Preface

As with each of my previous introductory structural equation modeling (SEM) books, my overall goal here is to provide my readers with a nonmathematical introduction to the basic concepts associated with this methodology, and to illustrate its basic applications using the Mplus program. All applications in this volume are based on Mplus 6, the most current version of the program at the time this book went to press. Although it is inevitable that newer versions of the program will emerge at some later date, the basic principles covered in this first edition of the book remain fully intact.

This book is specifically designed and written for readers who may have little to no knowledge of either SEM or the Mplus program. It is intended neither as a text on the topic of SEM, nor as a comprehensive review of the many statistical and graphical functions available in the Mplus program. Rather, my primary aim is to provide a practical guide to SEM using Mplus. As such, readers are “walked through” a diversity of SEM applications that include confirmatory factor analytic (CFA) and full latent variable models tested on a wide variety of data (single/multiple-group; normal/non-normal; complete/incomplete; continuous/ordinal), and based on either the analysis of covariance structures, or on the analysis of mean and covariance structures. Throughout the book, each application is accompanied by numerous illustrative “how to” examples related to particular procedural aspects of the program. All of the data files used for the applications in this book can be downloaded from http://www.psypress.com/9781848728394 or http://www.psypress.com/sem-with-mplus/datasets. In summary, each application is accompanied by the following:

• statement of the hypothesis to be tested
• schematic representation of the model under study
• full explanation bearing on related Mplus model specification input files
• full explanation and interpretation of related Mplus output files
• published reference from which the application is drawn
• illustrated use and function of the Language Generator option
• data file upon which the application is based

The book is divided into four major sections. A brief outline of each section and of its encompassing chapters is as follows:

Section I

This initial section comprises two introductory chapters. In Chapter 1, I introduce you to the basic concepts associated with SEM methodology, and then familiarize you with SEM symbol notation, the process of model-building, visible and nonvisible model components, and the formulation of covariance and mean structure modeling. Finally, I provide you with a general overview of Mplus notation and illustrate its application to three very simple models.

Chapter 2 focuses solely on the Mplus program. Here, I familiarize you with the lexicon and structuring of Mplus input files, identify the 10 key commands used in building these files, note the availability of several analytic and output options, acquaint you with the Mplus Language Generator facility, identify important default settings, and alert you to the provision of a graphics module designed for use in displaying observed data and analytic results. Along the way, I address the issue of model (or statistical) identification, demonstrate the calculation of degrees of freedom based on covariance, as well as mean structures, and distinguish between over-, just-, and under-identified models and their resulting implications. Finally, because I consider it imperative that you fully comprehend the link between model specification as described in the Mplus input, as well as schematically portrayed in its related model, I walk you through specifications associated with three simple, albeit diverse model examples: (a) a first-order CFA model, (b) a second-order CFA model, and (c) a full latent variable model.

Section II

This section is devoted to applications involving only single-group analyses. Chapter 3 involves specification of a first-order CFA model representing the multidimensional structure of a theoretical construct and outlines the steps involved in testing the validity of its postulated structure. In this first application, I walk you through each stage involved in building the related Mplus input file based on use of the Language Generator, illustrate how to run the job, and provide a detailed explanation of all results presented in the output file. Included here is a description of each goodness-of-fit statistic reported by Mplus, together with a complete breakdown of its computation.

In Chapter 4, we examine another first-order CFA model, albeit this time, as it relates to the hypothesized structure of a well-established measuring instrument. In contrast to the model tested in Chapter 3, the one tested here exhibits strong evidence of model misspecification. As such, I subsequently walk you through various stages of the post hoc model-fitting process in an attempt to establish a better-fitting model that is not only statistically viable and substantively meaningful, but also addresses the issue of scientific parsimony. Important caveats are noted throughout this model-fitting phase. Given that the data used here are non-normally distributed, I alert you to use and specification of the appropriate estimator and also outline how to obtain values related to skewness and kurtosis in Mplus.

Chapter 5 introduces you to the testing of a second-order CFA model representing the hypothesized structure of a measuring instrument based on data that are both ordinal and non-normally distributed. The basic concepts, underlying theory and statistical assumptions associated with the analysis of categorical data are described, and the general analytic strategies involving various estimators are reviewed. In the interest of both completeness and relevance (given a review of SEM applications in the literature), I walk you through two analyses of the same data: (a) with the categorical variables treated as if they are of a continuous scale, and (b) with the categorical nature of the variables taken into account. Along the way, I introduce you to the presence of a condition code error and illustrate how and why it relates to a problem of model identification.

Chapter 6 presents the final single-group application in this section. Here, I walk you through the various stages involved in testing the validity of a full latent variable model that represents an empirically-derived causal structure. In dissecting component parts of this model, I introduce and briefly address the topic of parceling. Finally, as part of a lengthy set of post hoc model-fitting analyses, I address the issue of model comparison and work through a detailed computation breakdown of the robust chi-square difference test that must be used when data are non-normally distributed.

