Do you want a statistics tool that is powerful; easy to learn; allows you to model complex data structures; combines the t test, analysis of variance, and multiple regression; and puts even more on top? Here it is! Statistics courses in psychology today often cover structural equation modeling (SEM), a statistical tool that allows one to go beyond classical statistical models by combining them and adding more. Let’s explore what this means, what SEM really is, and SEM’s surprising parallels with the hippie culture!

This article might be particularly helpful to those of you who did not learn about SEM who do not remember much. Our aim is to provide an enriching perspective on the not-so-trivial question, “What is SEM?” To tackle this question, we explore major developments in the world of music and statistics in the late 1960s. We review the concept of latent variables and the connected topic of what it actually means to get rid of measurement error, something often pointed out as one of the main strengths of SEM. Finally, we will point out conceptual issues of models including latent variables, and we will see whether it can be “done Bayesian.” The following is our view on some developments in music and statistics; we invite you to leave comments to discuss and add your own enriching views!

History: The Roots of Structural Equation Modeling

Let us start with music. Well before our world reached the stage of rock legends strutting on stage in school uniforms and children’s toys perverted into horrifying metal clowns, in the mid ’60s benign milksop duos were considered devilish rock music and a little raucous singing was considered quite wild. Hard rock, combining the power of the electric guitar with bluesy elements, was emerging at this time also, although it was still limited to rather traditional pop music structures.

Similarly, in statistics, researchers in the mid ’60s could choose from a limited number of major techniques that were useful but quite specific and limited in their scope. First, in the beginnings of the 20th century, Spearman (1904, 1907) had developed factor analytic techniques that he first used to examine his assumption of a general factor of intelligence. This technique allowed reducing a large number of variables to a smaller number of underlying constructs that are represented by nonobserved or latent variables, which we will discuss later. Second, Pearson (1908) and Fisher (1922, 1925) had developed multiple regression and related techniques that allowed estimating the relation between two constructs while controlling for the otherwise biasing effects of confounding variables. Third, Wright (1918, 1921, 1934) had developed path analytic techniques in which causal relations between various variables could be estimated at the same time. In Figure 1, an overview of this status quo in the ’60s is depicted, using simplified path diagram visualizations that are often used to depict SEMs.

Figure 1. A graphical depiction of three major statistical techniques available in the 1960s.

Based on this status quo, thrilling developments were to follow, both in music and in statistics. In the mid- to late ’60s, the hippies finally broke out of traditional pop music structures, and some of the first SEM related techniques were introduced, which would allow researchers to break out of the boundaries of traditional statistical methods.

In music, the hippies took the logical next step and merged blues and rock music with progressive, sometimes substance-induced thinking. This yielded a unifying, more general, and thereby more liberating framework that is known as psychedelic rock. In statistics, similarly but perhaps with lower substance intake, some genius researchers took the logical next step and merged the logics of factor analysis, multiple regression, and path analysis with progressive thinking. This yielded the unifying, more general framework of covariance structure modeling (Jöreskog, 1969, 1970), which is the core method underlying the SEM framework. This method got its name because instead of modeling all individual data points of a dataset, one takes only the variables’ variances and covariances (relations). These are combined into fewer variables that are related to each other, yielding a structural equation model.

To visualize this technique, a “full” structural equation model is depicted in Figure 2. Full means that the model has measurement parts and a structural part. The model combines aspects of all three mentioned classical statistical approaches. In the measurement parts of the model, which translate collected data into psychological constructs represented as latent variables, structures similar to factor analysis are modeled. The structural part of the model depicts how these psychological constructs relate to each other. Here, structures similar to multiple regression and path analysis are modeled.

Figure 2. A depiction of a “full” structural equation model, including three measurement parts (one entangled in red) and a structural part (entangled in green). The latent variables η1 – η3 represent theoretical constructs that are estimated from the covariance matrix of a dataset that consists of y and z variables. Dashed arrows indicate in which parts of the model the three formerly discussed, traditional techniques (factor analysis, multiple regression, path models) are merged and represented.

