Asymptotic Corrections of Standardized Extended Caution Indices

Sato and Tatsuoka suggested the caution index, several extended caution indices (ECIs), and their standardized versions. Among these indices, the standardized versions of the second and fourth ECIs (denoted as $$ and $$ , respectively, by Tatsuoka) are arguably the most popular and have been used by several researchers to assess person fit. While Tatsuoka stated that there is satisfactory evidence that $$ and $$ may approximately follow the normal distribution under no person misfit, she also stated that it would be ideal if their theoretical distributions can be derived algebraically. This article derives the "asymptotic" (or "theoretical large-sample") null distributions of $$ and $$ and suggests corrected versions of them that have asymptotic standard normal null distributions. The derivations are based on the asymptotic correction of person-fit statistics suggested by Snijders. A simulation study shows that the Type I error rate and power of the corrected versions are mostly superior compared with those of the corresponding uncorrected versions. A real data illustration follows. The suggested corrected versions appear to be satisfactory tools for assessing person fit.