Conformal novelty detection for replicate point patterns with FDR or FWER control

Monte Carlo tests are widely used for computing valid $p$-values without requiring known distributions of test statistics. When performing multiple Monte Carlo tests, it is essential to maintain control of the type I error. Some techniques for multiplicity control pose requirements on the joint distribution of the $p$-values, for instance independence, which can be computationally intensive to achieve, as it requires simulating disjoint null samples for each test. We refer to this as naïve multiple Monte Carlo testing. We highlight in this work that multiple Monte Carlo testing is an instance of conformal novelty detection. Leveraging this insight enables a more efficient multiple Monte Carlo testing procedure, avoiding excessive simulations by using a single null sample for all the tests, while still ensuring exact control over the false discovery rate or the family-wise error rate. We call this approach conformal multiple Monte Carlo testing. The performance is investigated in the context of global envelope tests for point pattern data through a simulation study and an application to a sweat gland data set. Results reveal that with a fixed simulation budget, our proposed method yields substantial improvements in power of the testing procedure as compared to the naïve multiple Monte Carlo testing procedure.