This book presents an introduction to structural equation modeling (SEM) and facilitates the access of students and researchers in various scientific fields to this powerful statistical tool. It offers a didactic initiation to SEM as well as to the open-source software, lavaan, and the rich and comprehensive technical features it offers.
Structural Equation Modeling with lavaan thus helps the reader to gain autonomy in the use of SEM to test path models and dyadic models, perform confirmatory factor analyses and estimate more complex models such as general structural models with latent variables and latent growth models.
SEM is approached both from the point of view of its process (i.e. the different stages of its use) and from the point of view of its product (i.e. the results it generates and their reading).

# Introduction

"The time of disjointed and mobile hypotheses is long past, as is the time of isolated and curious experiments.

From now on, the hypothesis is synthesis."

Gaston Bachelard *Le Nouvel Esprit scientifique*, 1934

"There is science only when there is measurement."

Henri Piéron, 1975

Predicting and explaining phenomena based on non-experimental observations is a major methodological and epistemological challenge for social sciences. Going from a purely descriptive approach to an explanatory approach requires a suitable, sound theoretical corpus as well as appropriate methodological and statistical tools. "It is because it is part of an already outlined perspective that structural equation modeling constitutes an important step in the methodological and epistemological evolution of psychology, and not just one of the all too frequent fads in the history of our discipline", wrote Reuchlin ([REU 95] p. 212). The point aptly described by Reuchlin applies to several other disciplines in the human and social sciences.

Since its beginning, there have been two types of major actors who have worked with structural equation modeling (SEM), sometimes in parallel to its development: those who were/are in a process of demanding, thorough, and innovative application of the method to their field of study, and those who were/are in a process of development and refinement of the method itself. The second group is usually comprised of statisticians (mathematicians, psychology psychostatisticians, etc.), whereas the first group is usually comprised of data analysts. While willingly categorizing themselves as data analysts, the authors of this book recognize the importance of the statistical prerequisites necessary for the demanding and efficient use of any data analysis tool.

This manual is a didactic book presenting the basics of a technique for beginners who wish to gradually learn structural equation modeling and make use of its flexibility, opportunities, and upgrades and extenxions. And it is by putting ourselves in the shoes of a user with a limited statistical background that we have undertaken this task. We also thought of those who are angry with statistics, and who, more than others, might be swayed by the beautiful diagrams and goodness-of-fit indices that abound in the world of SEM. We would be proud if they considered the undoubtedly partial introduction to SEM we give here as insufficient. As for those who find the use of mathematics in humain and social sciences unappealing, those who have never been convinced by the utility of quantitative methods in these sciences, it is likely that, no matter what we do, they will remain so forever. This manual will not concern them. It is hardly useful to focus on the fact that using these methods does not mean conceiving the social world or psychological phenomena as necessarily computable and mathematically formalizable systems. Such a debate has lost many a time to the epistemological and methodological evolution of these sciences. This debate is pointless.

In fact, computer software has made it possible to present a quantitative method in a reasonably light way in mathematical formulas and details. Currently, it is no longer justifiable to present statistical analysis pushed up to its concrete calculation mode, like when these calculations were by hand in the worst cases or done with the help of a simple calculator in the best cases. But the risk of almost mechanically using such programs, which often gives users the impression they are exempted from knowing the basics of technical methods and tests that they use, is quite real.

We have tried to limit this risk by avoiding making this book a simple SEM software user's guide. While recognizing the importance of prerequisite statistics essential to a demanding and efficient use of any data analysis tool, we reassure readers: less is more. Let us be clear from the outset that our point of view in this book is both methodological and practical and that we do not claim to offer a compendium of procedures for detailed calculations of SEM. We have put ourselves in the shoes of the user wishing to easily find in it both a technical introduction and a practical introduction, oriented towards the use of SEM. It is not a "recipe" book for using SEM that leads to results that are not sufficiently accurate and supported. Implementing it is difficult, as it also involves handling SEM software, thus following the logic of a user's guide.

In the first chapter following this introduction, the founding and fundamental concepts are introduced and the principle and basic conventions are presented and illustrated with simple examples. The nature of the approach is clearly explained. It is a confirmatory approach: first, the model is specified, and then tested. Handling the easy-to-learn lavaan software constitutes the content of the second chapter. Developed by Rosseel [ROS 12], the open-source lavaan package has all of the main features of commercial SEM software, despite it being relatively new (it is still in its beta version, meaning that is still in the test and construction phase). It has a remarkable ease of use.

Chapter 3 of this manual presents the main steps involved in putting a structural equation model to the test. Structural equation modeling is addressed both from the point of view of its process, that is, the different steps in its use, as well as from the point of view of its product, that is, the results it generates and their reading. Also, different structural equation models are presented and illustrated with the lavaan syntax and evaluation of the output: path models analysis and the Actor-Partner Interdependence Model (APIM). Similarly, the two constituent parts of a structural general equation model are detailed: the measurement model and the structural model. Here again, illustrations using the lavaan syntax and evaluation of the output make it possible for the reader to understand both the model and the software.

Any model is a lie as long as its convergence with the data has not been confirmed. But a model that fits the data well does not mean that it represents the truth (or that it is the only correct model, see equivalent models); it is simply a good approximation of reality, and hence a reasonable explanation of tendencies shown by our data. Allais [ALL 54] was right in writing that "*for any given level of approximation, the best scientific model is the one which is most appropriate* [italicized by the author]. In this sense, there are as many true theories as given degrees of approximation" (p. 59). Whatever it may be, and more than ever, the replication of a model and its cross-validation are required.

The fourth chapter is dedicated to what has been called "the more or less recent extensions of SEM". Here, the term "extensions" means advances and progress, because the approach remains the same, regardless of the level of complexity of the specified models and the underlying degree of theoretical elaboration. The aim here is to show the use of the power and flexibility of SEM through some examples. Its potential is immense and its opportunities multiple. Its promises are rich and exciting. However, it was not possible to go through them all. It seemed wise to focus on those that are becoming unavoidable. Moreover, some analyses have become so common that they could cease to be seen as a mere extension of basic equation models. One can think of multigroup analyses that offer the possibility to test the invariance of a model through populations, thus establishing the validity, or even universality, of the theoretical construct of which it is the representation. Latent state-trait models, which refer to a set of models designed and intended to examine stability of a construct over time (temporal), are more recent, and it is to them that we have dedicated a chapter that is both technical and practical. Finally, latent growth models that find their place in longitudinal, rare, and valuable data never cease to be of interest to researchers. Using them with the help of models combining covariance structure analysis and mean structure modeling is one of the recent advances in SEM.

We suggest that the reader acquires a progressive, technical introduction to begin with by installing the free software lavaan with no further delay. The second chapter of this book will help in getting started with this software. It is in the reader's interest to follow step-by-step the treatment of data in the book in order to replicate the models presented, and not move to the next step until they get the same results. These data will be available on a website dedicated to this manual.

We started this introduction by paraphrasing Reuchlin, We would like to conclude our introduction citing Reuchlin once again when he accurately said that SEM "are tools whose usage is not possible, it is true, unless there is some knowledge and some psychological hypotheses about the functioning of the behaviors being studied. It would be paradoxical for psychologists to consider this constraint as a disadvantage" [REU 95]. One could even say that they would be wrong to consider it in this way. And Hair, Babin, and Krey [HAI 17], marketing and advertising specialists, would agree with Reuchlin. In fact, in a recent literature review examining the use of SEM in articles published in the *Journal of Advertising* since its first issue in 1972, these authors acknowledge that the attractiveness of structural equation modeling among researchers and advertisers can be attributed to the fact that the method is proving...