The estimation of test score “gaps” and gap trends plays an important role in monitoring educational inequality. Researchers decompose gaps and gap changes into within- and between-school portions to generate evidence on the role schools play in shaping these inequalities. However, existing decomposition methods assume an equal-interval test scale and are a poor fit to coarsened data such as proficiency categories. This leaves many potential data sources ill-suited for decomposition applications. We develop two decomposition approaches that overcome these limitations: an extension of V, an ordinal gap statistic, and an extension of ordered probit models. Simulations show V decompositions have negligible bias with small within-school samples. Ordered probit decompositions have negligible bias with large within-school samples but more serious bias with small within-school samples. More broadly, our methods enable analysts to (1) decompose the difference between two groups on any ordinal outcome into portions within- and between some third categorical variable, and (2) estimate scale-invariant between-group differences that adjust for a categorical covariate.
State testing programs regularly release previously administered test items to the public. We provide an open-source recipe for state, district, and school assessment coordinators to combine these items flexibly to produce scores linked to established state score scales. These would enable estimation of student score distributions and achievement levels. We discuss how educators can use resulting scores to estimate achievement distributions at the classroom and school level. We emphasize that any use of such tests should be tertiary, with no stakes for students, educators, and schools, particularly in the context of a crisis like the COVID-19 pandemic. These tests and their results should also be lower in priority than assessments of physical, mental, and social–emotional health, and lower in priority than classroom and district assessments that may already be in place. We encourage state testing programs to release all the ingredients for this recipe to support low-stakes, aggregate-level assessments. This is particularly urgent during a crisis where scores may be declining and gaps increasing at unknown rates.