The choice of the ability estimate with asymptotically correct standardized person-fit statistics

Snijders (2001, Psychometrika, 66, 331) suggested a statistical adjustment to obtain the asymptotically correct standardized versions of a specific class of person-fit statistics. His adjustment has been used to obtain the asymptotically correct standardized versions of several person-fit statistics including the lz statistic (Drasgow et al., 1985, Br. J. Math. Stat. Psychol., 38, 67), the infit and outfit statistics (e.g., Wright & Masters, 1982, Rating scale analysis, Chicago, IL: Mesa Press), and the standardized extended caution indices (Tatsuoka, 1984, Psychometrika, 49, 95). Snijders (2001), van Krimpen-Stoop and Meijer (1999, Appl. Psychol. Meas., 23, 327), Magis et al. (2012, J. Educ. Behav. Stat., 37, 57), Magis et al. (2014, J. Appl. Meas., 15, 82), and Sinharay (2015b, Psychometrika, doi:10.1007/s11336-015-9465-x, 2016b, Corrections of standardized extended caution indices, Unpublished manuscript) have used the maximum likelihood estimate, the weighted likelihood estimate, and the posterior mode of the examinee ability with the adjustment of Snijders (2001). This paper broadens the applicability of the adjustment of Snijders (2001) by showing how other ability estimates such as the expected a posteriori estimate, the biweight estimate (Mislevy & Bock, 1982, Educ. Psychol. Meas., 42, 725), and the Huber estimate (Schuster & Yuan, 2011, J. Educ. Behav. Stat., 36, 720) can be used with the adjustment. A simulation study is performed to examine the Type I error rate and power of two asymptotically correct standardized person-fit statistics with several ability estimates. A real data illustration follows.