Boom! It is the end of the rocking ’60s, a technique called covariance structure modeling enters the field and turns out to be a blast, apparently capable of integrating various major approaches!

Intermediate Result: What is SEM?

So far, we have described that SEM is based on covariance structure modeling, which is able to combine the described capabilities of various prior techniques: Large numbers of variables can be reduced to smaller numbers of factors or latent variables. These can be set in complex relations to each other, and at the same time covariates can be controlled for, all in the same model. This allows testing complex hypotheses and theoretical models that involve various constructs within the same statistical model.

There’s more. Apart from these classical techniques, SEM can accommodate a baffling myriad of further traditional and modern types of statistical models. For example, SEM can be used for modeling basics such as variances, means, t tests, and ANOVAs (Green & Thompson, 2012); it is greatly suited for group comparisons (van de Schoot et al., 2012), categorical variables, latent class models (Edelsbrunner et al., 2015), multilevel models (Televantou et al., 2015), and item response theory models (Glockner-Rist & Hoijtink, 2003); and there are various SEM techniques for longitudinal data resembling repeated-measures t tests (Coman et al., 2013) and repeated-measures ANOVAs (Barker, Rancourt, & Velalian, 2014), cross-lagged models (Little, 2011), change-score models (McArdle & Nesselroade, 2014), and growth curve models (Little, 2011). Besides these modeling capabilities, it is often pointed out that SEM offers model fit indices that can help detect statistical misspecifications and theoretical misassumptions (see e.g., Geiser, 2012; Little, 2013). SEM also allows relaxing various strong statistical assumptions of more traditional models that often do not hold, for example by modeling varying variances between groups or over time, correcting standard errors and model fit values for non-normality in the data, including cases with missing data in the analysis, and modeling parameter heterogeneity and multimodality by using multiple group and mixture models (Agan et al., 2015; Hoyle, 2012; Little, 2013).

Knowing these modeling capabilities is useful but does not really answer our main question: “What is SEM?” As mentioned above, SEM is usually based on covariance structure modeling, which offers a convenient framework for modeling latent variables. For arriving at a proper definition, let’s dig into this main characteristic of SEM.

Latent Variables: Chanting away measurement error

Psychological characteristics are not directly measurable. We can ask people questions, we can measure their heart rate or brain activity, we can observe them in social interactions, but the resulting data do not directly tell us about people’s psychological characteristics. Rather, we assume that our data relate to people’s psychological characteristics in specific ways that we try to model. In SEM, we do this by assuming that various variables correlate because they all indicate the level of the same psychological construct.

For example, in the assessment of creativity, people are often asked to suggest many novel and useful ideas about what to do with everyday objects such as a can, a car tire, or a brick (e.g. Benedek et al., 2014). We can count how many ideas people come up with for each of these items. How should we use the resulting numbers to determine how creative people are? The statistician’s answer to this question is the latent variable, the idea of which is depicted in Figure 3.

Figure 3. Depiction of a latent variable (η1) in SEM and how it represents correlations between indicators (y1–y3) and their relations with other variables (z1). Following the paths depicted in grey, it is apparent that the .3-relations (in green) of the y variables with the z variable are modeled and thus summarized by the construct’s correlation with the z1 variable (.42) multiplied with the indicator variables’ loading (.71). The indicator variables’ intercorrelations (.5) are modeled and thus explained through their multiplied factor loadings (.71*.71 = .5).

In Figure 3, a covariance matrix is depicted, showing that people’s number of novel ideas is positively correlated across the three items (variables y1–y3): The more ideas people have for the can, the more ideas they tend to have as well for what to do with the car tire and the brick. Why is this so? The creativity researcher’s assumption is, surprise surprise, because some people are more creative than others! From a psychological point of view, this assumption entails that people’s creativity is the psychological construct underlying their ideas on all three items. Translating this into statistics, creativity is a latent variable that explains the correlations between the three items because it caused the variation underlying them. In the figure, this is indicated by arrows pointing from the latent variable representing creativity toward the variables representing people’s number of ideas on the three items.

