Flexible control of the median of the false discovery proportion

We introduce a multiple testing procedure that controls the median of the proportion of false discoveries in a flexible way. The procedure only requires a vector of p-values as input and is comparable to the Benjamini–Hochberg method, which controls the mean of the proportion of false discoveries. Our method allows free choice of one or several values of alpha after seeing the data, unlike the Benjamini–Hochberg procedure, which can be very anti-conservative when alpha is chosen post hoc. We prove these claims and illustrate them with simulations. Our procedure is inspired by a popular estimator of the total number of true hypotheses. We adapt this estimator to provide simultaneously median unbiased estimators of the proportion of false discoveries, valid for finite samples. This simultaneity allows for the claimed flexibility. Our approach does not assume independence. The time complexity of our method is linear in the number of hypotheses, after sorting the p-values.