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When conducting large scale inference, such as genome-wide associationstudies or image analysis, nominal $p$-values are often adjusted to improvecontrol over the family-wise error rate (FWER). When the majority of tests arenull, procedures controlling the False discovery rate (Fdr) can be improved byreplacing the theoretical global null with its empirical estimate. However,these other adjustment procedures remain sensitive to the working modelassumption. Here we propose two key ideas to improve inference in this space.First, we propose $p$-values that are standardized to the empirical nulldistribution (instead of the theoretical null). Second, we propose modelaveraging $p$-values by bootstrap aggregation (Bagging) to account for modeluncertainty and selection procedures. The combination of these two key ideasyields bagged empirical null $p$-values (BEN $p$-values) that oftendramatically alter the rank ordering of significant findings. Moreover, we findthat a multidimensional selection criteria based on BEN $p$-values and baggedmodel fit statistics is more likely to yield reproducible findings. Are-analysis of the famous Golub Leukemia data is presented to illustrate theseideas. We uncovered new findings in these data, not detected previously, thatare backed by published bench work pre-dating the Gloub experiment. Apseudo-simulation using the leukemia data is also presented to explore thestability of this approach under broader conditions, and illustrates thesuperiority of the BEN $p$-values compared to the other approaches.

Bagged Empirical Null p-values: A Method to Account for Model Uncertainty in Large Scale Inference