Longitudinal data analysis using structural equation models

LongitudinaL data anaLysis using structuraL Equation ModELs... nesselroade LongitudinaL data anaLysis using structuraL Equation ModELs... Longitudinal data analysis using structural eq

LongitudinaL data anaLysis using structuraL Equation ModELs

A M E R I C A N P S Y C H O L O G I C A L A S S O C I A T I O N

W A S H I N G T O N , D C

John J Mcardle John r nesselroade

LongitudinaL

data anaLysis

using structuraL Equation ModELs

Copyright © 2014 by the American Psychological Association All rights reserved Except

as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, including, but not limited to, the process of scanning and digitization, or stored in a database or retrieval system, without the prior written permission of the publisher.

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Library of Congress Cataloging-in-Publication Data

McArdle, John J.

Longitudinal data analysis using structural equation models / John J McArdle and John R Nesselroade.

pages cm Includes bibliographical references and index.

ISBN-13: 978-1-4338-1715-1 ISBN-10: 1-4338-1715-2

1 Longitudinal method 2 Psychology—Research I Nesselroade, John R II Title.

BF76.6.L65M33 2014 150.72'1—dc23 2013046896

British Library Cataloguing-in-Publication Data

A CIP record is available from the British Library.

Printed in the United States of America First Edition

http://dx.doi.org/10.1037/14440-000

Preface ix

Overview 3

I Foundations 15

Chapter 1 Background and Goals of Longitudinal Research 17

Chapter 2 Basics of Structural Equation Modeling 27

Chapter 3 Some Technical Details on Structural

Equation Modeling 39

Chapter 4 Using the Simplified Reticular Action Model Notation 59

Chapter 5 Benefits and Problems Using Structural Equation Modeling in Longitudinal Research 67

CONTENTS

II Longitudinal SEM for the Direct Identification

of Intraindividual Changes 73

Chapter 6 Alternative Definitions of Individual Changes 75

Chapter 7 Analyses Based on Latent Curve Models 93

Chapter 8 Analyses Based on Time-Series Regression Models 109

Chapter 9 Analyses Based on Latent Change Score Models 119

Chapter 10 Analyses Based on Advanced Latent Change Score Models 133

III Longitudinal SEM for Interindividual Differences in Intraindividual Changes 141

Chapter 11 Studying Interindividual Differences in Intraindividual Changes 143

Chapter 12 Repeated Measures Analysis of Variance as a Structural Model 151

Chapter 13 Multilevel Structural Equation Modeling Approaches to Group Differences 159

Chapter 14 Multiple Group Structural Equation Modeling Approaches to Group Differences 167

Chapter 15 Incomplete Data With Multiple Group Modeling of Changes 177

IV Longitudinal SEM for the Interrelationships in Growth 185

Chapter 16 Considering Common Factors/Latent Variables in Structural Models 187

Chapter 17 Considering Factorial Invariance in Longitudinal Structural Equation Modeling 207

Chapter 18 Alternative Common Factors With Multiple Longitudinal Observations 221

Chapter 19 More Alternative Factorial Solutions for Longitudinal Data 231

Chapter 20 Extensions to Longitudinal Categorical Factors 239

V Longitudinal SEM for Causes (Determinants)

of Intraindividual Changes 253

Chapter 21 Analyses Based on Cross-Lagged Regression and Changes 255

Chapter 22 Analyses Based on Cross-Lagged Regression in Changes of Factors 271

Chapter 23 Current Models for Multiple Longitudinal Outcome Scores 281

Chapter 24 The Bivariate Latent Change Score Model for Multiple Occasions 291

Chapter 25 Plotting Bivariate Latent Change Score Results 301

VI Longitudinal SEM for Interindividual Differences in Causes (Determinants) of Intraindividual Changes 305

Chapter 26 Dynamic Processes Over Groups 307

Chapter 27 Dynamic Influences Over Groups 315

Chapter 28 Applying a Bivariate Change Model With Multiple Groups 319

Chapter 29 Notes on the Inclusion of Randomization in Longitudinal Studies 323

Chapter 30 The Popular Repeated Measures Analysis of Variance 329

VII Summary and Discussion 331

Chapter 31 Contemporary Data Analyses Based on Planned Incompleteness 333

Chapter 32 Factor Invariance in Longitudinal Research 345

Chapter 33 Variance Components for Longitudinal Factor Models 351

Chapter 34 Models for Intensively Repeated Measures 355

Chapter 35 Coda: The Future Is Yours! 367

References 373

Index 401

About the Authors 425

George Orwell wrote a lot of important books At one point, he also considered the reasons why people write books at all One conclusion he reached was that this task was typically undertaken to deal with some demon

in the author’s life If this is true, and we have no reason to doubt Orwell so far, then we thought it might be useful to consider the demons that drive us

to take on this seemingly thankless task The best explanation we have come

up with involves at least three motives

We have led a workshop on longitudinal data analysis for the past decade, and participants at this workshop have asked many questions Our first motive in writing this book is to answer these questions in an organized and complete way

Second, the important advances in longitudinal methodology are too often overlooked in favor of simpler but inferior alternatives That is, cer-tainly researchers have their own ideas about the importance of longitudinal structural equation modeling (LSEM), including concepts of multi ple factorial invariance over time (MFIT), but we think these are essential ingredients of

useful longitudinal analyses Also, the use of what we term latent change scores,

which we emphasize here, is not the common approach currently being used

by many other researchers in the field Thus, a second motive is to distribute

PREFACE

knowledge about MFIT and the latent change score approach Most of the instruction in this book pertains to using computer programs effectively.

