DIF refers to differences in the functioning of items across groups, oftentimes demographic, which are matched on the latent trait or more generally the attribute being measured by the items or test. It is important to note that when examining items for DIF, the groups must be matched on the measured attribute, otherwise this may result in inaccurate detection of DIF. In order to create a general understanding of DIF or measurement bias, consider the following example offered by Osterlind and Everson (2009). In this case, Y refers to a response to a particular test item which is determined by the latent construct being measured. The latent construct of interest is referred to as theta (θ) where Y is an indicator of θ which can be arranged in terms of the probability distribution of Y on θ by the expression "f"(Y)|θ. Therefore, response Y is conditional on the latent trait (θ). Because DIF examines differences in the conditional probabilities of Y between groups, let us label the groups as the "reference" and "focal" groups. Although the designation does not matter, a typical practice in the literature is to designate the reference group as the group who is suspected to have an advantage while the focal group refers to the group anticipated to be disadvantaged by the test. Therefore, given the functional relationship formula_1 and under the assumption that there are identical measurement error distributions for the reference and focal groups it can be concluded that under the null hypothesis: