Model Modification in Covariance Structure Modeling: A Comparison among Likelihood Ratio, Lagrange Multiplier, and Wald Tests

Model comparisons in covariance structure modeling have traditionally been carried out by the likelihood ratio difference (D) test. Two more convenient approaches are reviewed and evaluated. The Lagrange Multiplier (LM) test evaluates the impact of model modification from a more limited model whereas the Wald (W) test makes the evaluation from a more general model. The empirical performance of the D, LM, and W tests under null and alternative hypotheses are compared in this study. The results indicate that both new tests performed as well as the D test for reasonable sample sizes. The nonmonotonicity of power function for the W test was discovered; however, it is not severe in this study. The LM test behaved as a central or noncentral XSUP2 variate under null or alternative hypotheses, as expected. However, when a correct null hypothesis was embedded in a composite hypothesis which was false, an incremental LM test tended to suggest more parameters than needed to be freed, especially at larger sample sizes. This incorrect behavior of the LM test was correctable by following up the LM test by a W test, that is, a combination of both LM and W tests seemed to provide a fairly satisfactory outcome in the process of model modification.