A third reason for writing this book is that we are enthusiastic about the possibilities for good uses of the longitudinal methods described here, some described for the first time and most never used in situations where we think they could be most useful In essence, we write to offer some hope to the next generation of researchers in this area Our general approach to scientific dis-course is not one of castigation and critique of previous work; rather than attack the useful attempts of others, we have decided to applaud all the prior efforts and simply lay out our basic theory of longitudinal data analysis We hope our efforts will spawn improved longitudinal research

Our weeklong workshop with the same basic title as this book has been sponsored every year since 2000 by the Science Directorate of the American Psychological Association (APA) This APA workshop is based on an earlier workshop on longitudinal methods started at the Max Planck Institute (MPI) for Human Development in Berlin in 1986 (at the invitation of the late Paul Baltes) This book presents the basic theory of longitudinal data analysis

used in the workshop A forthcoming companion book, titled Applications of

Longitudinal Data Analysis Using Structural Equation Models, will present the

data examples and computer programs used in the workshop

The current LSEM workshop continues to be sponsored by APA and

is now led by Dr Emilio Ferrer and Dr Kevin Grimm at the University of California at Davis each summer And near the end of each new workshop, one of the authors of this book still gives an invited lecture In this way, the features of our original LSEM workshop live on

We have found that the concepts of “change analysis” are wide ing We know this book-length treatment will not be definitive, and we just hope it is viewed as another step along the way These particular steps were developed during a time when both authors were teaching faculty of Human Development at The Pennsylvania State University (PSU) and subsequently colleagues at the Department of Psychology at the University of Virginia (UVa), the dates of this collaboration ranging from about 1985 to 2005 Dur-ing that time, we taught 10 summers of APA-sponsored workshops at UVa (2000–2009), and we learned some valuable lessons about longitudinal research

rang-For example, we learned we needed to separate the basic concepts of “growth and change” in our own approach to data analysis (see McArdle, 2009) We also learned about the importance of adequate measurement models (see McArdle

& Nesselroade, 2003) It also became apparent that the more the computer programs changed, the more they stayed the same (to be discussed) Perhaps

it is obvious that our collaboration would not have been possible unless we were encouraged to work together, so we thank both PSU and UVa for all the time they allowed us to think about these issues

There are many specific people to thank for our collaboration At the top of this list we must mention our wives, Carol Prescott and Carolyn Nes-selroade These two unusual people gave us critical support that helped us produce this book, and they continue to allow us the time to work on these matters Of course, we have tried to incorporate their ideas into this text

as best we can, and about all we can say now is “thanks.” We know this is really not enough

Next in line we thank our wonderful colleagues and graduate students

at UVa, including (in alphabetic order) Steven Boker, Sy-Minn Chow, Ryan Estabrook, Emilio Ferrer, Kevin Grimm, Paolo Ghisletta, Fumiaki Hama-gami, Thomas Paskus, Nilam Ram, Lijuan (Peggy) Wang, and Zhiyong (Johnny) Zhang Many of these students suggested changes in the materials, and we tried to use everything they suggested In particular, Aki helped us edit the prose you see here and provided most of the figures Other important students include Ulman Lindenberger (now MPI, Berlin director) and Karl Ulrich Mayr (now full professor, University of Oregon) We make special mention of Drs Ferrer and Grimm, who have contributed to this material in more ways than one and who, as mentioned earlier, lead the LSEM workshop (now at the University of California at Davis) each summer There are many others who deserve credit for their comments and questions All these gradu-ate students are making important contributions on their own right now, and this is especially rewarding for us

As stated earlier, our professional colleagues at UVa were essential to this effort, and the short list of important supporters includes Richard Bell, Mavis Heatherington, Richard McCarty, Dennis Profit, Jerry Clore, and Sandra Scarr Other colleagues who supported our efforts and made an important difference in our thinking were many well-known scientists—Paul Baltes (PSU, MPI), Ray Cattell (UI, UH), John Horn (DU, USC), Ron John-son (UH), Bill Meredith (UCB), Rod McDonald (OISE, UI), and Bill Rozeboom (UCA) There is simply no way to adequately thank this unique group of scientists for taking the time to confide in us things they had just discovered and what they actually thought about these things These people are no longer alive, but we hope their research and their thoughts live on here We also thank many members of the Society of Multivariate Experi-mental Psychology for their continuing support of the development of these ideas Finally, we thank all the many participants of our APA-sponsored workshops and the APA Science Directorate and APA Books staff for the time and effort they put in toward challenging us to produce a coherent piece about clearing up some basic concepts about longitudinal research

In sum, this book is intended as a tribute to the many contributions and the ideas of many, many others

LongitudinaL data anaLysis using structuraL Equation ModELs

Longitudinal Data Analysis Using Structural Equation Models, by J J McArdle and J R Nesselroade

Copyright © 2014 by the American Psychological Association All rights reserved.

OVERVIEW

Longitudinal data are difficult to collect, but longitudinal research is popular And this popularity seems to be growing The reasons why research-ers now appear to be enamored with this approach will be questioned, but there is no doubt the collection of longitudinal data is on the rise With this comes the subsequent need for good data analysis methods to analyze these special kinds of data Structural equation modeling (SEM) is a valu-able way to analyze longitudinal data because it is both flexible and useful for answering common research questions However, the most appropriate SEM strategy to use will depend on the specific question you are trying to answer

Baltes and Nesselroade’s (1979) seminal chapter identifies five basic questions or purposes of longitudinal SEM (LSEM):

We focus on the big picture approach rather than the algebraic details

We realize that excessive amounts of linear SEM algebra can get in the way

of big picture thinking, so we have tried to minimize the required algebra and calculus herein Thus, we have limited the algebra and calculus to separate exhibits We think the defining equations can be studied in some depth

as our main message is presented To facilitate student learning, a ing companion book will give several fully worked out examples, including computer scripts

forthcom-The remainder of this overview introduces basic topics that are central

to this book We begin by briefly explaining our approach as developmental methodologists and how this informs the design and timing of our measures

Next, we discuss the purpose of SEM in general and LSEM in particular

Finally, we explain how the rest of this book is organized in relation to Baltes and Nesselroade’s (1979) five purposes of LSEM

OUR APPROACH AS DEVELOPMENTAL METHODOLOGISTSWho are developmental methodologists anyway? And why should any-one listen to them? These are two questions that have been of interest to us for a long time, probably because we fit into this small category of scientists!

