Using item mean squares to evaluate fit to the Rasch model

Throughout the mid to late 1970's considerable research was conducted on the properties of Rasch fit mean squares. This work culminated in a variety of transformations to convert the mean squares into approximate t-statistics. This work was primarily motivated by the influence sample size has on the magnitude of the mean squares and the desire to have a single critical value that can generally be applied to most cases. In the late 1980's and the early 1990's the trend seems to have reversed, with numerous researchers using the untransformed fit mean squares as a means of testing fit to the Rasch measurement models. The principal motivation is cited as the influence sample size has on the sensitivity of the t-converted mean squares. The purpose of this paper is to present the historical development of these fit indices and the various transformations and to examine the impact of sample size on both the fit mean squares and the t-transformations of those mean squares. Because the sample size problem has little influence on the person mean square problem, due to the relatively short length (100 items or less), this paper focuses on the item fit mean squares, where it is common to find the statistics used with sample sizes ranging from 30 to 10,000.