between
Two Latent Means 73
3.4 Four Examples of the MIMIC Approach to the Mixed
Factorial ANOVA 82
4.1 Autoregression with Observed Variables 92
4.2 Autoregression with Latent Variables 104
4.3 Specific Factors Model 105
4.4 Cross-Lagged Effect with Observed Variables 107
4.5 Difference Score Prediction or “Unconditional” Model of Change 109
4.6 Regression Toward the Mean for Two Chosen Extreme Groups 111
4.7 Cross-Lagged Effect with Latent Variables 113
5.1 Cross-Lagged Panel Model with Two Observed Variables 123
5.2 Cross-Lagged Panel Model with Two Latent Variables 126
5.3 Cross-Lagged Panel Model with Time-Invariant Covariate 129
5.4 Synchronous Covariate at One Time Point 131
5.5 Synchronous Covariate at Two Time Points 131
5.6 Latent Fixed Effects Model 133
5.7 Perfect Simplex Model 134
5.8 Quasi-Simplex Model 135
5.9 Cross-Lagged Panel Model with Three Time Points 138
5.10 Cross-Lagged Panel Model with Static Unmeasured “Phantom” Variable 140
5.11 Synchronous Common Factor Model 141
5.12 Unmeasured Common and Synchronous Factor Hybrid Model 141
5.13 Three-Variable Cross-Lagged Panel Model 142
5.14 Longitudinal Mediation Model with Two Waves 143
5.15 Longitudinal Mediation Model with Three Waves 144
6.1 Latent State-Trait Model 154
6.2 Latent State-Trait Model with Method Factors 155
6.3 Trait-State-Error Model 158
6.4 Trait-State-Occasion Model with Multiple Indicators 161
6.5 Multistate-Multitrait Model (with Single Trait) 163
7.1 Hypothetical Linear Growth for a Single Case 172
7.2 Hypothetical Linear Growth for Several Cases 173
7.3 Latent Growth Curve Model 174
7.4 Four Possible Hypothetical Relationships between Intercept and Slope
for a Growth Model of Perceived Economic Security 176
7.5 Sample of 20 Predicted Growth Curves for Change in BMI Over 10 Years 184
7.6 Latent Growth Curve Model with Time-Invariant Covariate 188
7.7 Latent Growth Curve Model with Time-Varying Covariate 193
7.8 Latent Growth Curve Model with Individually Varying Time Scores 199
7.9 Example of Trajectories with Varying Start and End Times: Revenue for
Cities Implementing Corporate Tax Breaks During Different Years 200
7.10 Second-Order Latent Growth Curve Model 202
8.1 Linear Piecewise Model 217
8.2 Plots Illustrating Piecewise Slope Patterns 218
8.3 Results from Piecewise Models of BMI 220
8.4 Quadratic Latent Growth Curve Model 222
8.5 Quadratic Effects with Varying Coefficient Values 222
8.6 Plot of Predicted Scores from the Quadratic Model of BMI 226
8.7 Cubic Growth Curve Model 230
8.8 Several Possible Cubic Trends 230
8.9 Plot of the Logistic Latent Growth Curve Results 234
8.10 Damped Oscillator Model 237
8.11 Example of an Oscillating Curve 238
8.12 Latent Basis Model 239
8.13 Plot of Average Line from the Latent Basis Model of BMI 241
9.1 Latent Difference Score Model for Two Waves 249
9.2 Latent Difference Score Model for Six Waves 251
9.3 Latent Difference Score Model with Added Intercept and Slope Factors 254
9.4 Dual Change Score Model 256
9.5 Latent Difference Score Model with Time-Invariant and Time-Varying
Covariates 257
9.6 Simultaneous Latent Difference Score Model 259
9.7 Latent Difference Score Model with Multiple Indicators 260
10.1 Latent Class Model with Binary Indicators 266
10.2 Latent Class Model with Continuous Indicators (Use Four Indicators,
Show Meas Resids) 269
10.3 Latent Class Variable Prediction of a Continuous Latent Variable;
a Single-Indicator Latent Variable; and an Observed Variable 271
10.4 Simple Latent Transition Model with Two Time Points 276
10.5 Latent (Hidden) Markov Chain Model 278
10.6 Latent Transition Model 280
10.7 Latent Transition Model with Covariates 282
10.8 Growth Mixture Model 285
11.1 Plot of Positive Affect Means from the Diabetes Diary Data 295
11.2 Autocorrelation Function (Correlogram) and Spectral Density
Function (sdf) 299
11.3 Autoregressive (Simplex) Model 300
11.4 First-Order Autoregressive Model with Random Shock Factors, AR(1) 301
11.5 Second-Order Autoregression Model, AR(2) 302
11.6 Conditional Positive Affect Means (Intercepts) for First-Order Stationary
Autoregressive Model 303
11.7 Pure First-Order Moving Average Model with Estimated Means, MA(1) 305
11.8 Positive Affect Means from First-Order Moving Average Model 306
