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Asymptotic goodness-of-fit methods in contingency table analysis can strugglewith sparse data, especially in multi-way tables where it can be infeasible tomeet sample size requirements for a robust application of distributionalassumptions. However, algebraic statistics provides exact alternatives to theseclassical asymptotic methods that remain viable even with sparse data. We applythese methods to a context in psychometrics and education research that leadsnaturally to multi-way contingency tables: the analysis of differential itemfunctioning (DIF). We explain concretely how to apply the exact methods ofalgebraic statistics to DIF analysis using the R package algstat, and wecompare their performance to that of classical asymptotic methods.

Using Exact Tests from Algebraic Statistics in Sparse Multi-way Analyses: An Application to Analyzing Differential Item Functioning