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The new conceptual framework distinguishes between theanchor classand theanchor selection

strategy. Firstly, the anchor classdescribes the pre-specification of the anchor characteristics

(such as a pre-defined anchor length). We review the all-other (used, e.g., by Cohenet al.1996; Kim and Cohen 1998), the constant anchor (used, e.g., by Thissen et al. 1988; Wang 2004; Shih and Wang 2009) and the iterative anchor class (referred to here as iterative backward and used, e.g., by Drasgow 1987; Candell and Drasgow 1988; Hidalgo-Montesinos and Lopez-Pina 2002). Secondly, the anchor selection strategy determines which items are chosen as anchor items.

5.3.1 Anchor classes

In our conceptual frameworkanchor classes describe characteristics of the anchor that answer the following questions: Is the anchor length pre-defined? If so, how many items are included in the anchor? Is the anchor determined by the anchor class itself or is an additional anchor selection strategy necessary? Are iterative steps intended to define the anchor?

The equal-mean and the all-other anchor class

In theequal-mean-difficultyanchor class (see, e.g., Wang, 2004, and the references therein) all items are restricted to have the same mean difficulty (typically zero) in both groups, whereas in

theall-other anchor class (used, e.g., by Cohenet al.1996; Kim and Cohen 1998) the sum of

all items – except the item currently tested for DIF – is restricted to be zero and the anchor set

Aj ={1, . . . ,k} \ jdepends on the studied item j=1, . . . ,k.

Both anchor classes have a pre-defined anchor length but no additional anchor selection is nec- essary as the items included in the restriction are already determined by the anchor class itself. The equal-mean-difficulty and the all-other class only differ in one anchor item and, therefore, essentially lead to similar results (cf. Wang, 2004) and, hence, only the all-other method is included in the following parts of this thesis.

The constant anchor class

Theconstantanchor class (used, e.g., by Thissenet al.1988; Wang 2004; Shih and Wang 2009)

includes a pre-defined number of the items (e.g., 1 or 4 items according to Thissenet al., 1988) or a certain proportion of the items (e.g., 10% or 20% according to Woods, 2009) as anchor.

5.3 A conceptual framework for anchor methods 85 The term constantreflects the pre-defined, constant anchor length. The constant anchor class needs to be combined with an explicit anchor selection strategy. For the constant single anchor class, the first item of the ranking order of candidate anchor items is used as anchor, whereas for the constant four anchor class, the first four items of the ranking order of candidate anchor items are used as anchor.

The iterative backward anchor class

Theiterative backwardanchor class (used, e.g., by Drasgow 1987; Candell and Drasgow 1988;

Hidalgo-Montesinos and Lopez-Pina 2002) includes a variety of iterative methods that have been suggested, discussed and combined with different statistical methods to assess DIF. Here, we focus on the commonly used re-linking procedure where one parameter estimation step suffices to conduct DIF analysis. Firstly, the scales of both groups are linked on (approximately) the same metric, e.g., by using the all-other anchor method. Then, the DIF-items are excluded from the current anchor, the scales are re-linked using the new current anchor, the DIF analysis is carried out and the steps are repeated until two steps reach the same results (e.g., Drasgow, 1987; Candell and Drasgow, 1988; Park and Lautenschlager, 1990; Kim and Cohen, 1995; Hidalgo-Montesinos and Lopez-Pina, 2002). This iterative procedure is referred to here as the

iterative backward anchor class, since the method includes the majority of items in the anchor

at the beginning. Then, it successively excludes items from the anchor.

A new anchor class will be suggested in Chapter 6 and systematically compared to the com- monly used anchor classes presented above.

5.3.2 Anchor selection strategies

The anchor selection strategies discussed in this thesis are based on preliminary item analyses. This means that – before the final DIF test is done – DIF tests are conducted to locate (ideally) DIF-free anchor items. The (non-statistical) alternative relying on expert advice and certain prior knowledge of DIF-free anchor items (Wang, 2004; Woods, 2009) will not often be possible in practice (for a literature overview where this approach fails see Frederickxet al., 2010).

The all-other anchor selection

An example of an anchor selection strategy is the rank-based strategy proposed by Woods (2009) that we term all-other (AO) anchor selection strategy. Initially, every item is tested for DIF using all other items as anchor. The ranking order of candidate anchor items is defined according to the lowest rank(s) of the resulting (absolute) DIF test statistics.

Other suggestions

Ideas on how to select anchor items without prior knowledge were also given by Wang (2004). His original suggestions and also an anchor selection strategy that simplifies his approach will be presented in Chapter 6. Another suggestion on how to locate anchor items by Shih and Wang (2009) will be discussed in Chapter 7 together with three new anchor selection strategies.

5.3.3 Anchor methods

Ananchor methodresults as a combination of an anchor class with an anchor selection strategy

(in cases where the latter is necessary). For example, the explicit anchor selection is necessary for the constant anchor class: Firstly, the anchor selection is carried out to determine a ranking order of candidate anchor items and the procedure defined by the anchor class is carried out to determine the final anchor. Secondly, the final anchor found in the first step is then used for the assessment of DIF. This procedure was termed DIF-free-then-DIF strategy by Wang et al.

(2012).

Anchor classes that do not require an anchor selection strategy are the equal-mean, the all-other and the iterative backward anchor class. The next two chapters will illustrate that anchor meth- ods that do not rely on an explicit anchor selection strategy are inadvisable for DIF analysis. A variety of anchor methods – including the all-other, the constant, the iterative-backward and the newly suggested iterative forward anchor classes together with two anchor selection strate- gies – will be compared in the next chapter.