Now, let’s see what it means to get rid of measurement error. In the correlation matrix in Figure 3 there is also the variable z1, which represents people’s scores on a questionnaire that assesses openness. People’s openness for new experiences is positively correlated with their creativity: Being open might free people’s minds, and free minds might be open! In the correlation matrix, we see that the three creativity items show correlations of .3 with z1, the openness scores. In SEM, instead of modeling all of these three correlations, we sum up the items’ variance that represents creativity into a latent variable, η1. In this latent variable, all the variance that the three items share is represented. All the variance that the items do not share is assumed to be measurement error, which does not represent people’s creativity but rather assessment artifacts and randomness, and is thus left out of the latent variable. This can be seen in the three paths from the η1 variable to the y variables. These paths are factor loadings that show how strongly each item represents the construct that it is supposed to measure, in our case creativity. The factor loadings are all .71, which is lower than 1, indicating that only parts of the y variables’ variances go into the latent variable.

In Figure 4, there is a depiction of how measurement error variance is cut out of the indicator variables and only the share of variance which supposedly really represents creativity goes into the latent variable. The initial correlation between any of the y variables and z is .30. This is represented as an overlap in the y and z variables’ variances in the upper part of Figure 4. In the latent variable, only a part of any of the y variables’ variances is represented. However, the overlap in variance with the z variable is still the same. Crucially, the three y variables’ correlations with the z variable are now all represented by the latent variables’ correlation with the z variable. However, for the latter two variables, the proportion of overlapping variance is larger. As a result, the correlation that the latent variable has with z1 is estimated to be .42. This is higher than the three y-variables’ individual correlation estimates with the z1 variable, because it is corrected for measurement error, by just cutting it out of the variance cake. This is the reason why you can get rid of measurement error by using SEM. This might look like magic, but it is not; it is a strong theory about a measured construct that has been translated into a SEM.

Figure 4. A variance cake illustration of the process of creating a latent variable. Grab your piece!

If you would like to reproduce the model and results from this example, run the following code in the free R software that you can easily download and install from here :

install.packages("lavaan") # Install lavaan package; only necessary once install.packages("semPlot") # Install semPlot package; only necessary once library(lavaan) # Package for SEM library(semPlot) # Package to draw SEMs # Define correlation matrix lower = ' 1 .5 1 .5 .5 1 .3 .3 .3 1 ' crea.cov = getCov(lower, names = c("y1", "y2", "y3", "z")) # Assign variable names # Define model crea.model = ' # latent variable eta =~ y1 + y2 + y3 # regression/correlation z ~ eta ' # Fit model fit = sem(crea.model, sample.cov = crea.cov, sample.nobs = 100) # Investigate model output summary(fit, standardized = TRUE) # Visualize model semPaths(fit, style = "lisrel", "std", rotation = 3, ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 install . packages ( "lavaan" ) # Install lavaan package; only necessary once install . packages ( "semPlot" ) # Install semPlot package; only necessary once library ( lavaan ) # Package for SEM library ( semPlot ) # Package to draw SEMs # Define correlation matrix lower = ' 1 .5 1 .5 .5 1 .3 .3 .3 1 ' crea . cov = getCov ( lower , names = c ( "y1" , "y2" , "y3" , "z" ) ) # Assign variable names # Define model crea . model = ' # latent variable eta =~ y1 + y2 + y3 # regression/correlation z ~ eta ' # Fit model fit = sem ( crea . model , sample . cov = crea . cov , sample . nobs = 100 ) # Investigate model output summary ( fit , standardized = TRUE ) # Visualize model semPaths ( fit , style = "lisrel" , "std" , rotation = 3 , )

Estimating SEMs consists of finding a set of parameters that fits the data best. For complex SEMs this is achieved by iterative algorithms, for example a maximum likelihood estimator. A once-over into the relations of the estimates to the covariance matrix is provided in Figure 3. The estimation of SEMs can be well understood by applying Wright’s path tracing rules (Wright, 1934; see also Little, 2013). To sum up the estimation, the correlations of the three creativity test items with each other are represented in the estimates of the factor loadings, and all three of the items’ correlations with the z variable are represented in the latent variables’ estimated correlation with the z variable, which is corrected for measurement error through the factor loadings. The application and interpretation of latent variables is not only more complicated than one might at first imagine, but it is also attached to statistical and philosophical topics that have been debated for decades. Let us quickly resume what we discussed so far and then get into some of these issues.

