1.5 Model Modification

In application of SEM one usually specifies a model based on theory or empirical findings then fit the model to the available data. Very often the tentative initial model may not fit data well. In other words, the initial model may be somewhat mis-specified. In such a case, the possible sources of lack of model fit need to be assessed to determine what is specifically wrong with the model specification, then modify the model and re-test it using the same data. This process is called ‘model specification search.’

To improve the initial model that does not fit the data satisfactorily, most often the modification indices (MIs) (Sörbom, 1989) that are associated with the fixed parameters of the model are used as diagnostic statistics to capture model mis-specfication. A MI indicates the decrease in model statistic with 1 df indicating if a particular parameter is freed from a constraint in the preceding model.

A high MI value indicates the corresponding fixed parameter should be freed to improve model fit. Although a drop in of 3.84 with 1 df indicates a significant model fit improvement at P = 0.05 level, no strict rules of thumb exist concerning how large MIs must be to warrant a meaningful model modification. In Mplus output MIs are listed by default if a drop in a corresponding is at least 10. If there are several parameters with high MIs, they should be freed one at a time, beginning with the largest MI because change in a single parameter in a model could affect other parts of the solution (MacCallum, Roznowski, and Necowitz, 1992). Freeing additional parameters may improve model fit, however, the model modification must be theoretically meaningful. Associated with MI is the expected parameter change (EPC) index for the expected change in the value of a parameter if that parameter was freed (Saris, Satorra and Sörbom, 1987). Mplus provides MIs, EPC, and standardized EPC for all parameters in the model that are fixed or constrained to be equal to other parameters (Muthén and Muthén, 1998–2010).

It must be emphasized that the model modification or re-specification should be both statistics-driven and theory-driven. Any model modification must be justified on a theoretical basis and empirical findings. Blind use of MIs for model modification should be avoided. Parameters should not be added or removed solely for the purpose of model fit improvement. Our goal is to find a model that fits data well from a statistical point of view, and importantly all the parameters of the model must have substantively meaningful interpretation.