Anchor Point Selection: Scale Alignment Based on an Inequality Criterion.

For detecting differential item functioning (DIF) between two or more groups of test takers in the Rasch model, their item parameters need to be placed on the same scale. Typically this is done by means of choosing a set of so-called anchor items based on statistical tests or heuristics. Here the authors suggest an alternative strategy: By means of an inequality criterion from economics, the Gini Index, the item parameters are shifted to an optimal position where the item parameter estimates of the groups best overlap. Several toy examples, extensive simulation studies, and two empirical application examples are presented to illustrate the properties of the Gini Index as an anchor point selection criterion and compare its properties to those of the criterion used in the alignment approach of Asparouhov and Muthén. In particular, the authors show that-in addition to the globally optimal position for the anchor point-the criterion plot contains valuable additional information and may help discover unaccounted DIF-inducing multidimensionality. They further provide mathematical results that enable an efficient sparse grid optimization and make it feasible to extend the approach, for example, to multiple group scenarios.

Anchor Selection Strategies for DIF Analysis: Review, Assessment, and New Approaches.

Differential item functioning (DIF) indicates the violation of the invariance assumption, for instance, in models based on item response theory (IRT). For item-wise DIF analysis using IRT, a common metric for the item parameters of the groups that are to be compared (e.g., for the reference and the focal group) is necessary. In the Rasch model, therefore, the same linear restriction is imposed in both groups. Items in the restriction are termed the ``anchor items''. Ideally, these items are DIF-free to avoid artificially augmented false alarm rates. However, the question how DIF-free anchor items are selected appropriately is still a major challenge. Furthermore, various authors point out the lack of new anchor selection strategies and the lack of a comprehensive study especially for dichotomous IRT models. This article reviews existing anchor selection strategies that do not require any knowledge prior to DIF analysis, offers a straightforward notation, and proposes three new anchor selection strategies. An extensive simulation study is conducted to compare the performance of the anchor selection strategies. The results show that an appropriate anchor selection is crucial for suitable item-wise DIF analysis. The newly suggested anchor selection strategies outperform the existing strategies and can reliably locate a suitable anchor when the sample sizes are large enough.

A Framework for Anchor Methods and an Iterative Forward Approach for DIF Detection.

In differential item functioning (DIF) analysis, a common metric is necessary to compare item parameters between groups of test-takers. In the Rasch model, the same restriction is placed on the item parameters in each group to define a common metric. However, the question how the items in the restriction-termed anchor items-are selected appropriately is still a major challenge. This article proposes a conceptual framework for categorizing anchor methods: The anchor class to describe characteristics of the anchor methods and the anchor selection strategy to guide how the anchor items are determined. Furthermore, the new iterative forward anchor class is proposed. Several anchor classes are implemented with different anchor selection strategies and are compared in an extensive simulation study. The results show that the new anchor class combined with the single-anchor selection strategy is superior in situations where no prior knowledge about the direction of DIF is available.

Rasch Trees: A New Method for Detecting Differential Item Functioning in the Rasch Model.

A variety of statistical methods have been suggested for detecting differential item functioning (DIF) in the Rasch model. Most of these methods are designed for the comparison of pre-specified focal and reference groups, such as males and females. Latent class approaches, on the other hand, allow the detection of previously unknown groups exhibiting DIF. However, this approach provides no straightforward interpretation of the groups with respect to person characteristics. Here, we propose a new method for DIF detection based on model-based recursive partitioning that can be considered as a compromise between those two extremes. With this approach it is possible to detect groups of subjects exhibiting DIF, which are not pre-specified, but result from combinations of observed covariates. These groups are directly interpretable and can thus help generate hypotheses about the psychological sources of DIF. The statistical background and construction of the new method are introduced by means of an instructive example, and extensive simulation studies are presented to support and illustrate the statistical properties of the method, which is then applied to empirical data from a general knowledge quiz. A software implementation of the method is freely available in the R system for statistical computing.