Unification Increasing Freedom: Theoretical Concerns and Bayesian Implementation

Our discussion and example showed that basically any assumptions related to correlations between variables can be modeled based on the principle of covariance structure modeling. This approach can be extended to include variables’ variances and means, to enable the full repertoire of models that SEM can cover. Then, SEM enables psychology students to go beyond the traditional statistical techniques that they usually learn in basic statistics courses: In covariance structure-based modeling, they can implement complex theoretical assumptions without necessarily considering whether their models represent t tests, ANOVAs, or other traditional models that they have learned about. Try SEM and enjoy the increased level of freedom!

Are there any concerns? Indeed, in various psychology-related fields, SEM has superseded more classical regression-related techniques that cannot include the explicit modeling of latent variables. Unfortunately, the great flexibility of SEM and power of latent variables have led to mindless and reductionist modeling habits in many fields. Large numbers of measured variables are often modeled in sparse latent variable models, supposedly to get rid of measurement error and redundant information. However, there are concerns that the necessary statistical or theoretical assumptions might not be adequately met (Goldstein, 1979; Heene, Bollmann, & Bühner, 2014). For example, based on the idea of our good old friend Charles Spearman who invented factor analysis, scores from various intelligence tests are often modeled as a single latent variable that is meant to represent general intelligence. After more than 100 years of intelligence research there is still no convincing basis for assuming this construct (van der Maas et al., 2006). Thus, this modeling habit might lead to erroneous theoretical conclusions, as the information from the individual tests is lost. Similarly, in educational research, both theory and statistics often point toward complex psychological constructs which in the rush of large-scale studies such as PISA are put into single latent variables, potentially inhibiting scientific advance (Edelsbrunner & Dablander, 2015).

While we acknowledge that parsimony is a guiding principle that is often useful in science, we share concerns with researchers emphasizing these and further very strong theoretical and statistical assumptions underlying latent variables. For example, when one models constructs such as creativity as a latent variable, arguably they assume that this psychological characteristic really exists in people’s minds. This assumption however is difficult to test (Borsboom, Mellenberg, & van Heerden, 2003). Also, one should be aware that summarizing information from many indicator variables in one single latent variable necessarily results in loss of the indicator variables’ differential information. Statistically, one usually assumes normal distributions for all observed and latent variables and linear relations between the latent and observed variables. These assumptions are seldom met and arguably quite arbitrary.

Finally, it is an erroneous habit to read out unfounded theoretical conclusions from the mere fitting of SEMs. In general, SEMs do not have the power to reveal great theory. In case one does not have strong theory that supports the application of such high level latent variable models, some alternative methods to the predominant habit of using SEM have been proposed. In those models, direct relations between variables are in the focus without going latent, which might be more supportive for well-founded theory development (Cramer, Waldorp, van der Maas, & Borsboom, 2010; van der Maas et al., 2006). However, this does not imply that SEM is useless; one should just be informed about its strong theoretical prerequisites before engaging in uninformed practices. Strong models demand strong theoretical assumptions!

The statistics hipsters among you might come up with the question, “Is there a way to do it Bayesian?” Bayesian statistics offers a way of estimating parameters and testing hypotheses that overcomes many conceptual problems related to the use of p-value hypothesis testing and frequentist statistics that psychology students usually learn. It is well on the way toward becoming the top statistics framework for many researchers (and even appears in the form of hippie music at conferences), and you can read about it in the JEPS Bulletin here and here. Luckily, SEM is not a family of statistical models connected to either frequentist or Bayesian statistics. You can estimate SEMs in any way you want, and if you want to do it Bayesian, the blavaan package in the free R software offers a comfortable solution. The Mplus software is slightly more complex but also more powerful, and for even more modeling opportunities you can write your own models for the JAGS package in R or in Stan. Bayesian SEM has been recommended for its high flexibility in managing complex data situations (Edelsbrunner & Schneider, 2013; Song & Lee, 2012). Recently, there has been some user-friendly literature on Bayesian SEM (Kaplan & Depaoli, 2012; van de Schoot et al., 2014).