Methodologists study the many ways researchers evaluate evidence It

is clear that some formal methods are better than others, and the key role of the methodologist is to point this out to others There is no real need for a methodologist to actually find any facts (i.e., collect empirical data), and this seems to put these people in a special category One would think this makes the task much easier But it is also clear that other people seem to find it very hard to listen to a person who does not know all the troubles and nuances of doing “real” research So, in this book, we will not use computer simulation

to prove our points here; we have done so elsewhere, but only to check on

the accuracy of the programs Instead, we will only present real data And it

is probably best if the person suggesting what to do has some experience with the specific problem at hand One of the clearest statements of our general approach has been written about before:

Tukey argued that there have to be people in the various sciences who concentrate much of their attention on methods of analyzing data and of interpreting results of statistical analysis They have to use scientific judgment more than they use mathematical judgment, but not the for-mer to the exclusion of the latter Especially as they break into new fields

of sciencing, they must be more interested in “indication procedures”

than “conclusion procedures.” (Cooley & Lohnes, 1971, p v)

So we view ourselves as data analysts and not as statisticians This will

become evident as you read more details here We now add the term

devel-opmental to this distinction There are many words used to define

develop-mental, but here we will simply concentrate on “changes over time.” These changes, of course, could come from a system where a very small amount of time has passed, such as the observation of a person over a few minutes, or some part of day, or the observation of a person over many weeks, or the obser-vations of many people over many years In this definition, the specific times are not critical, and they can occur in early childhood, or late adulthood, or both What is important is the nature of change that can be captured and what aspects of change cannot

So we will also try to represent what is known in the field of mental methodology by being very explicit about the design and timing of our measures, and we will apply the methods we advocate to real data Our hope is that some of you will use these methods right now and others will improve these methods for future work Either way is certainly okay with us

develop-WHY USE SEM?

So why do we use SEM, or SEM programs, at all? Well, it is not true that

we use SEM because of the path diagrams—many traditional models can be represented using path diagrams, and, although it is not clear to many, we really do not need to use SEM tools to use SEM concepts Also, some of the standard multivariate tools are now very easy to use But there are three good reasons why we use SEM for data analysis

First, we use SEM because we have a priori ideas about individuals and groups that we want to examine in real data Some of these ideas are well beyond the standard analysis of variance (ANOVA) and the so-called general linear model framework We would certainly like to know if we are

wrong about these a priori ideas, so we certainly appreciate much of the ing work on SEM estimators, statistical indices, and overall goodness-of-fit indices.

ongo-Second, we use SEM because we want to consider the inclusion of unobserved variables in our models, that is, latent variables We often think about latent variables in our theories—variables that are not directly mea-sured or measurable—and we want to represent them in our models of these theories In this sense, the inclusion of latent variables is for clarity, not for obscurity It is also clear that the accurate representation of observed variable distributions may require more complex measurement models than the typi-cal normality assumptions

Third, we use SEM because we would like to have empirical assistance selecting the “true” or “correct” model, or at least an adequate model, for a set

of data We believe we can tell we have found an adequate model when we estimate parameters that do not differ with different samplings of person or variables or occasions (i.e., the parameters are invariant) In the terms of lin-ear regression, we do not always want the model with the highest explained variance for the current set of data, but we do want the model that is most likely to replicate over and over again

Since these three goals seem reasonable and continue to be part of most behavioral science research, SEM combined as both a concept and a tool is now very popular, and multivariate data analysis is likely to remain this way for a long time to come But perhaps it is now clear that the typical SEM was not specially defined for the issues of longitudinal data analysis, much less dynamic SEM, so what we offer in this book is a different variety of SEM

than that typically displayed in the SEM journals (e.g., see the journal

Structural Equation Modeling).

And, although we do add much here about reporting indices of fit (e.g., chi-square, root mean square error of approximation), we do so for the dual purposes of clarity and communication, and we also think very little of the typical search for the “best-fitting model.” We think that this kind of search can be of benefit to a single publication effort, but it holds little promise for evaluating the replicability or invariance of effects over multiple experi-ments Additionally, we view the consideration of a measurement model to

be a fundamental aspect of experimental design, and a poor measurement model can lead to failures that often go undetected Our experiences suggest that any model is best built up from component parts, and this leads us to consider the separation of different aspects of any model into submodels for evaluation Thus, instead of searching for the best-fitting model, we favor an SEM approach designed to carry many results from one analysis to the next

We carry out SEM analyses by fitting all models in pieces rather than starting with a simultaneous solution

WHY USE LSEM?

