Power Prediction in Large Scale Multiple Testing: A Fourier Approach

A problem that is frequently found in large-scale multiple testing is that, in the present stage of experiment (e.g. gene microarray, functional MRI), the signals are so faint that it is impossible to attain a desired level of testing power, and one has to enroll more samples in the follow-up experiment. Suppose we are going to enlarge the sample size by a times in the follow-up experiment, where a > 1 is not necessary an integer. A problem of great interest is, given data based on the current stage of experiment, how to predict the testing power after the sample size is enlarged by a times.

We consider test z-scores and model the test statistics in the current experiment as X_j ~ N(μ_j , 1), 1 ≤ j ≤ n. We propose a Fourier approach to predicting the testing power after n replicates. The approach produces a very accurate prediction for moderately large values of a ( a ≤ 7). Finally, we discuss potential applications of this method on real data with emphasis on gene microarray data.