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 test equating scores on different test forms. This paper argues, however, that the use of item response models does not require any test score equating. Instead, it involves the necessity of parameter linking due to a fundamental 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 separate item calibrations. Our main theorems characterize the formal nature of these linking functions for monotone, continuous response models, derive their specific shapes for different parameterizations of the 3PL model, and show how to identify them from the parameter values of the common items or persons in different linking designs.

Parameter linking in item response theory is generally necessary to adjust for differences between the true values for the same item and ability parameters due to the use of different identifiability restrictions in different calibrations. The research reported in this article explores a precision-weighted (PW) approach to the problem of estimating the linking functions for the common dichotomous logistic response models. Asymptotic standard errors (ASEs) of linking for the new approach are derived and compared to those of the mean/mean and mean/sigma linking methods to which it has a superficial similarity and to the Haebara and Stocking and Lord response function methods. Empirical examples from a few recent linking studies are presented. It is demonstrated that the new approach has smaller ASE than the mean/mean and mean/sigma methods and comparable ASE to the response function methods. However, when some of the item parameters have large estimation error relative to the others, all current methods appear to violate the rather obvious requirement of monotone decrease in ASE with the number of common items in the linking design while the ASE of the PW method demonstrates monotone decrease with the number of common items. The PW method also has the benefits of simple calculation and an ASE which is additive in the contribution of each item, useful for optimal linking design. We conclude that the proposed approach to estimating linking parameters holds promise and warrants further research.

Linking functions adjust for differences between identifiability restrictions used in different instances of the estimation of item response model parameters. These adjustments are necessary when results from those instances are to be compared. As linking functions are derived from estimated item response model parameters, parameter estimation error automatically propagates into linking error. This article explores an optimal linking design approach in which mixed-integer programming is used to select linking items to minimize linking error. Results indicate that the method holds promise for selection of linking items.

A growing body of empirical research indicates that structured writing interventions produce substantial immediate and long-term benefits, including improvements in cognitive, emotional, behavioral, and physiological functioning. To date, many of these studies have used non-clinical college student populations. In contrast, the present study involves a woman suffering anxiety symptoms resulting from domestic violence, and adopts the structured writing intervention as a self-administered adjunct to her clinical treatment. It uses a single-system design with multiple measures to track progress in reducing these symptoms.

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