Structural Equation Modeling with lavaan

By Kamel Gana and Guillaume Broc

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).

• Language: English
• Publisher: Wiley
• Release date: Dec 31, 2018
• ISBN: 9781119578994

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Preface

The core content of this book was written 20 years ago, when I began giving my first courses on structural equation modeling at the François Rabelais University in Tours, France. Having never been left pending, the manuscript has been constantly updated for the needs of my university courses and numerous introductory workshops to SEM that I was conducting in various foreign universities. These courses and workshops were both an introduction to the statistical tool and an adoption of a software without which this tool would be obscure, abstract and disembodied. To put it directly and bluntly, any introduction to structural equation modeling (SEM) compulsorily includes adopting a SEM software. Among LISREL, Amos, EQS, Mplus Sepath/Statistica, and Calis/SAS, there is no dearth of options. These commercial programs no doubt helped in demystifying structural equation modeling and have thus given it an actual popularity that continues to grow.

Writing a book, in this case a practical handbook of structural equation modeling, requires introducing one or more of these commercial software that are admittedly quite expensive. That is where the problem lies. Not that they do not deserve it, but picking one is inevitably advertising that software. I cannot and will not consent to this. Moreover, access to these commercial software remains, for many students and young researchers, an obstacle and constraint that is often insurmountable. I have often experienced the challenge of teaching SEM in African universities where it was impossible to have SEM commercial software. I happened to use a restricted student version of a commercial program to demonstrate in my class.

When R, free open-source software, was developed, the situation changed. R is made of packages dedicated to all kinds of data analysis and processing. The lavaan package, provided by Rosseel [ROS 12], is dedicated to SEM. It achieved immediate success as it has all the features proposed by commercial software, and it offers such a disconcerting ease of use. As with any statistical tool, practice remains the best way to master SEM. There is no better way to do this than by having software at hand. R and lavaan have changed our way of teaching statistical tools as well as the way in which students can become familiarized with, adapt, and use them. Our level of demand on them changes as the students' view of these tools evolves. Understanding, learning, and especially practicing without any limits (apart from that of having a computer): this is what R and its lavaan package offer to students. This book certainly contributes to it.

Without the impetus and the decisive and meaningful contribution of Guillaume Broc, author of the book Stats faciles avec R (De Boeck), this manual would not have seen the light of day. We share the following belief: access to science and its tools must become popular and be available to everyone. We think that this manual, devoted to structural equation modeling with lavaan, fully contributes to this purpose.

It is because this book aims to be a didactic handbook and a practical introduction to SEM meant for students and users who do not necessarily need complex mathematical formulae to adopt this tool and be able to use it wisely, that we submitted the first draft to some novice students in SEM in order to assess its clarity and comprehensibility. Their careful reading and their judicious and pertinent comments allowed for a substantial improvement of this manual. They are very much thanked for this. In fact, they are fully involved in this project. However, we retain and accept full responsibility for mistakes, shortcomings, or inadequacies that may be present in this manual.

Kamel GANA

October 2018

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 to be an excellent tool for testing advertising theories, and they admit bluntly that the increasing use of structural equation modeling in scientific research in advertising has contributed substantially to conceptual, empirical, and methodological advances in the science of advertising.

Is it not an epistemological evolution necessary to any science worthy of the name to go from the descriptive to the explanatory? This is obviously valid for a multitude of disciplines where SEM is already used: agronomy, ecology, economy, management, psycho-epidemiology, education sciences, sociology, etc.

1

Structural Equation Modeling

Structural Equation Modeling (SEM) is a comprehensive and flexible approach that consists of studying, in a hypothetical model, the relationships between variables, whether they are measured or latent, meaning not directly observable, like any psychological construct (for example, intelligence, satisfaction, hope, trust¹). Comprehensive, because it is a multivariate analysis method that combines the inputs from factor analysis and that of methods based or derived from multiple regression analysis methods and canonical analysis [BAG 81, KNA 78]. Flexible, because it is a technique that allows not only to identify the direct and indirect effects between variables, but also to estimate the parameters of varied and complex models including latent variable means.

Mainly of a correlational nature, structural models are both linear statistical models, whose normal distribution of variables is almost necessary, and statistical models in the sense that the error terms are considered to be partly related to the endogenous variables (meaning predicted). We say almost necessary because the success of structural equation modeling is such that its application extends, certainly with risks of error, to data obtained through categorical variables (ordinal or even dichotomous) and/or by clearly violating the multivariate normal distribution. Considerable mathematical advances (like the so-called robust estimation methods) have helped currently minimize these risks by providing some remedies to the non-normality of the distribution of variables and the use of data collected by the means of measurement scales other than that normally required for structural equation models, namely interval scales [YUA 00]. We will discuss more on that later.

Our first goal is to introduce the reader to the use of structural equation models and understand their underlying logic; we will not delve too much into mathematical and technical details. Here, we will restrict ourselves to introducing, by way of a reminder, the concepts of correlation, multiple regression, and factor analysis, of which structural equation modeling is both a summary and a generalization. We will provide to the reader some details about the concept of normality of distribution, meaning with linearity, a basic postulate of structural equation modeling. The reader will find the mathematical details concerning the basic concepts briefly recalled here in any basic statistical manual.

1.1. Basic concepts

1.1.1. Covariance and bivariate correlation

Both covariance and correlation measure the linear relationship between two variables. For example, they make it possible to learn about the relationship between two items of a test or a measure (e.g., a questionnaire) scale. Figure 1.1 provides a graphic illustration of the same.

Figure 1.1. Covariance/correlation between two variables (the small curved left-right arrow indicates the variance)

Covariance, which measures the variance of a variable with respect to another (covariance), is obtained as follows:

[1.1]

where:

– M = mean;

– N = sample size.

