Fit analysis in latent trait measurement models

The analysis of fit, whether viewed from the prospective of the fit of the data to the measurement model, or the fit of the measurement model to the data, is an important part of using latent trait models. In the case of the Rasch model, all of the desirable characteristics of the model (interval item and person measures, asymptotic standard errors, parameter invariance across subsets of persons or items, to name a few) are predicated on the requirement that the data fit the model. To the extent that the data do not fit the model, these properties hold to a lesser degree. The analysis of fit is of primary importance if the interpretation of the calibration results is to be useful. This article explores the nature of fit and provides a historical overview of fit indices. It then focuses on a particular family of fit indices that are based on the Pearsonian chi-square approach to fit, in an attempt to show why it is necessary to use a family of standardized fit indices to completely understand the relationship between the data and the model.