A Note on How to Quantify and Report Whether Irt Parameter Invariance Holds: When Pearson Correlations are Not Enough

Based on seminal work by Lord and Hambleton, Swaminathan, and Rogers, this article is an analytical, graphical, and conceptual reminder that item response theory (IRT) parameter invariance only holds for perfect model fit in multiple populations or across multiple conditions and is thus an ideal state. In practice, one attempts to quantify the degree to which a lack of invariance is likely to be present through repeated calibrations of item and examinee parameters. Motivated by two recent studies on item parameter invariance, this article shows how a seemingly intuitive measure such as Pearson’s Product-Moment Correlation Coefficient (PPMCC) is insufficient for that purpose, as it is not sensitive to restrictive linear relationships such as identities, which are required for parameter invariance to hold. It thus misses, for example, additive group-level effects, which are observed in practice with translated instruments or with large-scale assessments such as TIMSS.