Resampling-based multiple testing with applications to microarray data analysis

In microarray data analysis, resampling methods are widely used to discover significantly differentially expressed genes under different biological conditions when the distributions of test statistics are unknown. When sample size is small, however, simultaneous testing of thousands, or even millions, of null hypotheses in microarray data analysis brings challenges to the multiple hypothesis testing field. We study small sample behavior of three commonly used resampling methods, including permutation tests, post-pivot resampling methods, and pre-pivot resampling methods in multiple hypothesis testing. We show the model-based pre-pivot resampling methods have the largest maximum number of unique resampled test statistic values, which tend to produce more reliable P-values than the other two resampling methods. To avoid problems with the application of the three resampling methods in practice, we propose new conditions, based on the Partitioning Principle, to control the multiple testing error rates in fixed-effects general linear models. Meanwhile, from both theoretical results and simulation studies, we show the discrepancies between the true expected values of order statistics and the expected values of order statistics estimated by permutation in the Significant Analysis of Microarrays (SAM) procedure. Moreover, we show the conditions for SAM to control the expected number of false rejections in the permutation-based SAM procedure. We also propose a more powerful adaptive two-step procedure to control the expected number of false rejections with larger critical values than the Bonferroni procedure.