# Structural Equation Modeling: Definition and Analysis

Bonani Bose 8 Min Read

Structural equation modeling is a statistical technique for testing and estimating causal relations using a combination of statistical data and qualitative causal assumptions.

This definition of SEM was articulated by the geneticist Sewall Wright, the economist Trygve Haavelmo and the cognitive scientist Herbert A. Simon, and formally defined by Judea Pearl using a calculus of counterfactuals.

**According to Hoyle, Structural equation modeling may also be explained as a comprehensive statistical approach to testing hypotheses exploring relations between observed and latent variables. It is a methodology for representing, estimating, and testing a theoretical network of (mostly) linear relations between variables (Rigdon, 1998).**

Structural Equation Modeling Has two main goals:

(i) To understand the patterns of correlation/covariance among a set of variables and

(ii) To explain as much of their variance as possible with the model specified

**Structural Equation Modelling: Definition**

Structural equation modeling may also be defined as a multivariate statistical analysis technique that is used for analyzing structural relationships. This technique may better be explained as a combination of factor analysis and multiple regression analysis.

Structural Equation Modelling is used to analyze the structural relationship between measured variables and latent constructs. Largely preferred by the researchers Structural Equation Modelling estimates the multiple and interrelated dependence in a single analysis.

To explain in simpler words, two types of variables are used: endogenous variables and exogenous variables. Endogenous variables are equivalent to dependent variables and are equal to the independent variable.

Structural equation models are inclusive of both confirmatory and exploratory modeling. Confirmatory modeling usually starts out with a hypothesis that gets represented in a causal model. The concepts used in the model must then be operationalized to allow testing of the relationships between the concepts in the model.

**Structural Equation Modeling Examples**

Structural Equation Modeling Examples can better be explained with Structural Equation models (SEM). Structural equation models are a subset of graphical models.

Each Structural equation model is associated with a graph that represents the causal structure of the model and the form of the linear equations, where there is a directed edge from X to Y (X→Y) if the coefficient of X in the structural equation for Y is nonzero (i.e., X is a direct cause of Y), and there is a bi-directed edge between the error terms ɛX and ɛY if and only if the covariance between the error terms is nonzero.

In path diagrams, latent substantive variables are enclosed in ovals, and measured variables are enclosed in rectangles.

In the structural modeling, theoretical latent constructs are represented with ovals or circles and their measured empirical indicators are represented with rectangles. The lines connecting indicators to constructs and constructs to each other carry numerical values that quantify the degree of covariation accounted for by the model components.

Whereas some investigators claim that Structural Equation Modeling explains variation among latent variables; we may explain in a better way by saying that SEMs predict or account for variation among model components.

Structural Equation Models are incapable of providing causal mechanism information because they were not designed to do so. The same is true for the path and regression models.

Structural Modeling, path and regression23 models cannot specify the required causal mechanism information because of the following reasons:

(a) They are all based on correlations that cannot prove causation;

(b) They cannot explain how and why the variances and covariances have their observed values. They are silent with regard to the causal sequences that produced the observed variances and covariances;

(c) They cannot specify the necessary and/or sufficient conditions for how their identified effects occur;

(d) Even if they identified necessary and sufficient conditions, statistical models cannot explain why those conditions and not others are necessary and/or sufficient.

Statistical moderators are not mechanisms. Moderators and mediators are not mechanisms. Statistical models are several times removed from explaining psychological phenomena via physical mechanisms.

In simpler terms, we may say that statistical models do not identify sequences of causal events that are either necessary or sufficient to bring about the imputed result, they do not clarify the ‘process by which something takes place or is brought about’ and therefore they do not provide proximal causal mechanism information.

Structural equation models are also restricted by the ever-present possibility of model misspecification. This means the alternative models might fit, and predict, as well or better than the advocated model.

**Structural Equation Model Types**

Structural Modeling falls into four broad categories. These structural equation models are Path Analysis, Latent Variable Structural Model, Growth Curve Model, and Latent Growth Model.

**1. Path Analysis**

Path Analysis, one of the major structural equation models in use is the application of structural equation modeling without latent variables.

The best part about Path Analysis is that it includes the relationships among variables that serve as predictors in one single model. A classic example of this may be a mediation model.

Path Analysis, another Structural Equation Model type, is an extension of the regression model. In a path analysis model from the correlation matrix, two or more casual models are compared. The path of the model is shown by a square and an arrow, which shows the causation. Regression weight is predicted by the model.

**2. Confirmatory Factor Analysis**

Confirmatory Factor Analysis, also known as CFA, is a way ahead of data reduction. CFA is also known within SEM as the measurement model because is the step taken to determine how the factors (ε1 and ε1) are measured by the indicators (x1 to x8).

The Confirmatory Factor Model in SEM treats intelligence as a latent variable which can be measured on the basis of test scores in four areas: reading, writing, math, and analysis.

This Structural Equation Modelling example shows how to estimate a confirmatory factor model. The primary benefit of Structural Modeling is that it is entirely free, requiring an R installation only.

Here is a simple path diagram of a two-factor CFA:

**3. Latent Variable Structural Model**

The next structural equation model for analysis is the Latent Variable Structural Model.

The next step is to fit the structural model, which is what you probably think of when you hear about SEM. It is mainly using the measured latent variables within the path analysis framework. Once you have declared the latent variables you can hypothesize and test their relationships.

**4. Growth Curve Models**

Another popular use of Structural Equation Modeling is longitudinal models, commonly referred to as Growth Curve Models. Let’s say for instance you have multiple observations of the same variable over time, you may declare an intercept and a slope for the subjects ‘papers over time as latent variables by constraining the path coefficients in a specific way.

Since the paths are constrained, we have to estimate on growth curve modeling the means of the latent variables. These means give us the overall intercept and the overall slope across all subjects.

Latent Growth Curve Models are related to and offer an alternative means of running Mixed Models on longitudinal data. These mixed models are often known as Individual Growth Curve Models.

Has my post sparked an interest in you about structural equation modeling? Does researching about structural equation modeling intrigue you? Then you must go for a career in structural equation modeling research.

With a thorough knowledge of structural equation modeling, you will be able to explore the connectedness of data through SEMs with the R programming language using the Lavaan package.

Structural Equation Modeling will also introduce you to latent and manifest variables and how to create measurement models, assess measurement model accuracy, and fix poor fitting models. You will also learn to discover classic SEM datasets, such as the Holzinger and Swineford.

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