Asymptotic goodness-of-fit methods in contingency table analysis can struggle with sparse data, especially in multi-way tables where it can be infeasible to meet sample size requirements for a robust application of distributional assumptions. However, algebraic statistics provides exact alternatives to these classical asymptotic methods that remain viable even with sparse data. We apply these methods to a context in psychometrics and education research that leads naturally to multi-way contingency tables: the analysis of differential item functioning (DIF). We explain concretely how to apply the exact methods of algebraic statistics to DIF analysis using the R package algstat, and we compare their performance to that of classical asymptotic methods.

翻译：列联表分析中的渐近拟合优度方法在处理稀疏数据时可能面临困难，尤其是在多维列联表中，满足分布假设稳健应用的样本量要求往往难以实现。然而，代数统计为这些经典渐近方法提供了精确的替代方案，即使在数据稀疏的情况下仍能保持可行性。我们将这些方法应用于心理学与教育研究领域一个自然涉及多维列联表的场景——差异项目功能（DIF）分析。本文具体阐述了如何通过R语言包algstat将代数统计的精确方法应用于DIF分析，并比较了其与经典渐近方法的性能表现。