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：在列联表分析中，渐近拟合优度方法在处理稀疏数据时可能遇到困难，尤其是在多维表中，满足稳健应用分布假设的样本量要求往往不切实际。然而，代数统计提供了这些经典渐近方法的精确替代方案，即使在稀疏数据下仍然有效。我们将这些方法应用于心理测量学和教育研究中的一个自然产生多维列联表的场景：项目功能差异（DIF）分析。我们具体说明了如何使用R包algstat将代数统计的精确方法应用于DIF分析，并将其性能与经典渐近方法进行了比较。