Although SEM has both benefits and limitations, there are many good motivations to use contemporary SEM for longitudinal analyses Among many benefits, we can see a required clarity of process definitions, ease of programming

of complex ideas, clear tests of parameter equivalence, and generally ate analyses of otherwise inaccessible ideas A few of these SEM techniques even lead to novel ideas for objective data analysis, including easy ways to deal with ordinal measurements For example, perhaps SEM path diagrams provide

appropri-a cleappropri-arer wappropri-ay to think appropri-about longitudinappropri-al appropri-anappropri-alyses Unfortunappropri-ately, appropri-among mappropri-any LSEM limitations that remain are the need for large and representative samples

of persons for reasonable estimation and also for a normal distribution of als (or uniquenesses) to create appropriate statistical tests These are not easy requirements to present graphically But, most importantly, we need to state what is to be examined about dynamic influences in advance of the data analysis, on an a priori basis In this way any SEM requires the kind of clear and penetrating thinking seldom achieved in behavioral science In common practice, we get stuck using the SEM models in ways that are far simpler than

residu-is possible if some ingenuity residu-is applied

The LSEM approach differs from earlier approaches to longitudinal data analysis Several decades ago, longitudinal analyses were based largely on principles derived from linear growth models and formalized in terms of ANOVA techniques (e.g., Pothoff & Bock, 1975; Roy, 1964) These con-cepts were extended with the creation of randomized blocks and Latin squares (see Fisher, 1925; Winer, 1962) designed for plants and nonhuman animals

Of course, this can be much more complicated in human research For these reasons, in research on human aging, we often separate cross-sectional “age differences” from longitudinal “age changes” (as in McArdle, 2009) Many recent analyses of developmental data analysis use information gathered from

both cross-sectional and longitudinal selections in what are termed panel

studies (Hsiao, 2005), but the questions seem much broader Nevertheless,

ANOVA-type analyses, based largely on plants and nonhuman animals, flourished in the behavioral sciences, and they still seem to do so today

Some recent presentations on longitudinal data analysis are based on statistical procedures that combine these seemingly separate estimates of age differences and age changes (for references, see McArdle, 2009) One way to

do this is to create an “expected trajectory over time,” where the expected values are maximum likelihood estimates and formal tests of hypotheses are encouraged (e.g., Hsiao, 2003; Miyaziaki & Raudenbush, 2000; cf McArdle

& Bell, 2000) New computer programs for what are termed latent curve or

mixed-effects modeling allow parameters to be estimated and are used in

mak-ing predictions and inferences This book, however, is based on our earlier

work on LSEM We need to have some ways to understand longitudinal data, and several theoretical statements have been formalized into “models” with useful statistical properties These models allow us to consider alternative substantive ideas, to fit these alternatives to data, and hopefully to make an informed choice about which alternatives are most useful Key failures of any kind of analysis come when a researcher reflexively applies methods without thinking about what they may mean and then by not recognizing when the model predictions fail to look like the data.

What we basically want is a longitudinal structural equation model (LSEM) that has a minimal number of estimated parameters and fits the observed data well In the case of longitudinal data, it would also be useful if the model made predictions about future behaviors of individuals and groups

There is no doubt that this can be a demanding task for anyone, and we may not succeed In fact, this entire SEM approach may fail, and we will simply conclude that it is not possible to do this

This also means, in our view, the current SEM approach is far less lutionary than the past work SEM is simply a natural generalization of most prior work However, we do think the new SEM techniques are not really much better than the prior ones (cf Raykov & Marcoulides, 2006) In this sense, the classical techniques still form basic guideposts, and there are many worthwhile aspects of the older classical calculations We will mainly use SEM because it can be used to make concepts clear and it is now very convenient

revo-One matter about which we want to be very clear at the outset is the distinct way we make inferences about causal influences when we use random-ized assignment to groups versus when we have nonrandomized assignment

to groups The former is now becoming more popular as randomized cal trial (RCT) research (see McArdle & Prindle, 2008), and this is rea-sonable because the RCT approach leads to exceedingly simple inferences:

clini-Fisher (1925) cleverly and definitively showed that when individuals were randomly assigned to conditions, the effect of the condition could be easily and unambiguously estimated from the mean differences This is a clear-cut case of the statistical model of inference being forced to be correct by design

(i.e., with large enough sample sizes, N, the error terms can be assumed to be

uncorrelated with the predictors, so all resulting parameters are unbiased)

The same cannot be said of situations where the data have been lected without regard to the person selection: the persons are simply observed

col-as they progress through time, so this data collection design is often termed observational data (Cochran & Cox, 1950) This basic distinction between randomized and observational data has separated many research traditions (see Cronbach, 1957) Unfortunately, this is almost always the case in lon-gitudinal data collections where no randomized conditions are used at all, for either practical or ethical reasons, but dynamic effects that would be

estimated from randomized conditions are still sought (see Shrout, 2011)

Obviously, this self-selection of persons is a big barrier for good longitudinal research But we agree with R B Cattell (1966c), who basically asserted that there exists a continuum of experiments when we have data and substantive questions for which we do not know the answer He suggested, in these cases, that we mainly need to put a premium on good measurement and good data analysis After all, we do think that astronomy is a science, which is chock-full of scientific experiments (as did Cattell, 1966d)

The LSEM approaches presented here are obviously not all that can

be done using the more generic SEM concepts We treat these as starting points for other ideas, and we leave many advanced issues for a subsequent presentation (see McArdle & Hamagami, 2014a) But the SEMs we do pre-sent are highlighted here because they are a reasonable match to the goals of longitudinal data analysis proposed by Baltes and Nesselroade (1979) With this in mind, we will try not forget the most important lessons of the past

At the same time, we do need to invent new ways to collect optimal sets of data required for powerful tests of hypotheses about development These are among the most important challenges for contemporary and future longitudi-nal research, and our hope is that some of the SEM ideas presented here can

be instrumental in reaching that goal

PREVIEW OF THIS BOOKThis book consists of seven parts with five chapters each Part I (Chapters 1–5) presents an in-depth discussion of the goals and other back-ground information on longitudinal research, including SEM and research designs Parts II through VI are the heart of the book and cover models for Baltes and Nesselroade’s (1979) five objectives of longitudinal research

