# The form of structural equation models

Latent Constructs In structural equation modeling, the key variables of interest are usually "latent constructs"--abstract psychological concepts such as "intelligence" or "attitude. A structural equation model may include two types of latent constructs--exogenous and endogenous. History[ edit ] Structural equation modeling, as the term is currently used in sociology, psychology, and other social sciences evolved from the earlier methods in genetic path modeling of Sewall Wright. Their modern forms came about with computer intensive implementations in the s and s.

SEM evolved in three different streams: Much of this development occurred at a time that automated computing was offering substantial upgrades over the existing calculator and analogue computing methods available, themselves products of the proliferation of office equipment innovations in the late 20th century.

The text Structural Equation Modeling: From Paths to Networks provides a history of the methods.

## Structural Equation Models

In particular, PLS-PA the Lohmoller algorithm has been conflated with partial least squares regression PLSR, which is a substitute for ordinary least squares regression and has nothing to do with path analysis.

PLS-PA has been falsely promoted as a method that works with small datasets when other estimation approaches fail. Westland decisively showed this not to be true and developed an algorithm for sample sizes in SEM.

Anderson and Rubindeveloped the limited information maximum likelihood estimator for the parameters of a single structural equation, which indirectly included the two-stage least squares estimator and its asymptotic distribution Anderson, and Farebrother Two-stage least squares was originally proposed as a method of estimating the parameters of a single structural equation in a system of linear simultaneous equations, being introduced by Theil a, b, and more or less independently by Basmann and Sargan Of these, two-stage least squares was by far the most widely used method in the s and the early s.

Systems of regression equation approaches were developed at the Cowles Commission from the s on, extending the transportation modeling of Tjalling Koopmans. Sewall Wright and other statisticians attempted to promote path analysis methods at Cowles then at the University of Chicago.

Freedman summarized these objections in path analyses: Advances in computers made it simple for novices to apply structural equation methods in the computer-intensive analysis of large datasets in complex, unstructured problems.

The most popular solution techniques fall into three classes of algorithms: Pearl  has extended SEM from linear to nonparametric models, and proposed causal and counterfactual interpretations of the equations.

For example, excluding a variable Z from the arguments of an equation asserts that the dependent variable is independent of interventions on the excluded variable, once we hold constant the remaining arguments. Nonparametric SEMs permit the estimation of total, direct and indirect effects without making any commitment to the form of the equations or to the distributions of the error terms.

This extends mediation analysis to systems involving categorical variables in the presence of nonlinear interactions. Bollen and Pearl  survey the history of the causal interpretation of SEM and why it has become a source of confusions and controversies.

SEM path analysis methods are popular in the social sciences because of their accessibility; packaged computer programs allow researchers to obtain results without the inconvenience of understanding experimental design and control, effect and sample sizes, and numerous other factors that are part of good research design.

Direction in the directed network models of SEM arises from presumed cause-effect assumptions made about reality. Social interactions and artifacts are often epiphenomena — secondary phenomena that are difficult to directly link to causal factors. An example of a physiological epiphenomenon is, for example, time to complete a meter sprint. A person may be able to improve their sprint speed from 12 seconds to 11 seconds, but it will be difficult to attribute that improvement to any direct causal factors, like diet, attitude, weather, etc.

The 1 second improvement in sprint time is an epiphenomenon — the holistic product of interaction of many individual factors. Model specification[ edit ] Two main components of models are distinguished in SEM: Exploratory and confirmatory factor analysis models, for example, contain only the measurement part, while path diagrams can be viewed as SEMs that contain only the structural part.

In specifying pathways in a model, the modeler can posit two types of relationships: A modeler will often specify a set of theoretically plausible models in order to assess whether the model proposed is the best of the set of possible models.

Not only must the modeler account for the theoretical reasons for building the model as it is, but the modeler must also take into account the number of data points and the number of parameters that the model must estimate to identify the model. An identified model is a model where a specific parameter value uniquely identifies the model, and no other equivalent formulation can be given by a different parameter value.

A data point is a variable with observed scores, like a variable containing the scores on a question or the number of times respondents buy a car. The parameter is the value of interest, which might be a regression coefficient between the exogenous and the endogenous variable or the factor loading regression coefficient between an indicator and its factor.

## Structural Equation Models

If there are fewer data points than the number of estimated parameters, the resulting model is "unidentified", since there are too few reference points to account for all the variance in the model.

The solution is to constrain one of the paths to zero, which means that it is no longer part of the model.

Estimation of free parameters[ edit ] Parameter estimation is done by comparing the actual covariance matrices representing the relationships between variables and the estimated covariance matrices of the best fitting model.It is increasingly common to test hypotheses combining moderation and mediation.

Structural equation modeling (SEM) has been the favored approach to testing mediation hypotheses. However, the biggest challenge to testing moderation hypotheses in SEM was the complexity underlying the modeling of latent variable interactions.

We discuss the latent moderated structural equation procedure (LMS. in a retrieval system, or transcribed, in any form or by any means—electronic, mechanical, photocopy, recording, or Tour of models 61 [SEM] Stata Structural Equation Modeling Reference Manual [SVY] Stata Survey Data Reference Manual [ST].

May 07,  · STRUCTURAL VERSUS REDUCED FORM MODELS: A NEW INFORMATION BASED PERSPECTIVE structural and reduced form. Structural models originated with Black and Scholes (), Merton Section 3 reviews structural models, and Section 4 reviews reduced form models.

Section 5 links the. The structural equation for a substantive variable Xi is a linear equation with Xi on the left-hand side of that represents the causal structure of the model and the form of the linear has focused on the development of alternative indices that provide relatively different perspectives on the fit of structural equation models.

The. •the ‘lavaan model syntax’ allows users to express their models in a compact, a regression formula has the following form: y ~ x1 + x2 + x3 + x4 •in lavaan, a typical model is simply a set (or system) of regression formulas, an R package for structural equation modeling and more14 / correlation structure models, which hypothesize that a correlation matrix has a particular form.

Most structural equation models can be expressed as path diagrams. Consequently even beginners to structural modeling can perform complicated analyses with a minimum of training.

The Form of Structural Equation Models