When does measurement invariance matter?

question whether observed differences in psychometric test scores can be attributed to differences in the properties that such tests measure is relevant in many research domains; examples include the proper interpretation of differences in intelligence test scores across different generations of people,1 gender differences in affectivity,2 and crosscultural differences in personality. This question also has generated some of the most conspicuous controversies in the social and life sciences, where the highest temperature in the many heated discussions around the topic has, without a doubt, been reached in the debate on IQ-score differences between ethnic groups in the United States.4'5 Such debates are often unproductive because of a lack of unambiguous characterizations of concepts like "biased," "incomparable," and "culture-fair. " Terms are easily coined, as is illustrated by Johnson's6 count of no less than 55 types of measurement equivalence; however, it is often less easy to spell out their meaning in terms of their empirical consequences. However, without at least some degree of precision in one's conception of a term like "equivalence," it is difficult to have a scientifically productive debate, or even to agree on what aspects of empirical data are relevant for answering the questions involved. It is for this reason that the establishment of concepts like measurement invariance and bias in an unambigous, formal framework with testable consequences7"9 represents a theoretical development of great importance. Through this work, it has become clear that differences in raw scores (eg, IQ-scores) of different groups (eg, blacks and whites) cannot be used to infer group differences in theoretical attributes (eg, general intelligence) unless the test scores accord with a particular set of model invariance restrictions. Namely, the same attribute must relate to the same set of observations in the same way in each group. Statistically, this means that the mathematical function that relates latent variables to the observations must be the same in each of the groups involved in the comparison.7'8 This idea has become known as the requirement of measurement invariance. The theoretical definitions of measurement invariance and bias are very general, and apply to different models, such as item response theory (IRT) and factor models, in roughly the same way.10'11 This does not hold for the empirical methods available for testing measurement invariance. In the past decades, psychometricians working on measurement invariance have produced many different statistical techniques to assess differential item functioning (DIF). These techniques usually employ different statistical assumptions, for instance, regarding the form of the relation between latent and observed variables and the shape of the population distribution on the latent variable, and employ different modeling strategies as well as selection criteria for flagging items as biased. For this reason, it is difficult to assess the consequences of choosing a particular technique; moreover, it is not always clear to what extent the choice of technique makes a difference with respect to the diagnosis of meaurement invariance and bias in applied situations. For this reason, the articles on DIF collected here (by Crane et al;12 Dorans and Kulick;13 Jones;14 Morales, Flowers, Gutierrez, Kleinman, and Teresi;15 Edelen Orlando et al16) represent a useful project in the application of bias detection methods. Each set of authors analyzes the Mini-Mental State Examination (MMSE) for measurement invariance using the same data, albeit with different methods. Together, the articles provide a