11.9 Combined First-Order Autoregressive and First-Order Moving Average
Model, ARMA(1,1) 307
11.10 Means from the Diabetes ARMA(1,1) Model 308
11.11 First-Order Differencing Model 309
11.12 Second-Order Differencing Model 310
11.13 Positive Affect Means from the First-Order Differencing Model 311
11.14 Combined First-Order Autoregressive, First-Order Differencing, and
First-Order Moving Average Model, ARIMA(1,1,1) 311
11.15 Plot of the Exogenous Means for the ARIMA(1,1,1) Model 316
11.16 Plot of Exogenous Means From the Multiple Indicator ARMA(1,1) Model 318
12.1 Plots of Hazard and Survivor Function for Diabetes in the Health and
Aging Data Set 326
12.2 Specifications of the Unconditional Discrete Time Survival Model
without Latent Classes or Continuous Factor and with a Single
Continuous Factor 329
12.3 Discrete Time Survival Models with Covariates 330
12.4 Proportional Hazard Constraint Using a Single Latent Variable 331
12.5 Inclusion of Latent Class Factors for Modeling Unobserved Heterogeneity 332
12.6 Diabetes Hazard and Survivor Functions Across Time 335
12.7 Structural Equation Model with Cox Regression 339
13.1 Analogue Representation of Missing Data Mechanisms 346
13.2 Inclusion of Auxiliary Variables 357
13.3 Selection Model for a Lagged Regression 361
13.4 Selection Model Specifications for a Latent Growth Curve Model 362
13.5 Pattern Mixture Model Specifications for a Latent Growth Curve 365
13.6 Example of a Shared Parameter Model with a Latent Class Factor 372
13.7 Summary Diagram of MNAR Modeling Concepts 373
13.8 Simple Representation of a Symmetric Pattern Model 374
13.9 Symmetric Parameter Simplex Model 375
13.10 Symmetric Parameter Latent Growth Curve Model 375
A.1 All-y LISREL Notation 386
A.2 Full LISREL Notation 388
B.1 Quadratic Curve with Tangent 392
Tables
1.1 Computational Examples Illustrating the Algebraic Equivalences
of Referent and Factor Identification Solutions (Social Exchanges Data Set) 5
1.2 Computation of Marginal Proportions from Threshold Estimates
with Binary Indicators, P(y = 1) = .923 21
1
2.1 Simulation Results Demonstrating Effects of Noninvariance on Latent
Means and Variances 46
3.1 Means for the Equally Weighted Composite Measure of Positive Affect 85
4.1 Repeated Measures Results for Health Event 99
8.1 Orthogonal Polynomials for T = 3 Through T = 7 225
8.2 Latent Quadratic Growth Curve Results for BMI using four Coding
Methods 227
10.1 Within-Class Item Means for the Positive Affect Scale 274
10.2 Transition Probabilities for the Two-Wave Latent Transition Model 277
11.1 Stationarity Tests for Autocovariances of lags 1, 2, and 3 (N = 129)
for the Positive Affect Diabetes Data 296
11.2 Time Series Estimates for Consumer Confidence Index Data
(T = 164, N = 1) 314
12.1 Observed Hazard and Survival Proportions for a Major Health
Diagnosis in the Social Exchanges Data Set 325
12.2 Event Indicator Coding for a Diabetes Diagnosis in a Sample of
20 Respondents 334
13.1 BMI Scores for a Sample of 20 Respondents in the Health and
Aging Data Set 352
13.2 Missing Data Patterns for BMI in the Health and Aging Data Set 352
13.3 Results from Three Specifications of Pattern Mixture Growth
Curve Models of BMI 370
A.1 All-y LISREL Notation 384
A.2 Full LISREL Notation 387
Preface
This book is a thorough examination of structural equation modeling (SEM) strategies for
longitudinal data. SEM is an analysis approach that combines path analysis and confirma-
tory factor analysis, and its ability to model complex hypotheses and account for meas-
urement error creates a truly remarkable, flexible method for analyzing data collected over
time. The book is intended for advanced students or researchers in behavioral and social
sciences and other related fields. Throughout the book readers will find examples relevant
for students and researchers in psychology, gerontology, public health, sociology, educa-
tion, social work, and economics. It is appropriate as a text for a second course in SEM
or as a reference for researchers conducting longitudinal research. Familiarity is assumed
with basic concepts of SEM, such as path analysis confirmatory factor analysis, model
identification and fit, and general structural models.