Section III

Three applications related to multiple-group analyses are presented in this section - two are rooted in the analysis of covariance structures and one in the analysis of mean and covariance structures. In Chapter 7, I introduce you to the basic notion of measurement and structural invariance (i.e., equivalence), as well as to the basic testing strategy. Based only on the analysis of covariance structures, I show you how to test for measurement and structural equivalence across groups with respect to a measuring instrument; we test for the equivalence of factor loadings and common error residuals, as well as factor variances and covariances. In addition, given a slightly different slant to the same idea, I once again walk you through computation of the robust chi-square difference test.

In Chapter 8, we work from a somewhat different perspective that encompasses the analysis of mean and covariance structures. Here, I first outline basic concepts associated with the analysis of latent mean structures and then continue on to illustrate the various stages involved in testing for latent mean differences across groups. Given that one group comprises incomplete data, I address the issue of missingness and outline the many estimation options available in Mplus. Along the way, I introduce you to the issues of partial measurement invariance, under-identification pertinent to the number of estimated intercepts, and evaluation of invariance based on a practical, rather than statistical, perspective.

Chapter 9 addresses the issue of cross-validation and illustrates tests for the equivalence of causal structure across calibration and validation samples. In this chapter, as well as elsewhere in the book, I discuss the issue of a Heywood case and, in the interest of parameter clarification, urge specification of a particular TECH option in the OUTPUT command.

Section IV

In this final section, we examine three models that increasingly are becoming of substantial interest to SEM practitioners and researchers. Chapter 10 addresses the issue of construct validity and illustrates the specification and testing of a multitrait-multimethod (MTMM) model. More specifically, we work through two CFA approaches to MTMM analyses based on (a) the general CFA approach, and (b) the correlated uniquenesses CFA approach.

Chapter 11 focuses on longitudinal data and presents a latent growth curve (LGC) model. Following an introduction to the general notion of measuring change over time, I walk you through the testing of a dual domain LGC model that is tested with and without predictor variables. Along the way, we examine application of the Plot option in the OUTPUT command, and interpret results from several different perspectives.

Finally, in Chapter 12, I acquaint you with the notion of multilevel SEM analyses. Following an overview of multilevel modeling (MLV) in general, comparison of single-level and multilevel approaches to the analyses, and review of recent advances in the estimation of MLVs, we test for the validity of a two-level model representing the hypothesized structure of a measuring instrument based on data for 27 geographically diverse cultural groups.

Although there are now several SEM texts available, the present book distinguishes itself from the rest in a number of ways. First, it is the only book to demonstrate, by application to actual data, a wide range of CFA and full latent variable models drawn from published studies and accompanied by a detailed explanation of each model tested, as well as the resulting output files. Second, it is the only book to incorporate applications based solely on the Mplus program. Third, it is the only book to literally “walk” readers through: (a) model specification, estimation, evaluation, and post hoc modification decisions and processes associated with a variety of applications, (b) competing approaches to the analysis of multiple-group and MTMM models, and of categorical/continuous data, based on the Mplus program, (c) illustrated applications of the optional Mplus Plot and Tech OUTPUT commands, together with interpretation of their results, and (d) a step-by-step application of the Mplus Language Generator facility. Overall, this volume serves well as a companion book to the Mplus user's guide (Muthén & Muthén, 2007-2010), as well as to any statistics textbook devoted to the topic of SEM.

In writing a book of this nature, it is essential that I have access to a number of different data sets capable of lending themselves to various applications. To facilitate this need, all examples presented throughout the book are drawn from my own research. Related journal references are cited for readers who may be interested in a more detailed discussion of theoretical frameworks, aspects of the methodology, and/or substantive issues and findings. It is important to emphasize that, although all applications are based on data that are of a social/psychological nature, they could just as easily have been based on data representative of the health sciences, leisure studies, marketing, or a multitude of other disciplines; my data, then, serve only as one example of each application. Indeed, I urge you to seek out and examine similar examples as they relate to other subject areas.

Although I have now written seven of these introductory SEM books pertinent to particular programs (Byrne, 1989, 1994c, 1998, 2001, 2006, 2009), including the present volume (2011), I must say that each provides its own unique learning experience. Without question, such a project demands seemingly endless time and is certainly not without its frustrations. However, thanks to the incredibly rapid and efficient support provided by Linda Muthén via the Mplus Support Services (http://www.statmodel.com/support), such difficulties were always quickly resolved. In weaving together the textual, graphical, and statistical threads that form the fabric of this book, I hope that I have provided my readers with a comprehensive understanding of basic concepts and applications of SEM, as well as with an extensive working knowledge of the Mplus program. Achievement of this goal has necessarily meant the concomitant juggling of word processing, “grabber”, and statistical programs in order to produce the end result. It has been an incredible editorial journey, but one that has left me feeling truly enriched for having had yet another wonderful learning experience. I can only hope that, as you wend your way through the chapters of this book, you will find the journey to be equally exciting and fulfilling.