# Chapter 4: Structural Equation Models

### Model selection and model assessment for SEMs

An identified structural equation model can be estimated using the same principles as for factor analysis models: the SEM implies also a model for the joint distribution of the observed variables, and the estimated parameters of the fitted model are chosen so as to give the best match between the model-implied distribution and the observed distribution of the sample data. Most often this is done using the method of maximum likelihood (ML) estimation. Examples of the computer commands for fitting SEMs are given in Examples 1 and 2 of this chapter.

For model selection for SEMs, we recommend the following two-step approach:

1. Identify a sufficiently well-fitting and interpretable measurement model separately for each set of observed indicators which you expect on theoretical grounds to be measures of one or more common factors. In other words, indicators which are definitely regarded as measures of different constructs, and which you would never consider combining together in one summary measure of anything, are examined separately at this stage. This step is carried out using methods of model assessment for factor analysis models, as discussed in Chapter 2.
2. Using the forms of the measurement models identified in Step 1 (i.e. fixing their patterns of zero and non-zero factors loadings to specify which indicators measure which factors), fit full structural equation models to estimate the parameters of both the measurement and structural models. Use likelihood ratio tests or z-tests of coefficients to compare different specifications for the structural model, in particular to test whether some paths (regression coefficients) in this model may be set to 0.

This approach avoids the common but rather unhelpful approach of using the kinds of overall goodness of fit statistics which were discussed in Chapter 2 (RMSEAs, CFIs, overall goodness of fit statistics, and so on) to assess the fit of the whole SEM at once. The problem with this approach is that if the model already includes all structural paths and adequate measurement models for all sets of indicators which belong together, and if model assessment statistics still appear unsatisfactory, the only way to improve the "fit" of the model would be to add factor loadings or error correlations between factors and/or indicators which refer to theoretically distinct constructs (quantities such as a factor loading between, say, effectiveness of the police and an indicator of co-operation in our example). This is not something that we would or should actually do in practice, so model assessment which could lead to it is not very constructive.