Residual-based fit statistics are among the most common indicators of fit to the Rasch model. There is considerable discussion in the literature of the efficacy of item fit statistics in detecting measurement disturbances. However, to date there has been no investigation of whether these fit statistics are robust to interactions between item discrimination and item difficulty. This study uses simulations to investigate whether interaction effects occur for fit statistics commonly used with the Rasch model. It is found that when the parameters are estimated with the Rasch model, the values of certain item fit statistics vary depending on the interaction between location and discrimination. Specifically, in the study, OUTFIT MNSQ and INFIT MNSQ provide a relatively consistent index of item discrimination across a range of item difficulties, whereas the t-statistic and the log transformed fit residual vary in a systematic fashion that depends on item location.