Researchers decompose test score “gaps” and gap-changes into within- and between-school portions to generate evidence on the role that schools play in shaping educational inequality. However, existing decomposition methods (a) assume an equal-interval test scale and (b) are a poor fit to coarsened data such as proficiency categories. We develop two decomposition approaches that overcome these limitations: an extension of V, an ordinal gap statistic (Ho, 2009), and an extension of ordered probit models (Reardon et al., 2017). 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. These methods are applicable to decomposing any ordinal outcome by any categorical grouping variable.
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