Karabatsos compared the power of 36 person-fit statistics using receiver operating characteristics curves and found the "H[superscript T]" statistic to be the most powerful in identifying aberrant examinees. He found three statistics, "C", "MCI", and "U3", to be the next most powerful. These four statistics, all of which are nonparametric, were found to perform considerably better than each of 25 parametric person-fit statistics. Dimitrov and Smith replicated part of this finding in a similar study. The present article raises some issues with the comparisons performed in Karabatsos and Dimitrov and Smith and points to literature that suggests that the comparisons could have been performed in a more traditional and more fair manner. The present article then replicates the simulations of Karabatsos and demonstrates in several ways that the parametric person-fit statistics l[subscript z] and "ECI4[subscript z]" (that were also considered by Karabatsos) are as powerful as are "H[superscript T]" and "U3" in identifying aberrant examinees in more traditional and fair comparisons. Two parametric person-fit statistics are shown to lead to similar results as "H[superscript T]" and "U3" in a real data example.