Exploratory analyses to improve model fit: Errors due to misspecification and a strategy to reduce their occurrence

Two Lagrange multiplier (LM) methods may be used in specification searches for adding parameters to models: one based on univariate LM tests and respecification of the model (LM-respecified method) and the other based on a partitioning of multivariate LM tests (LM-incremental method). These methods may result in extraneous parameters being included in models due to either sampling error or the model being misspecified. A 2-stage specification search may be used to reduce errors due to misspecification. In the 1st stage, parameters are added to models based on LM tests to maximize fit. Second, parameters added in the 1st stage are deleted if they are no longer necessary to maintain model fit. Illustrations are presented to demonstrate that errors due to misspecification occur with the LM-respecified method and are even more likely with the LM-incremental approach. These illustrations also show how the deletion stage can help eliminate some of these errors.