  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.
