Optimal tests shrinking both means and variances applicable to microarray data analysis

As a consequence of the "large p small n" characteristic for microarray data, hypothesis tests based on individual genes often result in low average power. There are several proposed tests that attempt to improve power. Among these, the FS test that was developed using the concept of James-Stein shrinkage to estimate the variances showed a striking average power improvement. In this paper, we establish a framework in which we model the key parameters with a distribution to find an optimal Bayes test which we call the MAP test (where MAP stands for Maximum Average Power). Under this framework, the FS test can be derived as an empirical Bayes test approximating the MAP test corresponding to modeling the variances. By modeling both the means and the variances with a distribution, a MAP statistic is derived which is optimal in terms of average power but is computationally intensive. An empirical Bayes test called the FSS test is derived as an approximation to the MAP tests and can be computed instantaneously. The FSS statistic shrinks both the means and the variances and has numerically identical average power to the MAP tests. Much numerical evidence is presented in this paper that shows that the proposed test performs uniformly better in average power than the other tests in the literature, including the classical F test, the FS test, the test of Wright and Simon, the moderated t-test, SAM, Efron's t test, the B-statistic and Storey's optimal discovery procedure. A theory is established which indicates that the proposed test is optimal in power when controlling the false discovery rate (FDR).