Part II (Chapters 6–10) explains how to analyze information about the

changes within a person Baltes and Nesselroade (1979) used the term

intra-individual differences, but we will speak of within-person changes (McArdle,

2009) Because we now think “change” is most easily and most appropriately indexed by the use of a simple change score, although others have questioned this logic, this approach is formalized in our first true model definition in Part II The careful reader will notice we make some effort to show how this simple change approach to data analysis can be useful, both in calculation and interpretation, and we try to separate the controversial aspects from the noncontroversial aspects This use of simple change score approach can be considered as our first suggestion about using longitudinal data analysis

Part III (Chapters 11–15) explains how to analyze differences between groups of persons in the way people change This terminology differs from that of Baltes and Nesselroade’s (1979) original text, which used the phrase

“intraindividual differences in interindividual differences.” Now there are at

least two reasons for altering the prior terminology: (1) The term within

per-sons is traditional ANOVA terminology to designate the separation of scores

from the same person (2) This is in contrast to components that are between persons By separating these terms into within and between persons, the term

differences can then be reserved for models that imply the separation of

differ-ent individuals In general, we will focus on concepts about “the differences

in the changes,” and we hope this generally confusing set of verbal terms will still make sense The representation of group differences in change processes can be accomplished using the variety of methods that the literature now refers to as latent curve, mixed-effects, multilevel, or even multiple group modeling We will try to clarify the options in these alternatives This sepa-ration of terms can be considered our second suggestion about using longi-tudinal data analysis

Part IV (Chapters 16–20) explains how to analyze multiple outcome ables This is not typically done by others in this area, and single variables seem like enough In this book, we have used this objective as a vehicle for discussing the important concepts of multivariate measurement The use of common fac-tors as a way to summarize information can be considered the third requirement

vari-of our longitudinal data analysis approach, and this was certainly our intention

Practical problems in the fitting of any statistical model with longitudinal data begin with scaling and metrics These ideas can be examined at the item level

by forming a scoring system for any desired construct using classical concepts from item response theory (see Lord, 1952; McArdle, Grimm, Hamagami, Bowles, & Meredith, 2009; McArdle & Hamagami, 2003) Other key scaling issues include the exact timing of observations (e.g., Boker & McArdle, 1995;

Gollob & Reichardt, 1987) because, as is often the case, any variable mations can alter the statistical patterns Optimal measurement collections are

transfor-a mtransfor-ajor chtransfor-allenge for new empirictransfor-al resetransfor-arch transfor-and transfor-are worthy of much further discussion (McArdle, 2011d) Indeed, the multivariate perspective (see Cattell, 1966b, 1966d) when combined with SEM permits tests of hypotheses of com-mon concepts we ordinarily take for granted We will try to show how recent developments in SEM can lead to useful concepts, such as scale-free measure-ment and invariant common factors, and we are especially keen on what can

be accomplished with models of MFIT We will also try to show how this can helps us achieve the error-free changes we seek This part emphasizes our com-mitment to the common factor model

Part V (Chapters 21–25) explains how to analyze the causes of individual changes This part suggests that there are many alternative lon-gitudinal models that can be fitted to different kinds of longitudinal data It will surprise many readers, but only a selected set of these models take into account the specific ordering of responses In these LSEMs, it will be important

intra-to recognize that models of individual differences among dynamic variables lead to hypotheses about individual trajectories over time That is, the dif-ference (or change model here) model leads to the trajectory expression in the same way the differential equation model leads to the integral expression (see Boker, 2001; Oud & Jansen, 2000) To make this tractable and practi-cally useful, we present models of discrete timing and pose several alternative models about time-based dynamics at the individual level; then we illustrate how we can turn these into structural equation models Any dynamic inter-pretation requires a focus on time-dependent parameters in the models rather than the time-independent correlation of the time-based scores, and this puts extra emphasis on the explicit definition of time and time lags (e.g., Gollob

& Reichardt, 1987; McArdle & Woodcock, 1997) This part emphasizes our suggested commitment to time precedence and a dynamic approach

Part VI (Chapters 26–30) explains how to analyze the causes of individual differences in intraindividual changes This part considers the possi-bility that there are group differences in the prior dynamic models Although this probably seems obvious, it is hardly ever examined, and we try to show how it could be done Once again, the representation of group differences in change processes can be accomplished using the variety of methods the lit-erature now referred to as latent curve, mixed-effects, multilevel, or multiple group modeling But in these LSEMs it is also important to recognize that models of group differences among dynamic variables do not necessarily reflect dynamic processes for the group The LSEM analyses presented here are limited because they make specific assumptions about latent changes in measured individuals and groups (as in McArdle & Prindle, 2008; Meredith

inter-& Tisak, 1990; Muthén inter-& Curran, 1997) The current literature also seems

to pursue a lot of exploratory data analysis models, including what are now

referred to as latent class mixture models or even SEM Trees Although these

analyses can be a very useful at appropriate times, we will not discuss these els here (but see McArdle & Hamagami, 2014b) Thus, Part VI emphasizes our commitment to, and suggested use of, allowing group difference in dynamics

mod-Finally, Part VII (Chapters 31–25) integrates the previous chapters and elaborates on several of the book’s main topics, including incomplete data designs, uses of individual time series, and meta-analysis problems but we do so without concrete examples We also suggest the use of recent computer programs

(see our forthcoming companion book, Applications of Longitudinal Data Analysis

Using Structural Equation Models), but these are fairly standard applications now,

so we focus on models where the computer code is publicly available (i.e., R) In our companion book we also will present code that we think is widely used (e.g., Mplus) This code will obviously change over the next few years, but we do not expect many alterations in the underlying ideas of either book

