Response model parameter linking

Abstract

With a few exceptions, the problem of linking item response model parameters from different item calibrations has been conceptualized as an instance of the problem of equating observed scores on different test forms. This thesis argues, however, that the use of item response models does not require any post hoc observed-score equating, but that the necessity of parameter linking is due to a problem inherent in the formal nature of these models---their general lack of identifiability. More specifically, item response model parameters need to be linked to adjust for the different effects of the identifiability restrictions used in different item calibrations. The research first characterizes the formal nature of linking functions for monotone, continuous, dichotomous response models. New definitions emerge for the linking parameters. Derivation of the linking function for the c parameter reveals an identity function, bringing into question response function linking methods that confound estimation error in the c parameter with linking function parameter estimates. Minimal elements for linking function identifiability for both traditional and slope-intercept parameterizations of the 3PL model are presented. Closed-form asymptotic standard errors (ASE) of the linking parameters are derived for single linking elements for the 3PL model and well-known polytomous models and a precision-weighted estimator for the linking function is proposed. The new estimator exhibits the desirable feature of monotone decrease in linking error as linking elements are added to the design. It is shown that the estimator outperforms mean/mean and mean/sigma linking approaches and facilitates optimal linking design. Several examples of optimal linking design are provided. The new estimator also enables exploration of the relationship between the number of response categories of an item and the ASEs of the estimated linking parameters. For the generalized partial credit model, analysis is presented that suggests that ASE of the slope parameter in the linking function will hardly change with the number of response categories of an item, while the ASE of the intercept parameter increases with the number of categories. Empirical examples are provided that concur with analytic results.