Reasons for Writing this Book
At the time this book was conceived there were no books devoted exclusively to a structural
modeling approach to longitudinal data. Although there were several books on growth
curve analysis, there had been nothing that documented how SEM might be applied to a
more complete range of questions about change over time. Introductory texts on SEM do
not often deal with the longitudinal case, understandably, and there are quite a number of
books on longitudinal analysis that use other statistical methods.
Since first being introduced to SEM in graduate school, I have been conducting longi-
tudinal research and teaching statistics and structural equation modeling courses for more
than 20 years. For the last several years, I have been teaching summer statistics workshops,
alternating between courses on SEM, multilevel regression, and a more general course on
longitudinal data analysis. In teaching the latter course I have included a compendium of
analytic approaches, and this course especially made me realize that analysts are faced with
an intimidating set of choices when analyzing longitudinal data. Even for the analysis of
just two time points, there are an astounding number of possible analyses – repeated meas-
ures ANOVA, MANOVA, ANCOVA, regression, difference scores, McNemar’s chi-square,
marginal homogeneity tests, conditional logistic, and loglinear models. When considering
just regression analysis, one has to decide whether to use a lagged model, controlling for
an early time point, or try to predict difference scores. During most of my career, I have
been reading about and making use of many impressive developments in SEM for ana-
lyzing longitudinal data, including time series analysis, survival analysis, latent state-trait
models, nonlinear growth curve models, and latent difference score models. In doing so,
I thought “Wouldn’t it be great to try to bring all of those longitudinal models together
under one roof to provide a resource for researchers who want to address any number of
longitudinal hypotheses with SEM?” Well, after more than a year of very intense work,
here it is: Longitudinal Structural Equation Modeling: A Comprehensive Approach.
In writing this book, I have learned an amazing amount about how the very large fam-
ily of longitudinal analyses strategies can fit together under one roof. My hope is that read-
ers will too. I hope that in reading this book you will not only see which modeling options
are available for which particular hypotheses but will also develop a deeper understand-
ing of structural modeling more generally. The early contributions of Jöreskog, Keesling,
and Wiley and others put decades of work on path analysis and factor analysis into one
system. In its current form, SEM is an even broader unifying system that encompasses an
enormous family of statistical analyses, involving discrete variables, latent class factors,
and aspects of multilevel analyses. Like other entities, biological organisms or sophisti-
cated electronics, SEM’s basic building blocks, simple and multiple regressions, are fully
reducible, but, in my opinion, they come together to form a statistical modeling system
that has some emergent properties. It is a powerful system for analyzing many kinds of
hypotheses, and this book is an attempt to paint a more complete picture of how this sys-
tem can be applied to analysis of longitudinal data.
Special Features
There are several special features in this book. Binary and ordinal variables are discussed
throughout, a topic that is far too rare in other SEM texts. For those without much back-
ground on structural models with these types of variables, the first chapter includes a
review of the most important concepts. Comment sections are included frequently within
chapters to give general overviews, evaluation of the topic from a balanced perspective,
and some practical guidance. So that readers can acquire further knowledge or gain other
views and instructors can supplement the text, an annotated list of recommended readings
is given at the end of each chapter. New terms are highlighted in the text and definitions
are summarized in the Glossary in the back of the book to facilitate use as a course text.