Being the dispersion around the mean, the variance is obtained as follows:

[1.2]

The values of a covariance have no limits. Only, it should be noted that the positive values of covariance indicate that values greater than the mean of a variable are associated with values greater than the mean of the other variable and the values lesser than the mean are associated in a similar way. Negative covariance values indicate values greater than the mean of a variable are associated with values lesser than the mean of the other variable.

Unlike covariance, correlation measures such a relationship after changing the original units of measurement of variables. This change, called standardization or normalization, involves centering-reducing (i.e. M = 0.00, standard deviation = 1.00) a variable (X) by transforming its raw score into z score:

[1.3]

where:

– M = mean of X;

– σ = standard deviation of X.

The standard deviation is simply the square root of the variance:

[1.4]

Remember that standard deviation is the index of dispersion around the mean expressing the lesser or higher heterogeneity of the data. Although standard deviation may not give details about the value of scores, it is expressed in the same unit as these. Thus, if the distribution concerns age in years, the standard deviation will also be expressed in the number of years.

The correlation between standardized variables X and Y (ZX and ZY, [1.3]) is obtained as follows:

[1.5]

Easier to interpret than covariance, correlation, represented, among others, by the Bravais-Pearson coefficient r, makes it possible to estimate the magnitude of the linear relationship between two variables. This relationship tells us what information of the values of a variable (X) provides information on the corresponding values of the other variable (Y). For example, when X takes larger and larger values, what does Y do? We can distinguish between the two levels of responses. First, the direction of the relationship: when the variable X increases, if the associated values of the variable Y tend overall to increase, the correlation is said to be positive. On the other hand, when X increases, if the associated values of Y overall tend to reduce, the correlation is called negative. Second, the strength of the association: if information of X accurately determines information of Y, the correlation is perfect. It corresponds respectively to + 1 or – 1 in our demonstration. In this case, participants have a completely identical way of responding to two variables. If information of X does not give any indication about values assumed to be associated with Y, there is complete independence between the two variables. The correlation is then said to be null. Thus, the correlation coefficient varies in absolute value between 0.00 and 1.00. The more it is closer to + 1.00 or – 1.00, the more it indicates the presence of a linear relationship, which can be represented by a straight line in the form of a diagonal². On the other hand, more the coefficient goes to 0, the more it indicates the lack of a linear relationship. A correlation is considered as being significant when there is a small probability (preferably less than 5%) so that the relationship between the two variables is due to chance.

As much as the covariance matrix contains information about the relationships between the two measures (scores) and their variability within a given sample, it does not allow for comparing, unlike a correlation matrix, the strength of the relationships between the pairs of variables. The difference between these two statistics is not trivial as the discussions on the consequences of the use of one or the other on the results of the analysis of structural equation models are to be taken seriously. We will discuss this further.

Furthermore, it is worth remembering that there are other types of correlation coefficients than the one that we just saw. In structural equation modeling, the use of tetrachoric and polychoric correlation coefficients is widespread, as they are suitable for measurements other than those of interval-levels. The first is used to estimate the association between two variables, called dichotomous; the second is used when there are two ordinal-level variables.

1.1.2. Partial correlation

The correspondence between two variables may result from various conditions that the calculation of correlation cannot always detect. Thus, assuming that only the welloff have the financial means to buy chocolate in a given country, even a very strong correlation observed between the two variables – consumption of chocolate and life satisfaction – in older people does not mean that the first is the cause of the second. First, we are right in thinking of the opposite, as a statistic expressed by a correlation can be read in two ways. In addition, we can think that these assumptions are all erroneous, and that it would perhaps come from a common cause, the milieu that similarly determines the two variables, which, due to this fact, prove to be correlated. In this case, life satisfaction in a category of the elderly does not come from consuming chocolate, but from the preferred milieu in which they live; it is this milieu that allows them to both consume chocolate and be happy. Here, we see a spurious (artificial) relationship that we will shed light on through the following illustration. Let us consider that the correlation matrix between these three variables measured from a sample of elderly people (Table 1.1).

Table 1.1. Correlation matrix (N = 101)

The use of partial correlations is useful here, as it will allow us to estimate the relationship between X and Y controlling for Z that will be hold constant. This correlation is written in this way, rXY.Z, and is calculated as follows:

[1.6]

Considering the numerator of this equation, we can observe how the milieu variable (Z) was hold constant: we simply removed the two remaining relationships from the relationship between chocolate consumption (X) and life satisfaction (Y), namely (rXZ) and (rYZ). If we apply this formula to the data available in the matrix [1.7], the partial correlation between X and Y by controlling Z will be as follows:

[1.7]

We realize that by controlling for the variable milieu, the relationship between chocolate consumption and life satisfaction fades, as it is likely an artificial one. The milieu takes the place of confounding factor, giving the relationship between chocolate consumption and life satisfaction an artificial nature, meaning spurious.

To conclude, we put emphasis on the fact that it is often unwise to interpret the correlation or partial correlation in terms of causality. Mathematics cannot tell us about the nature of the relationship between two variables. It can only tell us to what extent the latter tend to vary simultaneously. In addition, the amplitude of a link between two variables may be affected by, among other things, the nature of this relationship (i.e. linear or non-linear), the normality of their distribution, and psychometric qualities (reliability, validity) of their measures.

As for the causality, it requires three criteria (or conditions): 1) the association rule, that is the two variables must be statistically associated; 2) the causal order between variables, the (quite often) temporal order where the cause precedes the effect must be determined without ambiguity and definitely with theoretical reasons that allow for assuming the order; 3) the non-artificiality rule, in which the association