The Deconstruction of Measurement Invariance (and DIF)

Measurement invariance holds, if the distribution of the observed variables (e.g., test items) is conditionally independent of group membership for every value of the latent variable. A DIF model describes the group-specific differences by group-specific item parameters. Based on a geometric perspective on measurement, it is argued that measurement invariance holds if the generalized true scores (i.e., the link function of conditional expectations) of all groups are spread across the same affine subspace (of all conceivable generalized true scores). A taxonomy of patterns of DIF and measurement invariance is introduced that relies on the degree of overlap and parallelism of the group-specific affine subspace associated with the group-specific measurement models. It is argued the each DIF model can be transformed in a higher-dimensional measurement model with measurement invariance and with constraints on the group-specific distribution of the latent variables. It turns out that DIF implies the existence of a latent variable that is constant in one group but varies across groups. Consequently, the DIF approach relies on postulating complete segregation of prespecified groups in latent space which is inherently discriminatory. It is concluded that DIF (and the idea of an absence of measurement invariance) is a chimera that does not rely on a viable conceptual basis but refers to extremely implausible limiting cases of the group-specific distributions in latent space.