Conclusion: What is SEM?

A little journey through the world of music and statistics in the ’60s and beyond brought us some interesting insights. We learned about expressing musical freedom, about modeling complex theories based purely on correlations between variables, about modeling things we don’t directly measure, about eliminating measurement error from the estimation of interesting parameters, and about the issues emerging from mindless applications of these procedures. We have seen a large number of models that can be accommodated within the framework of SEM.

In music, merging blues and rock with creativity multiplied the possibilities. The musical borders were broken, and any ideas that had to do with rock music instruments could be expressed in psychedelic rock. The similarity to SEM is astonishing: when you break out of specific models and measurement error, any ideas that have to do with covariances, variances, means, and latent variables can be expressed in structural equation modeling. Cheers to musical and statistical freedom!

Many definitions of SEM just mention some of the models and possibilities that SEM offers. For the conclusion of this article, we acknowledge the dazzling multitude of SEM related models and opportunities to arrive at a quite general definition that we find appealing: “Structural Equation Modeling (SEM) is a comprehensive statistical approach to testing hypothesis about relations among observed and latent variables” (Hoyle, 2012). For this means, we believe it a framework that will stick around and enrich our research lives for many years to come.

Suggested Readings

Software based introductions to SEM:

Geiser, C. (2012). Data analysis with Mplus. Guilford Press.

Beaujean, A. A. (2014). Latent variable modeling using R: A step-by-step guide. Routledge.

Advanced and longitudinal SEM:

Little, P. T. D. (2013). Longitudinal structural equation modeling. Guilford Press.

McArdle, J. J., & Nesselroade, J. R. (2014). Longitudinal data analysis using structural equation models. American Psychological Association.

Theoretical background of latent variables:

Borsboom, D. (2008). Latent variable theory. Measurement: Interdisciplinary Research & Perspective, 6, 25–53. http://doi.org/10.1080/15366360802035497

Borsboom, D., Mellenbergh, G. J., & van Heerden, J. (2003). The theoretical status of latent variables. Psychological Review, 110, 203–219. http://doi.org/10.1037/0033-295X.110.2.203

More history and background of SEM?

Skrondal, A., & Rabe-Hesketh, S. (2007). Latent variable modelling: A survey. Scandinavian Journal of Statistics, 34, 712-745.

Matsueda, R. L., & Press, G. (2012). Key advances in the history of structural equation modeling. In: Hoyle, R. (Ed.) Handbook of structural equation modeling (pp. 17-42). Guilford New York, NY.

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Barker, D. H., Rancourt, D., & Jelalian, E. (2014). Flexible models of change: Using structural equations to match statistical and theoretical models of multiple change processes. Journal of Pediatric Psychology, 39, 233–245.

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About the authors

Peter Edelsbrunner is currently doctoral student at the section for research on learning and instruction of ETH Zurich in Switzerland. He is a graduate of the Department of Psychology of the University of Graz in Austria. Peter is interested in conceptual knowledge development and the application of flexible mixture models to developmental research. Since 2011 he has been active in the Junior Researcher Programme and related activities.

Christian Thurn is studying psychology in the beautiful city of Konstanz by Lake Constance, Germany. He met Peter Edelsbrunner in his internship at the section for research on learning and instruction of ETH Zurich and supported him in writing this blog article. Currently, he is planning his bachelor’s thesis on group decisions in economic situations. He is tutor for statistics and a big fan of the R statistics software.

This post was edited by Jonas Haslbeck and Altan Orhon.

Peter Edelsbrunner Peter is currently doctoral student at the section for learning and instruction research of ETH Zurich in Switzerland. He graduated from Psychology at the University of Graz in Austria. Peter is interested in conceptual knowledge development and the application of flexible mixture models to developmental research. Since 2011 he has been active in the EFPSA European Summer School and related activities. More Posts