Controlling two-dimensional false discovery rates by combining two univariate multiple testing results with an application to mass spectral data

Mass spectral data exhibit a small number of signals (true peaks) among many noisy observations (signals or true peaks) in a high-dimensional space. This unique aspect of mass spectral data necessitates solving the problem of testing for many composite null hypotheses simultaneously. In this study, we develop a new procedure to control the false discovery rate of simultaneous multiple hypothesis tests, consisting of many “bivariate” composite null hypotheses. Two types of composite null hypothesis, the intersection-type and the union-type null, are considered separately. The proposed procedure comprises two stages. In the first stage, we simultaneously test each “univariate” simple hypothesis of “bivariate” composite hypotheses at the pre-decided false discovery rate. In the second stage, we combine the marginal univariate test results so that the two-dimensional false discovery rate for the “bivariate” composite null hypotheses is less than the desired significance level α. The new procedure provides a closed-form decision rule on the bivariate test statistics, unlike existing methods for controlling the two-dimensional local false discovery rate (2d-fdr). We numerically compare the performance of our procedure to existing 2d-fdr control methods in different settings. We then apply the procedure to the problem of differentiating the origins of herbal medicine using gas chromatography-mass spectrometry.