All notations in this book are based on LSEM and are listed in Exhibit 1

F1[t] Measurement factor score 1 at time t

F2[t] Measurement factor score 2 at time t

L or L2 Likelihood as a single index of model fit

t, T t = specific time; T = total times (occasions of measurement)

X[t] Observed score at time t of variable X (usually defining independent variables) Y[t] Observed score at time t of variable Y (usually defining dependent variables)

j[t + 1]

normality assumption

(continues)

(occasion of measurement) V[2] - Verbal composite score at Time 2

Abbreviations for types of models

time

variance

i

Longitudinal Data Analysis Using Structural Equation Models, by J J Mcardle and J R nesselroade

Copyright © 2014 by the american Psychological association all rights reserved.

1

this first quote above, by John stuart Mill, was supplied by our edgeable colleague, steven M Boker, for a chapter written some years ago (nesselroade & Boker, 1994) this is an apt opening quote for this chapter for a couple of reasons first, it simultaneously recognizes the concept of sta-bility and the concept of change Both concepts are important; indeed, they are both critical Without a concept of stability there would be no need for a concept of change and vice versa one way the reader may have sometimes heard the second quote phrased as “We do not know who discovered water, but we are pretty sure it wasn’t a fish.” this is intended to mean that humans are “tuned” to appreciate variation rather than sameness the former registers

knowl-on our sensory mechanisms and stimulates us, whereas in its extreme form we soon become unaware of the latter silence is a wonderful adjunct to sound sleep but, judging from the plethora of audio devices in operation, it is a major bane of today’s teenager’s waking existence

BaCKGRound and GoaLs of LonGitudinaL REsEaRCH

a party of order or stability, and a party of progress or reform, are both necessary elements of a healthy state of political life

—J s Millone thing about which fish know exactly nothing is water, since they have no anti-environment which would enable them to perceive the element they live in

—Marshall McLuhan, War and Peace in the Global Village, 1964

More to the point, if there were nothing but constancy, there would be

no need to develop and use longitudinal methods an emphasis on variability, coupled with an acceptance of the fact that the individual is the primary unit

of analysis in studying behavior virtually “forces” the researcher to seek some form of longitudinal (repeated measurements) information How else can one access variability in behavior at the individual level (nesselroade & Molenaar, 2010a, 2010b, 2010c)?

the second, and more important sense in which the Mill quote is apt,

is that it establishes a dynamic between the two concepts: political “reality”

resides in neither party; rather it is in the relationships the two parties bear to each other Much of the longitudinal modeling about which you will be read-ing may tend to emphasize either change or stability over the other, mainly because that is where our science is historically But there will be chapters explicitly devoted to presenting how one models the more dynamic features

of longitudinal data this is an important distinction and the direction in which we believe the future of behavioral science modeling lies

dEfinition of LonGitudinaL REsEaRCH

“the study of phenomena in their time-related constancy and change is the aim of longitudinal methodology.” Baltes and nesselroade (1979) advanced this sentiment in a book that one of us edited and for which a historical overview chapter was written this definition and the five rationales for lon-gitudinal research that Baltes and nesselroade developed help to define the scope of longitudinal research and provide key underpinnings for the remain-der of this book We repeat these principles now because we think they are important

When Baltes and nesselroade (1979) set out to delimit what longitudinal research was about, they encountered considerable uncertainty in the litera-ture a couple of examples of some of the comments regarding longitudinal research they found were “there is no hard and fast definition of what consti-tutes a longitudinal study” (Hindley, 1972, p 23) and “Longitudinal is a general term describing a variety of methods” (Zazzo, 1967, p 131) these expressions

of ambiguity were in some ways discouraging but, on the other hand, it was challenging to try to provide a more systematic and positive working definition

they produced the following:

Longitudinal methodology involves repeated, time-ordered observation

of an individual or individuals with the goal of identifying processes and causes of intraindividual change and of interindividual patterns of intra-individual change [in behavioral development] (Baltes & nesselroade,

1979, p 7)

as we move toward developing and presenting the five rationales for longitudinal research, this definition fleshes out the essential nature

of longitudinal research that we eventually settled on a little bit more in the present book, we consider quite a wide range of longitudinal research possibilities—from some single subject designs to panel designs of various kinds We have found the above definition a helpful guide in trying to answer the question “Just what are the main arguments for why longitudinal research

is important?”

for Baltes and nesselroade, longitudinal research was closely tied to developmental questions, and we have “lifted” the following figures from something that Baltes wrote (see figure 1 in Baltes & nesselroade, 1979) We chose them because they illustrate well the theoretical ideas about what one might be observing longitudinally in the study of development

there are several features of figures 1.1 and 1.2 that we want to call

to your attention the idea being expressed in figure 1.1 is that, across a life span, for many different kinds of behavioral phenomena, there does appear to be considerable evidence for an increase in between-persons vari-ability with age at a very practical level, people at younger ages, on average, are a lot more like each other in many respects than they are at older ages

at the same time, there are some variables for which individuals’ ries tend to go up with age, others for which they tend to go down, and some that are more complicated than that But the point is, if one puts together

trajecto-a lot of the informtrajecto-ation thtrajecto-at we now htrajecto-ave thtrajecto-at is descriptive trajecto-about the wtrajecto-ays people change over time, there is a wide variety of possibilities obviously,

in the Study of Behavior and Development (p 16), by J R Nesselroade and

P B Baltes, 1979, New York, NY: Academic Press Copyright 1979 by Elsevier

Adapted with permission.

