# Structural equation model (SEM)

By Soumya Srivastava on March 20, 2018

Structural equation model is a statistical modelling technique. Structural equation model (SEM) tests estimate or establish relationships between variables. It is a multivariate statistical data analysis technique. SEM analyzes the structural relationships or establishes causal relationships between variables.

It has various names like path analysis, latent variable analysis, path modelling, and analysis of covariance structures, causal modelling and linear structural relations (LISREL).

SEM conducts confirmatory modelling rather than exploratory modelling. It tests the validity of the model rather than finding or creating a suitable model. Confirmatory modelling here refers to theory testing. Thus SEM tests a model conceptually derived beforehand from a theory. In other words, SEM tests if the theory fits the data collected.

SEM is a combination of factor analysis and a series of multiple regression analyses. It can simultaneously test both the measurement model and structural relationship specified in the model.

## Basis of structural equation model Covariance

Covariance is the basic statistics of SEM. It is a measure of how much two random variables change together. It presents the strength of association between two variables that is the un-standardized correlation. Here un-standardized is used since it takes into account the standard deviation of two variables. Due to association of covariance with SEM facilitates in:

• Understanding different patterns of correlation among the set of variables in the model.
• Explaining as much variance as possible by the specified model in SEM.

SEM analyzes the covariance matrix of the measures. The covariance matrix also called variance-covariance matrix or dispersion matrix is a matrix that contains the covariance of the element x and element y at the xth and yth position of a random vector in the matrix wherein a random vector is a random variable that has many dimensions.

A covariance matrix is created after estimating the parameters on the model. The matrix is then compared with the covariance matrix of the data collected. In case matrices are consistent with each other, SEM  presents the possible explanation for the relationship between different measures in the model specified.

## Theory fit for model fit

As discussed above, SEM tests the model fit for the derived model. Model is derived from a theory based on a concept. A fit theory creates a fit model. Thus it is necessary to meet but insufficient condition for the theory has to be valid. The condition talks about the theory to be able to reproduce the correlation values that were actually observed in the data. Thus the covariance matrix of the theory (or model) should be equal or similar to the one obtained from the data.

## Structural equation model (SEM) versus Regression analysis

Regression analysis finds the impact of independent variables on the dependent variable. Furthermore, the Structural equation model (SEM) presents a model based on coefficients presenting the relation and association between the independent and the dependent variables. But SEM is better than regression because of following reasons:

• SEM allows multiple dependent variables while regression allows only one dependent variable. This is because it runs a series of multiple regression analysis simultaneously.
• It takes into consideration the error terms as well while regression assumes the measurements to be perfect. Due to this reason, it is a more accurate model.
• SEM correlates the variables. Regression adjusts the variables in the model. In other words, regression controls the variables in the model while it does not control any of the variables in the model to establish or test the relationship between independent and dependent variables.

## 3 prerequisites of structural equation model (SEM)

• Replace the missing values in the data
• Remove the outliers in the data
• Meet the assumptions of normality, linearity, and independence in the data.

## Advantage of structural equation model

The biggest advantage of SEM over other model-driven statistical techniques is that it is capable of simultaneously testing the measurement relationship and structural relationship between the set of variables taken into consideration. The reason for the same is the fact that it is a combination of regression analysis and factor analysis.

## Four uses of structural equation model

• Tests the theory that examines the association or the strength of relationships amongst the variables in the model. Further, it takes into consideration multiple dependent variables in the model along with testing the fit of the model.
• Tests for mediation that is indirect effects.
• Longitudinal tests using longitudinal data.
• Hierarchical or multilevel nested model.

## Three types of structural equation model

• Confirmatory factor analysis.
• Path analysis with observed variables.
• Path analysis with latent variables.

It involved the relation of covariance with SEM and the concept of theory fit. The article also compared and contrasted it with regression analysis and presented the advantages, uses, and types of SEM.