Adaptive combination of dependent tests

The construction of a multivariate two sample test is considered. An attractive approach to this problem, for instance when the data contain missing values or the number of variables is large, is to form an overall test by combining the componentwise test statistics. This can be done via their pp-values or some other transformation. An important problem is how to perform the combination, as the relative power of a given combination will depend on the unknown true alternative. Recently, an approach has been proposed that makes use of the data to identify an appropriate combination. The method forms a pool of potential combinations of the componentwise pp-values, setting the overall test statistic to the minimum pp-value across the pool. One drawback of the approach, however, is that it does not utilize dependence between the componentwise tests, and thus potentially ignores valuable information. This issue is addressed, and two approaches are described that make use of the data to (1) determine which tests to combine; and (2) how best to utilize the between test statistic dependence. Simulations show that the proposed methods can lead to a substantial increase in power. An application to a dietary intervention study is given.