The feature that makes item response theory (IRT) models the models of choice for many psychometric data analysts is parameter invariance, the equality of item and examinee parameters from different populations. Using the well-known fact that item and examinee parameters are identical only up to a set of linear transformations specific to the functional form of a given IRT model, violations of these transformations for unidimensional IRT models are algebraically investigated and coefficients are derived for some violations. Since a lack of invariance constitutes item parameter drift (IPD) at the individual item level or item-set level, the magnitude and types of biases introduced by IPD along with their impact on examinee true scores can be algebraically derived, and these connections are demonstrated with results from a recently published simulation study (C. Wells, M. Subkoviak, and R. Serlin, 2002). This paper faciliates a deeper understanding of different types of lack of parameter invariance and their practical consequences for decision making through a framework that combines analytical, numerical, and visual perspectives on parameter invariance as a fundamental property of measurement. An appendix provides bias coefficients. (Contains 6 figures and 21 references.) (Author/SLD)