In some areas, e.g., statistical genetics, it is common to apply a maximum test, where the maximum of several competing test statistics is used as a new statistic, and the permutation distribution of the maximum is used for inference. Here, it is shown that maximum tests are special cases of adaptive permutation tests. The 30-year old idea of adaptive statistical tests is more flexible than previously thought when permutation tests are used, and the selector statistic is calculated for every permutation. Because the independence between the selector and the test statistics is no longer needed, the test statistics themselves can be used as selectors. Then, the maximum tests fit into the concept of adaptive tests. In addition to the gained flexibility, maximum tests can be more powerful than classical adaptive tests.
Neuhäeuser, Markus and Hothorn, Ludwig A.
"Maximum Tests are Adaptive Permutation Tests,"
Journal of Modern Applied Statistical Methods: Vol. 5
, Article 4.
Available at: http://digitalcommons.wayne.edu/jmasm/vol5/iss2/4