Double truncation method for controlling local false discovery rate in case of spiky null

Many multiple test procedures, which control the false discovery rate, have been developed to identify some cases (e.g. genes) showing statistically significant difference between two different groups. However, a common issue encountered in some practical data sets is the presence of highly spiky null distributions. Existing methods struggle to control type I error in such cases due to the “inflated false positives," but this problem has not been addressed in previous literature. Our team recently encountered this issue while analyzing SET4 gene deletion data and proposed modeling the null distribution using a scale mixture normal distribution. However, the use of this approach is limited due to strong assumptions on the spiky peak. In this paper, we present a novel multiple test procedure that can be applied to any type of spiky peak data, including situations with no spiky peak or with one or two spiky peaks. Our approach involves truncating the central statistics around 0, which primarily contribute to the null spike, as well as the two tails that may be contaminated by alternative distributions. We refer to this method as the “double truncation method." After applying double truncation, we estimate the null density using the doubly truncated maximum likelihood estimator. We demonstrate numerically that our proposed method effectively controls the false discovery rate at the desired level using simulated data. Furthermore, we apply our method to two real data sets, namely the SET protein data and peony data.