This article extends the work by Armstrong and Shi on CUmulative SUM (CUSUM) person-fit methodology. The authors present new theoretical considerations concerning the use of CUSUM person-fit statistics based on likelihood ratios for the purpose of detecting cheating and random guessing by individual test takers. According to the Neyman-Pearson Lemma, the optimality of such statistics relies on how accurately normal and aberrant behaviors are modeled. General and specific models for cheating and random guessing are investigated. The detection rates of several statistics are compared using simulated data. Results showed that the likelihood-based CUSUM statistics that use the proposed models for aberrant behavior performed better than some of the more commonly used statistics, especially for cheating behavior. (Contains 4 tables and 11 figures.)