Asymptotically Normally Distributed Person Fit Indices for Detecting Spuriously High Scores on Difficult Items

Abstract

Snijders developed a family of person fit indices that asymptotically follow the standard normal distribution, when the ability parameter is estimated. So far, $l_z^{*}$ , U*, W*, ${\mathrm{ECI}2}_z^{*}$ , and ${\mathrm{ECI}4}_z^{*}$ from this family have been proposed in previous literature. One common property shared by $l_z^{*}$ , U*, and W* (also ${\mathrm{ECI}2}_z^{*}$ and ${\mathrm{ECI}4}_z^{*}$ in some specific conditions) is that they employ symmetric weight functions and thus identify spurious scores on both easy and difficult items in the same manner. However, when the purpose is to detect only the spuriously high scores on difficult items, such as cheating, guessing, and having item preknowledge, using symmetric weight functions may jeopardize the detection rates of the target aberrant response patterns. By specifying two types of asymmetric weight functions, this study proposes SHa(λ)* (λ = 1/2 or 1) and SHb(β)* (β = 2 or 3) based on Snijders’s framework to specifically detect spuriously high scores on difficult items. Two simulation studies were carried out to investigate the Type I error rates and empirical power of SHa(λ)* and SHb(β)*, compared with $l_z^{*}$ , U*, W*, ${\mathrm{ECI}2}_z^{*}$ , and ${\mathrm{ECI}4}_z^{*}$ . The empirical results demonstrated satisfactory performance of the proposed indices. Recommendations were also made on the choice of different person fit indices based on specific purposes.