Applications of Asymptotic Expansion in Item Response Theory Linking

Abstract

The purpose of this chapter is to have asymptotic expansions of the distributions of the estimators of linking coefficients using item response theory (IRT; see, e.g., Bock & Moustaki, 2007) in the common-item nonequivalent groups design. In IRT linking, usually item parameters are available only as their estimates. Consequently, the parameters in IRT linking, that is, linking coefficients, are subject to sampling variation. So, it is important to see the magnitudes of the estimates considering this variation. One of the typical methods to evaluate their sizes is using the asymptotic standard errors (ASEs) of the estimators of linking coefficients. The ASEs of the coefficient estimators by the methods using moments of item parameters (hereafter referred to as moment methods; Loyd & Hoover, 1980; Mislevy & Bock, 1990; Marco, 1977; see also Kolen & Brennan, 2004, Ch. 6) were derived by Ogasawara (2000). The corresponding ASEs for the methods using response functions (Haebara, 1980; Stocking & Lord, 1983; see also Kolen & Brennan, 2004, Ch. 6) were obtained by Ogasawara (2001b). The ASEs in equating methods for true and observed scores were obtained by Lord (1982a) and Ogasawara (2001a, 2003), whereas the standard errors by the bootstrap were investigated by Tsai, Hanson, Kolen, and Forsyth (2001; see also Kolen & Brennan, 2004, Ch. 7). The ASEs by kernel equating (see von Davier, Holland, & Thayer, 2004b) were obtained by Liou, Cheng, and Johnson (1997) and von Davier et al. by different methods.