Conditional observed score (COS) and latent trait (LT) definitions of differential item functioning (DIF) are explored to determine when they are equivalent. COS methods rely solely on observed measurements, and LT methods model the response to an item as a function of an unobserved hypothetical latent ability or trait. For the case of dichotomous test items, the COS approach defines DIF by population differences in the conditional probabilities of responding correctly to the item, conditioning on the observed ability measure. The LT approach defines DIF by population differences in the conditional probabilities of responding correctly to the item, conditioning on the latent trait. Although DIF detection methods are usually applied to dichotomous ability or achievement test items, the discussion focuses on a more general level, and the results may have applications for general questions of measurement invariance in multiple populations. It is concluded that the conditions under which COS and LT definitions are equivalent are quite specialized. Equivalence generally requires invariance of the conditional densities. Other conditions of equivalence are discussed. It appears that precise invariance of prior densities can rarely be assumed in practice. (TJH)