Structural equation modeling

Structural equation modeling (SEM) is a statistical technique for testing and estimating causal relationships using a combination of statistical data and qualitative causal assumptions. This view of SEM was articulated by the geneticist Sewal Wright (1921), the economists Trygve Haavelmo (1943) and Herbert Simon (1953), and formally defined by Judea Pearl (2000) using a calculus of counterfactuals.

SEM encourages confirmatory rather than exploratory modeling; thus, it is suited to theory testing rather than theory development. It usually starts with a hypothesis, represents it as a model, operationalises the constructs of interest with a measurement instrument, and tests the model. The causal assumptions embedded in the model often have falsifiable implications which can be tested against the data. With an accepted theory or otherwise confirmed model, SEM can also be used inductively by specifying the model and using data to estimate the values of free parameters. Often the initial hypothesis requires adjustment in light of model evidence, but SEM is rarely used purely for exploration.

Among its strengths is the ability to model constructs as latent variables (variables which are not measured directly, but are estimated in the model from measured variables which are assumed to 'tap into' the latent variables). This allows the modeler to explicitly capture the unreliability of measurement in the model, which in theory allows the structural relations between latent variables to be accurately estimated.

In SEM, the qualitative causal assumptions are represented by the missing variables in each equation, as well as vanishing covariances among some error terms. These assumptions are testable in experimental studies and must be confirmed judgmentally in observational studies.

An alternative technique for specifying Structural Models using partial least squares has been implemented in software such as LVPLS (Latent Variable Partial Least Square), PLSGraph and SmartPLS (Ringle et al. 2005). Some feel this is better suited to data exploration. More ambitiously, The TETRAD project aims to develop a way to automate the search for possible causal models from data.

Model specification
Since SEM is a confirmatory technique, the model must be specified correctly based on the type of analysis that the modeler is attempting to confirm. There are usually two main parts to SEM: the structural model showing potential causal dependencies between endogenous and exogeneous variables, and the measurement model showing the relations between the latent variables and their indicators. Confirmatory factor analysis models, for example, contain only the measurement part; while path diagrams (to be distinct from linear regression) can be viewed as an SEM that only has the structural part. Specifying the model delineates causal (in fact counterfactual) relationships between variables that are thought to be possible (and therefore want to be 'free' to vary) and those relationships between variables that already have an estimated relationship, which can be gathered from previous studies (these relationships are 'fixed' in the model).

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.

Estimation of free parameters
Parameter estimation is done comparing the actual covariance matrices representing the relationships between variables and the estimated covariance matrices of the best fitting model. This is obtained through numerical maximization of a fit criterion as provided by maximum likelihood, weighted least squares or asymptotically distribution-free methods.

This is best accomplished by using a specialized SEM analysis program, such as SPSS' AMOS, EQS, LISREL, Mplus, Mx, the sem package in R, or SAS PROC CALIS (more information on SAS PROC CALIS: see UCLA or UCR).

Assessment of fit
Using a SEM analysis program, one can compare the estimated matrices representing the relationships between variables in the model to the actual matrices. Formal statistical tests and fit indices have been developed for this purposes. Individual parameters of the model can also be examined within the estimated model in order to see how well the proposed model fits the driving theory. Most, though not all, estimation methods make such tests of the model possible.

However, the model tests are only correct provided that the model is correct. Although this problem exists in all statistical hypothesis tests, its existence in SEM has led to a large body of discussion among SEM experts, leading to a large variety of different recommendations on the precise application of the various fit indices and hypothesis tests.

For each measure of fit, rules of thumb have evolved regarding what represents good fit between model and data. These rules of thumb often need to be updated based on contextual factors such as the sample size, the ratio of indicators to factors, and the overall size of the model. Measures of fit differ in several ways. Some of them reward more parsimonious models (i.e., those with more constrained parameters). Because different measures of fit capture different elements of the fit of the model, it is appropriate to report a selection of different fit measures.

Some of the more commonly used measures of fit include:
 * Chi-square: A fundamental measure of fit used in the calculation of many other fit measures. Conceptually it is a function of the sample size and the difference between the observed covariance matrix and the model covariance matrix.
 * Root Mean Square Error of Approximation (RMSEA)
 * Standardized Root Mean Residual (SRMR)
 * Comparative Fit Index (CFI)

Model modification
The model may need to be modified in order to improve the fit, thereby estimating the most likely relationships between variables. Many programs provide modification indices which report the improvement in fit that results from adding an additional path to the model. Modifications that improve model fit are then flagged as potential changes that can be made to the model. In addition to improvements in model fit, it is important that the modifications also make theoretical sense.

Interpretation and communication
The model is then interpreted and claims about the constructs are made based on the best fitting model.

Caution should always be taken when making claims of causality even when experimentation or time-ordered studies have been done. The term causal model must be understood to mean: "a model that conveys causal assumptions," not necessarily a model that produces validated causal conclusions. Collecting data at multiple time points and using an experimental or quasi-experimental design can help rule out certain rival hypotheses but even a randomized experiment cannot rule out all such threats to causal inference. Good fit by a model consistent with one causal hypothesis does not rule out equally good fit by another model consistent with a different causal hypothesis. However careful research design can help distinguish such rival hypotheses.

Replication and revalidation
All model modifications should be replicated and revalidated before interpreting and communicating the results.

Comparison to other methods
In machine learning, SEM may be viewed as a generalization of Linear-Gaussian Bayesian networks which drops the acyclicality constraint, i.e. which allows causal cycles.

Advanced uses

 * Invariance
 * Multiple group comparison: This is a technique for assessing whether certain aspects of a Structural Equation Model or Confirmatory Factor Analysis is the same across groups (e.g., gender, different cultures, test forms written in different languages, etc).
 * Latent growth modeling
 * Relations to other types of advanced models (multilevel models; item response theory models)
 * Alternative estimation and testing techniques
 * Robust inference
 * Interface with survey estimation
 * Multi-Method Multi-Trait Moddels