this variety of form and function places a tremendous burden on ers to measure, to model, and to represent adequately such a rich variety

research-of phenomena therefore, we need a lot research-of different tools to model and understand longitudinally what development or other kinds of processes across time involve

figure 1.2 reminds us of there are numerous, complex reasons why we change over time developmentally some of these reasons are orchestrated

by our own selves, some by societal institutions, and still others are seemingly unorchestrated but still occur, and with lasting effects

We now turn to the five main rationales for conducting longitudinal research that were presented by Baltes and nesselroade (as listed earlier)

these five rationales, we argue, are the primary reasons why one engages in longitudinal research the structural equation modeling analyses are simply ways to carry out these analyses the terminology used then is admittedly somewhat cumbersome now, but it is usefully descriptive and will be referred

to often in the remainder of this book Let us examine each of the five in turn

in a slightly different way

Figure 1.2 Illustration of three major influence systems on development: normative age graded, normative history graded, and nonnormative From Longitudinal Research in the Study of Behavior and Development (p 19), by J R Nesselroade

and P B Baltes, 1979, New York, NY: Academic Press Copyright 1979 by Elsevier

Adapted with permission.

NORMATIVE AGE GRADED

NORMATIVE HISTORY GRADED

NONNORMATIVE INFLUENCES ON DEVELOPMENT (Biological and Environmental)

TIME

fiVE RationaLEs foR ConduCtinG LonGitudinaL REsEaRCH

Direct Identification of Intraindividual Change

as noted earlier, there is one way to get at change (variation) at the individual level, and that is by measuring the same individual at least twice it

is an obvious point in some ways but, for various reasons, it was a good place

to begin in developing these rationales for conducting longitudinal research

intraindividual change occurs in many forms for example, it may involve changes in level, amplitude, or frequency or, it may involve interbehavioral changes for instance, physical abuse of others may be supplanted by verbal abuse as one ostensibly becomes more socialized

still other kinds of changes are found in the literature such as are ignated by terms such as the differentiation hypothesis of cognitive devel-opment (olsson & Bergman, 1977; Reinert, 1970) the theory holds that

des-as organisms age, their cognitive structure becomes more differentiated so, for instance, instead of being equally good or bad at all cognitive tasks, an individual may be a standout in verbal behavior and only mediocre in math-ematics Pertinent to the focus of this book, even though differentiation is

an individual development concept, it is typically studied via cross-sectional designs and group data analyses

as the reader is likely to be aware, this may or may not be defensible

the relative merits of substituting cross-sectional research for longitudinal efforts have long been argued in order for cross-sectional comparisons to be good approximations to what one can study longitudinally, either the cross-sectional data must meet the condition that different-aged subjects must come from the same parent population at birth, it must be possible to match subjects across age levels, or different-aged subjects must have experienced identical life histories or else these conditions must be shown to be irrelevant

in most cases, it seems to us that longitudinal designs will not be supplanted easily

Direct Identification of Interindividual Differences

in Intraindividual Change

the basic idea here is relatively simple if one is studying vidual differences in the traditional sense, one is, by definition, making com-parisons across individuals But when one’s focus is on comparing patterns

interindi-of intraindividual change rather than single scores or levels across als, longitudinal information is necessary for defining those change patterns among which differences are sought

individu-Let us briefly illustrate something that is going to come up again and again in the material to follow Consider the mathematical expression in Exhibit 1.1 this expression was taken from one of the earliest articles now recognized as a progenitor of latent curve modeling, tucker (1966) in this article, tucker presented what he called generalized learning curve analysis

We point this out for the following reason: this is a way of identifying taneously both the general group structure and the individual differences struc-ture within one framework this was originally presented by tucker (1958, 1966) as a principal component analysis of the raw data, rather than of the deviation scores or covariances, because he wanted to preserve the trial-to-trial differences (gains) in performance What you will see later under the topic of latent curve modeling, including latent growth curve modeling and latent change score modeling, offers a variety of refinements on this general approach

simul-Analysis of Interrelations in Behavioral Change

this third rationale pays homage to the multivariate perspective on behavior (Baltes & nesselroade, 1973; Cattell, 1966; nesselroade & ford, 1987), which recognizes that determinants have multiple consequences and consequences have multiple determinants a meaningful way to try to study these manifold relationships within a multivariate perspective involves the formation of various kinds of linear combinations according to some crite-rion by which the defining weights are chosen Canonical correlation, linear discriminant function analysis, and other multivariate analysis techniques rest on this basic principle factor analysis, including its use in creating mea-surement models, involves simultaneously examining the interrelations

of multiple factors and multiple variables Clearly, if one wants to study the interrelations in behavioral change—how behaviors change together over time—one needs longitudinal information

EXHIBIT 1.1The Expression of a Generalized Learning CurveConsider the mathematical expression that follows (data from Tucker, 1958, 1966):

[ ] = ω [ ] [ ]Q + ω [ ] [ ]Q + + ω [ ] [ ]Q +e[ ]

This decomposition leads to a way of talking about what is happening on the hand side (the actual performance Y[t]) in terms of general patterns of change, that

left-is, intraindividual change patterns—the unknown group weights (w[k]), but also the

individual level information in the latent variables (Q [k]).

