This article proposes several statistics for assessing individual fit based on two unidimensional models for continuous responses: linear factor analysis and Samejima's continuous response model. Both models are approached using a common framework based on underlying response variables and are formulated at the individual level as fixed regression models. From this formulation, different global statistics, and statistics at the individual-item-score level are developed, extended, or adapted. These statistics are intended to be used mainly as detection tools, and for obtaining clues about the possible causes of the detected misfit. The behavior of the global statistics is assessed by means of simulation studies. The use of all the proposed statistics as well as their interpretation and limitations is illustrated with an empirical example in the personality domain. (Contains 2 tables and 2 figures.)