# Structural Equation Modeling - Steps in Performing SEM Analysis - Assessment of Model and Model Fit

Assessment of Model and Model Fit

Having estimated a model, analysts will want to interpret the model. Estimated paths may be tabulated and/or presented graphically as a path model. The impact of variables is assessed using path tracing rules (see path analysis).

It is important to examine the "fit" of an estimated model to determine how well it models the data. This is a basic task in SEM modeling: forming the basis for accepting or rejecting models and, more usually, accepting one competing model over another. The output of SEM programs includes matrices of the estimated relationships between variables in the model. Assessment of fit essentially calculates how similar the predicted data are to matrices containing the relationships in the actual data.

Formal statistical tests and fit indices have been developed for these 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.

Of course as in all statistical hypothesis tests, SEM model tests are based on the assumption that the correct and complete relevant data have been modeled. In the SEM literature, discussion of fit has led to a variety of different recommendations on the precise application of the various fit indices and hypothesis tests.

There are differing approaches to assessing fit. Traditional approaches to modeling start from a null hypothesis, rewarding more parsimonious models (i.e. those with fewer free parameters), to others such as AIC that focus on how little the fitted values deviate from a saturated model (i.e. how well they reproduce the measured values), taking into account the number of free parameters used. 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-Squared 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.
• Akaike information criterion (AIC)
• A test of relative model fit: The preferred model is the one with the lowest AIC value.
• where k is the number of parameters in the statistical model, and L is the maximized value of the likelihood of the model.
• Root Mean Square Error of Approximation (RMSEA)
• Another test of model fit. RMSEA values <.05 are considered to indicated good fit. An RMSEA of .1 or more is often taken to indicate poor fit.
• Standardized Root Mean Residual (SRMR)
• The SRMR is a popular absolute fit indicator. A good model should have an SRMR smaller than .05.
• Comparative Fit Index (CFI)
• In examining baseline comparisons, the CFI depends in large part on the average size of the correlations in the data. If the average correlation between variables is not high, then the CFI will not be very high. A CFI value of .90 or higher is desirable.

For each measure of fit, a decision as to what represents a good-enough fit between the model and the data must reflect other contextual factors such as sample size (for instance very large samples make the Chi-squared test overly sensitive), the ratio of indicators to factors, and the overall complexity of the model.

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