Within every chapter, there is extensive focus on how to apply and interpret each type
of model. Every major modeling approach introduced within a chapter is also illustrated
with one or more realistic data sets. The data sets used for the analyses are derived from
four longitudinal studies that illustrate a variety of measures and study designs. Measures
consisting of continuous, binary, and ordinal variable types assess constructs related to
social relationships, psychological factors, physical health, and consumer confidence. Study
designs capture change across various time frames – three semi-annual waves, six biennial
waves, twenty-four daily diary assessments, and a single-case study with 164 monthly
economic index values. More details about the data sets are included in the following sec-
tion titled “Example Data Sets” An accompanying website at www.longitudinalsem.com
includes all data sets used in the book and syntax for Mplus and lavaan, the R package,
software programs for every example. This book is not about how to use a particular soft-
ware program, however. There are just too many SEM programs that researchers use these
days to tie explanations to just one program.
Many efforts have been made for the material to be detailed and thorough while
remaining highly accessible. The book is intended for behavioral and social scientists
without any extensive math background, not for statisticians (though I hope some more
statistically minded readers will also find it valuable). Perhaps most importantly, model
specifications are conveyed through extensive illustration of each major approach.
The figures also include the notation for parameters in the model that aid connections
between the model specification and the equations. I believe it is essential to present
the equations associated with the fundamental concepts in order to engender a solid
understanding of each model. This can be accomplished with simple algebraic formu-
las, however, and little more math than what most readers learned in high school (or
even eighth or ninth grade!) is required. I avoid matrix algebra in all but a few places
to increase accessibility of the material. There are a few exceptions when matrices are
presented simply to give common matrix notation associated with a model. I use the
LISREL notation system in this book, because this notation is widely used in the stat-
istical literature, and I want readers to be able to learn beyond this book. I realize that
many readers have been introduced to SEM without being introduced to this notation
system. Anticipating that many readers may be unfamiliar with LISREL notation, each
symbol in each formula is clearly defined and Appendix A presents a detailed introduc-
tion to the notation. Readers should not be deterred by the notation. At least as imple-
mented in this book, it represents little more than replacing English letters used in other
texts with Greek letters (e.g., β instead of b and η instead of F). Knowledge of calculus
is not needed either, although Appendix B gives a very brief and very basic introduction
to some of the concepts of derivatives that may help supplement an understanding of
curves that represent change over time.
Outline of the Book
The order of chapters in this book has been carefully considered, and though they can be
read out of order, more will be gained if the order of the chapters is followed. The first
four chapters certainly should be read before the others, because nearly every later chap-
ter refers back to them. But readers will benefit most by keeping within the order after
that as well, as nearly all chapters make use of concepts introduced in prior chapters.
For example, the latent transition chapter (Chapter 10) refers back frequently to simplex
models from the cross-lagged panel chapter. And the missing data chapter (Chapter 13)
requires some knowledge of mixture modeling introduced in Chapter 10.
Chapter 1 reviews some of the key concepts of latent variables, such as latent vari-
able variance and mean identification and analysis of binary and ordinal variables.
Chapter 2 applies much of the information from Chapter 1 to assessing longitudinal
measurement invariance. SEM tests of dependent means and proportions over few time
points are explored in Chapter 3, and basic concepts of stability and change, difference
scores, and lagged regression are covered in Chapter 4. The remaining chapters are
each primarily devoted to one major type of longitudinal structural equation model.
Chapter 5 builds on the preceding chapter by exploring full cross-lagged panel models
and simplex models in depth. Chapter 6 focuses on modeling stability with several ver-
sions of state-trait models. Chapters 7 and 8 are devoted exclusively to latent growth
curve models, with one chapter on linear change and the other chapter on nonlinear
change. Latent difference score models are the topic of Chapter 9. Chapter 10 intro-
duces latent class analysis concepts generally and then applies them to latent transition
models and growth mixture models. Structural modeling analysis of time series for
multiple cases and single cases are discussed in Chapter 11. Chapter 12 shows how
SEM can be used to for survival analysis, the approach to modeling observations of
events that are censored. Missing data issues are covered in Chapter 13 where they can
be discussed in the context of many of the preceding models.