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.

测验分数“差距”和差距趋势的估计在监控教育不平等方面起着重要作用。研究人员将差距和差距变化分解为学校内部和学校之间的部分，以证明学校在塑造这些不平等方面的作用。但是，现有的分解方法假定测试间隔相等，并且不适合粗化的数据（例如熟练程度类别）。这使得许多潜在的数据源不适合分解应用程序。我们开发了两种克服这些限制的分解方法：V的扩展，有序间隙统计量和有序概率模型的扩展。模拟显示V分解对学校内部的小样本偏差可忽略不计。有序的概率分解对学校内样本的偏见可以忽略不计，但对于学校内小样本的偏见则更为严重。更广泛地说，我们的方法使分析人员能够（1）将任何序数结果上的两组之间的差异分解为某些第三类别变量之内和之间的部分，以及（2）估计针对类别协变量进行调整的组间尺度差异。