  We introduce the saddlepoint approximation-based conditional randomization
test (spaCRT), a novel conditional independence test that effectively balances
statistical accuracy and computational efficiency, inspired by applications to
single-cell CRISPR screens. Resampling-based methods like the distilled
conditional randomization test (dCRT) offer statistical precision but at a high
computational cost. The spaCRT leverages a saddlepoint approximation to the
resampling distribution of the dCRT test statistic, achieving very similar
finite-sample statistical performance with significantly reduced computational
demands. We prove that the spaCRT p-value approximates the dCRT p-value with
vanishing relative error, and that these two tests are asymptotically
equivalent. Through extensive simulations and real data analysis, we
demonstrate that the spaCRT controls Type-I error and maintains high power,
outperforming other asymptotic and resampling-based tests. Our method is
particularly well-suited for large-scale single-cell CRISPR screen analyses,
facilitating the efficient and accurate assessment of perturbation-gene
associations.