Analysis of Causes (Determinants) of Intraindividual Changes

Here the essential idea is that almost any notions we have of causal nections involve some kind of time-dependent observations at a minimum, one must observe the entity/entities at least twice in time, preferably more,

con-to draw inferences regarding determinants of change We realize this is a statement about observational data, and we will not make any special claims

to understanding causality (but see Part V of this book) in fact, this concept holds at both the individual level but also in examining what causes people to differ in the way they are Models for this goal are relatively novel, and they

are termed dynamic Because this is a main reason for this book, this goal will

not be pursued now

Analysis of Causes (Determinants) of Interindividual Differences in Intraindividual Change

all individuals do not change in the same way Why? the search for explanations for why there are interindividual differences in intraindividual change patterns also requires us to attend to causes or determinants Here, too, in order to make inferences regarding causal relations one needs at a minimum two different observations Whether the same repeated measure-ment occasions used to define the intraindividual changes can also be the basis for inferences regarding causation is a matter of the research design used and the nature of the resulting data these matters also will be examined in detail

in the remainder of this book

LinKaGE BEtWEEn tHEoRY and MEtHodoLoGYthe conduct of rigorous longitudinal research is a demanding enterprise, and keeping the five rationales in mind provides a way for us to approach more systematically the discussion of the finer points of longitudinal methods and modeling the problems and pitfalls to be avoided in designing longitu-dinal work are discussed in detail in many different sources (see, e.g., Baltes, Reese, & nesselroade, 1977; and Laursen, Little, & Card, 2012)

a useful criticism that Campbell and stanley (1963) pointed out in

their original article about simple longitudinal studies is that they are

pre-experimental that is taken by many as a pejorative term, especially if one

is trained in classical experimentation Because the fact is that for much of the work behavioral scientists do, participants cannot be randomly assigned

to treatment conditions this is certainly true in traditional longitudinal designs, so one has to be aware at least of the various threats to the validity

of such designs in relation to the Campbell and stanley (1963) arguments, however, the complicating problem is that for process and developmental questions, often what is seen as a threat to internal validity in simple longitu-dinal designs is also the very phenomenon of interest to us (i.e., maturation)

this issue will be discussed at various points in this book

in one of his last articles, the late Joachim f Wohlwill (1991) identified his view of the relationships between method and theory in developmental research as partial isomorphism Here is his basic point and we think it is a crucial one: Wohlwill observed that many of us were taught to believe that theory should drive method and people who knew only methods ought to have some sense of chagrin for their lack of appreciation of theory Wohlwill then pointed out that what really happens is that sometimes theory is out ahead, demanding new methods, but sometimes method is out ahead, push-ing theory He described it very much in the same dynamic terms as suggested

in the quote by John stuart Mill at the beginning of this chapter the reader may want to keep this in mind as so much of this book is devoted to methods

as valuable as they are, methods do not get us anywhere by themselves

Progress in a discipline eventually comes through the interplay of forces rather than because one dominates the other Even if you have no interest in theory,

it is still incumbent on you, the reader, to “push” those who do care about ory just as it is the theoretician’s obligation to demand newer, more appropri-ate methods be made available for testing theoretical propositions one way

the-of stimulating this dialog is to provide methods—along with the suggestion that there is something interesting to do with the data Here is something that has the potential of showing this kind of relationships or detailing that kind of relationships as scientists, we think that is the duty of all of us, not just those trying to write better theories, to try to keep this interchange moving ahead

finally, before jumping into the deeper water, we want to say just a bit about the concept of causal lag for the simple reason that, during the course

of making his or her way through this book, the reader is going to repeatedly encounter the concept there will be discussions at various points about such notions as intervals for example, how often should one measure over a given span of time? if one wants to “tie down” a process of some sort, are there optimal points of measurement? if one is really trying to connect putative causes with putative effects, how do you do that, in terms of the sequence of observations

of your measurement? the reader will see that these represent a complicated set of issues, but there are at least some aspects that can be dealt with in a pretty straightforward way

the idea of lag now becomes very important in the longitudinal research and modeling context, perhaps much more so than we have realized in the past it is complicated in part by the fact that when one considers multivari-ate systems and latent variables, the possibility arises that various pertinent

variables do not have the same optimal spacing or optimal lag in regard to their dynamics and their relationships to other variables We cannot promise you that you will reach the end of the book with the answers to these ques-tions, but we can promise you that you will be more sensitive to them.

Let us try to summarize a few key ideas at the beginning our task in this chapter was partly to get readers up to speed first of all, the longitudinal ideal is emphasizing a temporal perspective in research questions involving one’s data that is one of the underlying “golden threads” running through the entire book Longitudinal data most directly provide the information

we need for making sure that a temporal perspective can be implemented

in thinking about the phenomena of interest to us Process and systematic change—these ideas do not stand a chance of being explored effectively, we will argue, without adopting a temporal perspective that includes some kind

of longitudinal orientation Longitudinal data provide exceptional tunities to study and model interesting phenomena they offer different opportunities than do other kinds of data, and at the same time, they present different kinds of challenges our focus for the book is on exploring many

oppor-of these opportunities provided by longitudinal data—and their associated challenges

Longitudinal Data Analysis Using Structural Equation Models, by J J McArdle and J R Nesselroade

Copyright © 2014 by the American Psychological Association All rights reserved.

2

One purpose of this chapter is to present an accessible overview of

recent research on what are termed structural equation models (SEM; following

McArdle & Kadlec, 2013) We will define our own notation (see Chapter 4), which we hope will be useful for longitudinal research at a level intended for graduate students in the behavioral sciences, some possibly taking a SEM class currently Thus, if the reader knows a lot about SEM already, this chap-ter (and the next two) can be skipped without much loss While formal train-ing in algebra or calculus is helpful, it seems it is not really required to apply such methods But we think it is important to provide more details on the rea-son we use SEM as a generic approach to deal with the current longitudinal problems We are not the first to come to this conclusion, and the concepts illustrated earlier by Goldberger (1973) seem reasonable now because they focused on primitive elements in SEM

BASICS OF STRUCTURAL EQUATION MODELING